Latent variable models are widely used to account for unobserved determinants of economic behavior. Traditional nonparametric methods to estimate latent heterogeneity do not scale well into multidimensional settings. Distributional restrictions alleviate tractability concerns but may impart non-trivial misspecification bias. Motivated by these concerns, this paper introduces a quasi-Bayes approach to estimate a large class of multidimensional latent variable models. Our approach to quasi-Bayes is novel in that we center it around relating the characteristic function of observables to the distribution of unobservables. We propose a computationally attractive class of priors that are supported on Gaussian mixtures and derive contraction rates for a variety of latent variable models.

§ INTRODUCTION

Many questions in economics involve inferring information on unobserved determinants of economic behavior. For instance, <cit.> and <cit.> study the importance of human capital development on child outcomes. In these models, observed variables such as test scores and height serve as imperfect proxies for latent ability and health. Similarly, studies on search unemployment <cit.> and labor income dynamics <cit.>   explicitly incorporate unobserved heterogeneity and measurement error in observables (e.g. observed wages, hours and earnings).

To account for the informational uncertainty that arises from unobservables, economic models that incorporate latent variables are widely used. In multidimensional settings, a common strategy to retain tractability is to impose distributional restrictions on observables and unobservables.[For instance, in their empirical specification, <cit.> and <cit.> fix the distribution of  unobservables to lie in a Gaussian family. This induces a Gaussian likelihood on observables, which facilitates the use of parametric maximum likelihood.] These approaches are attractive in that they provide a researcher with a tractable distribution (e.g. Gaussian) to sample from for counterfactual analysis. On the downside, they may impart non-trivial misspecification bias through the distributional restrictions.[In low dimensional settings, deconvolution based approaches (e.g. ; ; ) are commonly used to estimate the distribution of unobservables. These methods require minimal distributional restrictions on unobservables but are computationally challenging and  highly sensitive to user chosen tuning parameters, especially so in multidimensional setups.] Motivated by these concerns, this paper introduces a novel quasi-Bayes approach to estimate a large class of multidimensional latent variable models.

Traditional quasi-Bayes (; ) combines a GMM objective function $Q_n(\theta)$   and a prior $\nu(\theta)$ to create a quasi-posterior distribution. Intuitively, the GMM objective function reweights the prior distribution to assign greater weight to areas of the parameter space where the objective function is small. As a consequence, posterior samples concentrate around minimizers of the GMM objective function. The usual quasi-posteror takes the form

\begin{align}
\label{cherhong} \nu(\theta|\mathcal{Z}) \propto  e^{- Q_n(\theta)} \nu(\theta) .
\end{align}

In this paper, we generalize the representation in (<ref>) to settings where $\theta$ represents a latent distribution of interest. While traditional quasi-Bayesian approaches in econometrics focus on moment restrictions that identify a finite dimensional parameter, our approach instead centers around identifying restrictions which relate the characteristic functions of observables to the distribution of unobservables. Our analysis begins with the observation that a large class of latent variable models are characterized by a collection of simple identifying restrictions on the characteristic function of the observables. We use these restrictions to build a pseudo-likelihood based objective function for the model. When combined with a prior, this provides a researcher with quasi-Bayes decision rules such as point estimators (e.g. posterior means) and credible sets.

We propose a class of priors that are supported on Gaussian mixtures. Our focus on this class is partially motivated from the observation that finite Gaussian mixture models are widely used  in a variety of empirical settings to model rich forms of heterogeneity. As Gaussian mixtures can be efficiently sampled from, they provide researchers with a convenient framework to perform counterfactual analysis. One consequence of building a framework around such a structure is that we obtain decision rules with particularly desirable theoretical guarantees in settings where the true data generating process closely resembles a (possibly infinite) Gaussian mixture, while at the same time remaining viable in the more general case. A further desirable property of focusing on Gaussian mixture based priors is that they greatly facilitate the interplay analysis between the prior and pseudo-likelihood. Importantly, Gaussian mixtures admit a simple characteristic function which in turn leads to a tractable pseudo-likelihood. Indeed, one of the main ideas behind our analysis is that priors on Gaussian mixtures translate effortlessly to priors on characteristic functions and vice versa.

The main contributions of this paper are as follows. We provide a unified treatment for three classes of latent variable models. These are $(i)$ models with classical measurement error, $(ii)$ models with repeated measurements and $(iii)$ linear multi-factor models. In each case, we derive characteristic function based identifying restrictions and use them to build a pseudo-likelihood for the model. We then combine this likelihood with a theoretically motivated class of priors and derive $L^2$ rates of convergence (posterior contraction rates) for the induced quasi-Bayes posterior. As our analysis is the first quasi-Bayesian approach to these classes of models, we expect that the general analysis may be of independent interest towards related extensions.  As a by product of our analysis, we also contribute to  a general understanding  of characteristic function based pseudo-likelihoods. Importantly, certain likelihoods may only lead to partial identification but can nonetheless be modified in a suitable way to achieve point identification.

To the best of our knowledge, our paper is the first nonparametric Bayesian approach to utilize a pseudo-likelihood based on characteristic functions. In our analysis, we use a Dirichlet process prior to induce a prior on Gaussian mixtures which in turn leads to an induced prior on characteristic functions. One consequence of this analysis is that we obtain novel $L^{\infty}$ posterior contraction results even for the special case of density estimation. Intuitively, convergence of characteristic functions implies risk bounds in stronger metrics.[For example, if $f_X$ and $f_{Y}$ have characteristic functions $\varphi_{X}$ and $\varphi_{Y}$, $ \| f_X - f_Y  \|_{L^{\infty}} \leq \| \varphi_X - \varphi_Y  \|_{L^1}. $] This approach is of independent interest and may be
useful in a variety of related extensions.

As a first application, we use data collected from the India Young Lives Survey  to estimate production functions for cognition and health for children aged 1-12
in India, revisiting <cit.>. As in the original analysis, we use the latent factor model of <cit.> to estimate the joint distribution of unobservables and observables. Our analysis deviates from the literature in that we estimate this joint distribution nonparametrically and do so without imposing a specific distribution on the measurement errors.[This refers  to estimation methods that explicitly depend on the specific form of the measurement error distribution in their implementation. For example, Gaussian measurement errors are frequently  used to induce a Gaussian likelihood on the observables. If the latent distribution is also assumed to be Gaussian, its mean and covariance structure can  be recovered through minimum distance least squares.] In line with previous results in the literature, our results show that cognition displays persistence across the development period and that parental investment is effective at developing cognition at all ages, with a higher return for younger children. Our results differ from the literature in that, while previous studies attribute cognition at later ages primarily to persistence, our results suggest a greater influence  of alternative factors. Specifically, we find that investment continues to  maintain a significant impact during adolescence, albeit still lower than in early childhood. As our sample size is relatively modest, and we do not consider estimation by relaxing only a single restriction, it is difficult to ascertain  whether our differences are due to nonparametrically modeling the latent distribution and/or leaving free the distribution of the measurement errors. To that end, we view our general methodology as complementary to the existing literature in that it may serve as a robustness check to possible violations of these restrictions.

As a secondary application, we apply our methodology to model individual earnings data from the Panel Study of Income Dynamics (PSID), revisiting <cit.>. Our results in this setting are consistent with the literature (e.g. ; ) in that the distribution of wage shocks in U.S data appear to be non-Gaussian and leptokurtic. Additionally, as our priors are supported on infinite Gaussian mixtures, our results in this setting also complement previous approaches (e.g. ; ) that advocate for Gaussian mixtures to model earnings dynamics. To assess the performance of our method, we compare the implied fit of the wage growth moments to alternative estimators and the observed data. Overall, we find that our approach is capable of reproducing the higher moments observed in U.S wage growth data.

The paper is organized as follows. Section <ref> briefly summarizes notation that frequently appears throughout the text. Section <ref> provides a brief review of Gaussian mixtures and the Dirichlet process prior. Section <ref> introduces the quasi-Bayes framework for all the latent variable models considered in this paper. Section <ref> develops the quasi-Bayes limit theory and main results. Section <ref> provides simulation evidence on the performance of our proposed methodology. Section <ref> presents the empirical application to human capital development in India. Section <ref> presents the empirical application to earnings dynamics with PSID data. Section <ref> concludes. Section <ref> contains proofs and auxiliary results for all the statements in the main text.

§.§ Related Literature

Latent variable models have a long history in econometrics and statistics. For a comprehensive overview, we refer to <cit.>. <cit.> establish the consistency of a sieve maximum likelihood estimator for the distribution of unobservables in a single spell duration model. For classical measurement error with a known error distribution, <cit.> establish  optimal  convergence rates of kernel deconvolution estimators. For early work on measurement error in panel frameworks, see <cit.>. For models with repeated measurements, <cit.> use Kotlarski's lemma <cit.> to build a deconvolution based estimator. For extensions and related results on models with repeated measurements or auxiliary information, see  <cit.>; <cit.>; <cit.>; <cit.>; <cit.>; <cit.>; <cit.>. For general multi-factor models with mutually independent factors, <cit.> propose a kernel deconvolution based estimator. For instrumental variable based approaches to models with measurement error, see <cit.>; <cit.>; <cit.>; <cit.>; <cit.>.

Our paper is also related to a growing literature on Bayesian inference in density estimation and deconvolution. For pure Bayesian approaches to density estimation, see <cit.>. For pure Bayesian approaches to deconvolution with a known error distribution, see <cit.>. To the best of our knowledge, our paper is the first to propose a quasi-Bayes framework for these models. As such, our paper also contributes to a growing literature on quasi-Bayes in nonparametric econometric models (e.g. ; ).

Quasi-Bayesian methods have a long history in econometric models  identified through moment restrictions. Traditional approaches (e.g. ; ) have typically focused on strongly identified parametric models. Recent work (e.g. ; ) in the literature also demonstrates that certain quasi-Bayesian decision rules have desirable properties in settings where identification is weak or non standard. Although our focus in this paper is not on settings with non standard identification, we do note that (in our nonparametric ill-posed inverse setting) finite sample identification strength is typically user determined by tuning parameters that determine the structure of the parameter space. A complex parameter space leads to objective function that are near-flat in substantial portions of the parameter space. As traditional estimators (e.g. deconvolution based) are typically based on directly inverting these objective functions, they are highly sensitive to the underlying structure.[Intuitively, an estimator that is based on inverting an objective function that is near-flat in substantial portions of a parameter space typically displays large variability.] By contrast, quasi-Bayes provides additional regularization to solve the ill-posed inverse problem through a prior and flexible choice of pseudo-likelihood.

§.§ Notation

Denote the standard Euclidean real coordinate space of dimension $d$ by $\R^d$. For complex coordinates, we use $\mathbb{C}^d$. Let $\mathbf{i}$ denote the imaginary unit. The magnitude of a complex valued function $f : \R^d \rightarrow \mathbb{C}$ is defined as $  \left| f(t) \right|^2 =  \big( \mathfrak{R}[f(t)] \big)^2 + \big( \mathfrak{I}[f(t)] \big)^2 $, where $\mathfrak{R}$ and $\mathfrak{I}$ denote the real and imaginary parts, respectively. Given a random vector $Z \in \R^d$, we denote its characteristic function by $  \varphi_{Z}(t) = \E[e^{ \mathbf{i} t' Z } ]$.  Given functions $f,g : \R^d \rightarrow \R$, we denote their convolution by $f \star g (x) = \int_{\R^d} f(x-z) g(z) dz $. Convolution of a function $g: \R^d \rightarrow \R$ with a Borel measure $\mu$ on $\R^d$ is denoted in a similar manner by  $g \star \mu (x) = \int_{\R^d} g(x-z) d \mu (z)$. The Fourier Transform of a function $f : \R^d \rightarrow \R$ and a Borel measure $\mu$ on $\R^d$ is denoted by $\mathcal{F}[f] (t) = \int_{\R^d} e^{  \mathbf{i} t' x} f(x) dx$ and $\mathcal{F}[\mu] (t) = \int_{\R^d} e^{  \mathbf{i} t' x} d \mu (x)    $, respectively.

Let $\E$ and $\mathbb{P}$ denote the usual expectation and probability operators. Let $\E_n$  denote the empirical analog of $\E$. We use $\| . \|$ and $\| . \|_{\infty}$ to denote the Euclidean and infinity norm on $\R^d$, respectively. We use $\mathbf{S}_{+}^d$ to denote the set of positive definite matrices on $\R^{d \times d}$. Given two measures $(\lambda,\mu) $,  we denote the product measure by $\nu = \lambda \otimes \mu$. Let $ \mathbf{H}^{p} = ( \mathbf{H}^p, \| . \|_{\mathbf{H}^p}) $ denote the usual $p$-Sobolev space of functions on $\R^d$. Given a symmetric positive definite matrix $\Sigma \in \mathbf{S}_+^d$, we denote its ordered eigenvalues by $\lambda_1(\Sigma) \leq \lambda_2(\Sigma) \leq \dots \leq \lambda_{d}(\Sigma)$.

We introduce more specific notation for multivariate Gaussian distributions to denote various dependencies that we make use of. Let $\mathcal{N}(0,\Sigma)$ denote the multivariate Gaussian distribution with mean zero and covariance matrix $\Sigma$. We use $\phi_{\Sigma}$ and $\varphi_{\Sigma}$ to denote the $\mathcal{N}(0,\Sigma)$ density and characteristic function, respectively.

§ REVIEW

We begin in Section <ref> with a brief review of Gaussian mixtures. In Section <ref>, we review the Dirichlet process and the prior it induces on Gaussian mixtures.

§.§ Gaussian Mixtures

Given a probability distribution $P$ on $\R^d$,  the Gaussian mixture with mixing distribution $P$ and covariance matrix $\Sigma \in \R^{d \times d}$ is denoted by
\begin{align}
\label{gmm}  \phi_{P,\Sigma} (x) = \phi_{\Sigma} \star P (x) = \int_{\R^d} \phi_{\Sigma}(x-z) d P(z).
\end{align}

If $P$ is  a discrete distribution, i.e $P = \sum_{j=1}^{\infty} p_j \delta_{\mu_j} $ which assigns probability mass $p_j$ to a point $\mu_j \in \R^d$, the preceding definition reduces to
\begin{align*}
\phi_{P,\Sigma}(x) = \sum_{j=1}^{\infty} p_j \phi_{\Sigma}(x- \mu_j).
\end{align*}

If $\mathcal{F}$ denotes the Fourier transform operator, the characteristic function (CF) of a Gaussian mixture is given by
\begin{align} \label{gmmcf}  \varphi_{P,\Sigma}(t) = \F[ \phi_{\Sigma} \star P ] (t) =  \F[\phi_{\Sigma}](t) \times \F[P](t) = e^{- t' \Sigma t /2} \F[P](t).   \end{align}

For a discrete distribution $P = \sum_{j=1}^{\infty} p_j \delta_{\mu_j} $, the preceding expression simplifies to

\begin{align*}   \varphi_{P,\Sigma}(t)  =    e^{- t' \Sigma t /2} \sum_{j=1}^{\infty} p_j e^{\mathbf{i} t' \mu_j}     .     \end{align*}

From the representation in $(\ref{gmm})$, it follows that a Gaussian mixture is completely characterized by its mixing distribution $P$ and covariance matrix $\Sigma$.

In practice, a researcher may choose to model the distribution of a random vector $X \in \R^d$ as an exact Gaussian mixture of the form in (<ref>). That is, there exists a positive definite matrix $\Sigma_0 \in \R^{d \times d}$ and a mixing distribution $F_0$ such that the density of $X$ can be expressed as
\begin{align}
\label{fxgaussm}  f_X(x) =   \phi_{\Sigma_0} \star F_0 (x)  = \int_{\R^d} \phi_{\Sigma_0}(x-z) d F_0(z).
\end{align}

As the mixing distribution $F_0$ may be continuously distributed and/or possess unbounded support, the representation in (<ref>) offers a flexible parametrization that lies somewhere between a parametric and fully nonparametric model. Furthermore, while the representation in (<ref>) may be misspecified, there always exists a mixing distribution and covariance matrix $(\Sigma_0, F_0)$ such that the misspecification bias can be made arbitrarily small.[To be specific, suppose $X \in \R^d$ admits a density $f_X \in  L^2(\R^d)$. A simple way to achieve a (not necessarily optimal) Gaussian mixture approximation is to let $P = f_X$  and $\Sigma_n = \sigma_n^2 I$ for some $\sigma_n^2 \downarrow 0$. As the Fourier transform preserves distance (up to a constant), it follows that

\begin{align*}
\| \phi_{P, \Sigma_n} (x) - f_X(x) \|_{L^2} = \| f_X \star \phi_{\sigma_n^2 I} - f_X        \|_{L^2} \asymp \| (e^{- \| t \|^2 \sigma_n^2 /2} - 1 ) \varphi_{X} (t)   \|_{L^2} \xrightarrow[n \rightarrow \infty]{} 0.
\end{align*}

§.§ Dirichlet Process Priors

The Dirichlet process is a distribution on the space of probability distributions on $\R^d$. That is, a random draw from the Dirichlet process is a distribution $P$ on $\R^d$. As the name suggests, the precise definition of the process is closely related to that of the finite dimensional Dirichlet distribution on the simplex. The following definition clarifies this.

A random distribution $P$ on $\R^d$ is a Dirichlet process distribution with base measure $\alpha$ if for every finite measurable partition of $\R^d = \bigcup_{i=1}^k A_i$, we have
\begin{align}
\label{dp01} (P(A_1),P(A_2), \dots , P(A_k)) \sim  \text{Dir}(k,\alpha(A_1),\dots,\alpha(A_k))\:,
\end{align}

where $\text{Dir}(.)$ is the Dirichlet distribution with parameters $k$ and $\alpha_1(A_1), \dots , \alpha_k(A_k) > 0$ on the $k$ dimensional unit simplex $\Delta = \{ x \in \R^k : x_i \geq 0 \; , \sum_{i=1}^k x_i =1 \}$.[The probability density of $\text{Dir}(k,\alpha_1,\dots,\alpha_k)$, with respect to the $k-1$ dimensional Lebesgue measure on the unit simplex $\Delta$, is proportional to $ \prod_{i=1}^k x_i^{\alpha_i-1}  $.]

An alternative definition that is especially useful in simulating the process is the following.

Fix $\beta > 0$ and a base probability distribution $\alpha$ on $\R^d$. Let $V_1,V_2 , \dots \stackrel{i.i.d}{\sim} \text{Beta}(1, \beta) $  and $\mu_1,\mu_2 , \dots \stackrel{i.i.d}{\sim} \alpha $. Define the weights
\begin{align*} & p_1 = V_1 \; \; , \;  p_j = V_j \prod_{i=1}^{j-1} (1 - V_i) \; \; \; \; \; \; \;  j \geq 2. \end{align*}

It can be shown <cit.> that $\sum_{i=1}^{\infty} p_i = 1$ almost surely. The Dirichlet process with concentration parameter $\beta$ and base measure $\alpha$ is the law of the random discrete distribution
$$ P = \sum_{i=1}^{\infty} p_i \delta_{\mu_i} \:, $$

where $\delta_{\mu_i}$ is the point mass at $\mu_i$. That is, $P$ takes value $\mu_i$ with probability $p_i$. Notationally, we suppress the dependence on $\beta$ and express this as $P \sim \text{DP}_{\alpha}$. As Figure <ref> illustrates, the parameter  $\beta$ controls the spread around the base measure $\alpha$, with larger values of $\beta$ leading to tighter concentration.

CDF Draws of $P \sim \text{DP}_{\alpha} $. Black solid line indicates CDF of base measure $\alpha =  N(0,1)$.

A Dirichlet process prior can be used to build a prior on Gaussian mixtures in (<ref>). Specifically, given a Dirichlet process prior $\text{DP}_{\alpha}$ and an independent prior $G$ on the set $\mathbf{S}_+^d$ of positive definite matrices, the induced prior on Gaussian mixtures is given by
\begin{align}
\label{dpgmm} \phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) d P(z) \;  , \; \; \;      (P, \Sigma) \sim \text{DP}_{\alpha} \otimes G.
\end{align}

In the remaining sections, we will frequently refer to the prior in (<ref>). For ease of notation, we denote this product prior by $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G $.

§ MODELS AND PROCEDURES

In this section, we introduce three commonly used latent variable models: $(i)$ classical measurement error[This is also known as the multivariate deconvolution model <cit.>], $(ii)$ models with repeated measurements and $(iii)$ linear independent multi-factor models. For each model, we consider identifying restrictions that relate the distribution of the latent variable to the characteristic function of observables. These restrictions are used to construct a quasi-likelihood. When combined with a prior as in (<ref>), we obtain a quasi-Bayes posterior that is supported on infinite Gaussian mixtures.

§.§ Classical Measurement Error

Consider a classical measurement error model where we observe a random sample from

\begin{align}
\label{mdeconv}   Y = X + \epsilon \; \; \; \; , \; \; \E[\epsilon] = 0.
\end{align}

Here, $Y \in \R^d $ is an observed vector, $X \in \R^d$ is an unobserved vector whose distribution is of interest and $\epsilon$ is an unobserved error that is
independent of $X$. We may view $Y$ as an imperfect proxy or error contaminated version of $X$.

We are interested in recovering the latent distribution of $X$. As the individual contributions of $X$ and $\epsilon$ cannot be separately identified from an observation of $Y$, identification of the latent distribution of $X$ generally requires some further auxiliary information on the distribution $F_{\epsilon}$ of the unobserved error $\epsilon$. In the deconvolution literature (e.g. ), it is common to assume that the distribution $F_{\epsilon}$ is completely known. As in <cit.>, we consider a slightly weaker requirement where a researcher has auxiliary information in the form of a random sample  ${\epsilon}_1, \dots , {\epsilon}_m \stackrel{i.i.d}{\sim} F_{\epsilon} $ of size $m$.[There are no restrictions on the dependence of the auxiliary sample with the original sample.] As we discuss in Section <ref>, one way to generate such auxiliary information is through repeated measurements of $Y$.

We start by observing that, as the unobserved error $\epsilon$ is independent from the latent vector $X$, the characteristic functions of $(Y,X,\epsilon)$ factor as
\begin{align}
\label{mdeconv-f} \varphi_{Y}(t) = \varphi_{X}(t)  \varphi_{\epsilon}(t) \; \; \; \; \; \forall \; t \in \R^d.
\end{align}

Our starting point towards constructing a quasi-likelihood is to observe that (<ref>) can be viewed as a collection of  identifying restrictions on the model. As the characteristic functions $\varphi_{Y}(t) = \E[e^{\mathbf{i}t' Y}]$ and $\varphi_{\epsilon}(t) = \E[e^{\mathbf{i}t' \epsilon}]$ depend on population expectations, it is infeasible to evaluate either side of (<ref>) directly. Towards that goal, given the observed data, we estimate the population expectations using their empirical analogs:

\begin{align}
\label{charde} &   \widehat{\varphi}_Y(t)  =  \E_n \big[ e^{\mathbf{i} t' Y}  ]    = \frac{1}{n} \sum_{j=1}^n e^{\mathbf{i} t' Y_j} \\ &  \label{charde2}  \widehat{\varphi}_\epsilon(t)  =  \E_m\big[ e^{\mathbf{i} t' \epsilon}   ] =  \frac{1}{m} \sum_{j=1}^m e^{\mathbf{i} t' {\epsilon}_j}.
\end{align}

As estimated characteristic functions do not have uniformly (over $t \in \R^d$) valid statistical guarantees, it is necessary to consider a bounded set of restrictions that grows with the sample size.[Under standard regularity conditions, the usual statistical guarantee (see Lemma <ref>) is
$$  \sup_{\| t \|_{\infty} \leq T} \left|  \widehat{\varphi}_Y(t) - \varphi_Y(t) \right| = O_{\mathbb{P}}( n^{-1/2} \sqrt{\log T}    ) . $$
] To that end, given a fixed $T > 0$, we denote the ball of radius $T$ and the $L^2$ magnitude (norm) of a function $f : \R^d \rightarrow \mathbb{C}$ over the ball by
\begin{align} \label{ball}
&  \mathbb{B}(T) = \{ t \in \R^d :  \| t \|_{\infty} \leq T  \} \; \; , \; \; \;  \|f \|_{\mathbb{B}(T)}^2 =  \int_{\mathbb{B}(T)}  \left| f(t) \right|^2 dt .
\end{align}

If $\theta$ represents a distribution with characteristic function $\varphi_{\theta}$, we aim to construct a quasi-likelihood for the latent distribution of $X$ using the quantity
\begin{align} \label{quaslik} L( \theta ) = -
\| \widehat{\varphi}_{Y} -  \varphi_{\theta}  \widehat{\varphi}_{\epsilon}  \|_{\mathbb{B}(T)}^2.
\end{align}

While in principle $L(.)$ can be evaluated at any distribution $\theta$, in the interest of computational tractability, it will be convenient to restrict our attention to distributions that admit a simple characteristic function. At a minimum, this ensures that the quasi-likelihood in (<ref>) can be easily evaluated at such distributions. Towards this goal, we consider candidate solutions in the class of infinite Gaussian mixtures:
\begin{align}
\label{gauss-mix-sol} \phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) dP(z).
\end{align}

As discussed in Section <ref>, a Gaussian mixture is completely characterized by its mixing distribution $P$ and covariance matrix $\Sigma$. Therefore, to induce a quasi-Bayes posterior we endow $(P,\Sigma)$ with a prior. As discussed in Section <ref>, we use a Dirichlet process $\text{DP}_{\alpha}$ prior for the mixing distribution $P$. We then choose an independent prior $G$ for the covariance matrix $\Sigma$. The choice of prior $G$ is flexible, with some leading to more straightforward computation in higher dimensions. General conditions on $G$ are specified in Section <ref>. In particular, an Inverse-Wishart prior is always permitted.

We define, as discussed in Section <ref>, a prior on Gaussian mixtures given by
\begin{align*}
\phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) d P(z) \;  , \; \; \;      (P, \Sigma) \sim \nu_{\alpha,G} =  \text{DP}_{\alpha} \otimes G.
\end{align*}

Let $\mathcal{Z}_n$ denote the observed data. If $\varphi_{P,\Sigma}$ denotes the characteristic function of a Gaussian mixture $\phi_{P,\Sigma}$, we define the deconvolution quasi-Bayes posterior by
\begin{align}
\label{qb-dec}  \nu_{\alpha,G}(  \varphi_{P,\Sigma} \;  | \:  \mathcal{Z}_n) =  \frac{\exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T)  }^2        \big)}{\int \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T)  }^2        \big) d \nu_{\alpha,G}(P,\Sigma) }.
\end{align}

As the quasi-Bayes posterior is constructed using $\widehat{\varphi}_{Y}$ and $\widehat{\varphi}_{\epsilon}$, the conditioning on $\mathcal{Z}_n$ is used to denote that the quasi-posterior is defined conditionally on the observed data. Our construction defines the posterior directly on characteristic space. However, as there is a one-to-one relation between $\phi_{P,\Sigma}$ and $\varphi_{P,\Sigma}$, without loss of generality we view $ \nu_{\alpha,G}(. \:| \:\mathcal{Z}_n) $ as a quasi-Bayes posterior over  distributions and their characteristic functions.

The quasi-Bayes posterior uses the  identifying restrictions in (<ref>) as a quasi-likelihood for the model. Intuitively, we expect samples from the posterior to concentrate on Gaussian mixtures that approximately satisfy the model's identifying restrictions. As the latent distribution $X$ uniquely satisfies the restrictions, these Gausian mixture samples are expected to concentrate around the distribution of $X$. This intuition is formalized in Section <ref>.

As a formal  estimator of the latent distribution, we use the expected density under the posterior distribution in (<ref>). Specifically, we denote the posterior mean by
\begin{align}
\label{posmean}   \E _{\nu_{\alpha,G}|\mathcal{Z}_n} \big(f \big) = \int  f d \nu_{\alpha,G}(f|\mathcal{Z}_n).
\end{align}

Our priors are, by construction, supported on infinite Gaussian mixtures. In theory, alternative priors such as the logistic Gaussian process priors  <cit.> or  log-sieve based priors <cit.> could be used as well. Unfortunately, realizations from these priors do not admit a closed form characteristic function. This considerably complicates evaluation of the likelihood $L(.)$ in (<ref>). Motivated by computational tractability, we focus on Gaussian mixtures as they admit a simple closed form characteristic function.

Pure Bayesian approaches to deconvolution have been studied in the literature <cit.>. In these frameworks, it is assumed that the density $f_{\epsilon}$ is known. From the model in (<ref>), the density of $Y$ is given by the convolution $ f_{Y}(y) = f_{X} \star f_{\epsilon}(y) =  \int_{\R^d} f_{X}(y-t) f_{\epsilon} (t) dt$. It follows that the pure-Bayes posterior distribution under a $ \nu_{\alpha,G} =  \text{DP}_{\alpha} \otimes G$ prior is given by

\begin{align}
\label{pure-bayes} \nu(\phi_{P,\Sigma} |\mathcal{Z}_n) = \frac{ \prod_{i=1}^n   \int_{\R^d} \phi_{P,\Sigma} (Y_i - t) f_{\epsilon}(t) dt      }{ \int  \prod_{i=1}^n   \int_{\R^d} \phi_{P,\Sigma} (Y_i - t) f_{\epsilon}(t) dt   d \nu_{\alpha,G} (P,\Sigma)  }.
\end{align}

In particular, even if the density $f_{\epsilon}$ is known, the convolution structure in the posterior likelihood makes it very challenging to compute in higher dimensions. By contrast, quasi-Bayes uses a loss function that is based on transforming the convolution structure $f_{X} \star f_{\epsilon} (t)$ into simple multiplication $ \varphi_{X} (t) \times \varphi_{\epsilon} (t) $ in the Fourier domain.[In a sense, this is similar to the idea behind Fast Fourier Transform (FFT) methods for convolution <cit.>. There, convolution is replaced with simple multiplication in the Fourier domain and the convolution is obtained by inversion.] This remains true in the more complicated setups studied in Section <ref> and <ref>.[We could not find any references that study nonparametric pure bayesian approaches to repeated measurements (Section <ref>) and multi-factor models (Section <ref>). ]

As our quasi-likelihoods are smooth functions of $(P,\Sigma)$, gradient based Markov chain Monte Carlo (MCMC) methods can be used  to construct a random sample from the quasi-Bayes posterior. In our implementation, we use the No-U-Turn Sampler (NUTS) version of Hamiltonian Monte Carlo <cit.>.

§.§ Models with Repeated Measurements

In this section, we continue our investigation into quasi-Bayes representations for latent variable models. In Section <ref>, we considered the setting where a researcher has auxiliary information on the error distribution that contaminates the latent variable. Specifically, we assumed the availability of a  random sample from the error distribution. In practice, this specific form of auxiliary information may not always be available. Here we examine the more common situation where a researcher observes several noisy measurements of the latent variable $X$. To fix ideas, suppose we observe $(Y_1,Y_2)$ from the model:

\begin{align}
\label{repm-model} &  Y_1 = X + \epsilon_1 \;  ,  \;  \E[\epsilon_1] = 0 ,  \\  & Y_2 = X + \epsilon_2 \;  ,  \;  \E[\epsilon_2] = 0 . \nonumber
\end{align}

Here, $X \in \R^d$ is a latent random vector whose distribution is of interest and $(\epsilon_1,\epsilon_2)$ are unobserved errors. As a baseline example, $X$ may denote latent child ability and $(Y_1,Y_2)$ measurements of test scores in different subjects (e.g. math, english).  Similar to the preceding section, it is not necessary for $Y_1$ and $Y_2$ to have the same number of observations $n$. Nonetheless, for simplicity and notational ease, we continue under this setting.

In the special case where $(\epsilon_1,\epsilon_2,X)$ are mutually independent and $\epsilon_2$ is a symmetric distribution around zero, the model in $(\ref{repm-model})$ can be reduced to the classical measurement error model in (<ref>) by differencing out the error. Specifically, consider the observed quantities
\begin{align} &
\bigg(   \frac{Y_1 + Y_2}{2}  \bigg) = X +  \bigg(   \frac{\epsilon_1 + \epsilon_2}{2}  \bigg)\: \label{repmeasaux0} , \\ & \bigg( \frac{Y_1 - Y_2}{2}   \bigg) = \bigg(  \frac{\epsilon_1 - \epsilon_2}{2}  \bigg)   \label{repmeasaux}
\end{align}

The main idea being that, under the appropriate symmetry and independence conditions, the observations from $(\ref{repmeasaux})$ can be used as an auxiliary sample for the error in (<ref>). As such, the analysis in the preceding section and the quasi-Bayes posterior in (<ref>) can be directly applied to this setting. This analysis, while convenient to implement, may not always be applicable. In particular, weaker restrictions on the joint distribution of $(X,\epsilon_1,\epsilon_2)$ may violate either the mutual independence of $(X_1,\epsilon_1,\epsilon_2)$ or the requirement of symmetrically distributed errors.

Under the weaker restriction that the errors satisfy $\E[\epsilon_1|X,\epsilon_2] = 0 $ and $\epsilon_2 \perp X$, it is known (; ) that the latent distribution of $X$ can be identified through the representation[The representation in (<ref>) belongs to a family of Koltarski type (; ) identities that often appear in the identification of econometric models with measurement error. For examples in auction  based frameworks, see (; ). ]
\begin{align}
\label{koltarski}   f_{X}(x) =  \frac{1}{(2 \pi)^d} \int_{\mathbb{R}^d}  e^{- \mathbf{i} t'x} \exp \bigg(  \int_{\mathcal{P}_t}  \frac{\E[\mathbf{i} Y_1 e^{\mathbf{i} \zeta' Y_2}  ]}{\E[ e^{\mathbf{i} \zeta' Y_2}  ]} d \zeta    \bigg) dt.
\end{align}

Here, $\int_{\mathcal{P}_t}$ is the path integral over the straight line segment joining the origin to $t \in \R^d$. The closed form representation in (<ref>) is convenient for theoretical analysis, but feasible analogs of it are very challenging to implement. This is especially true in higher dimensions where both the path and outer integral are difficult to evaluate precisely. This is further complicated by the fact that the empirical analog of the ratio appearing inside the path integral is volatile  and highly sensitive to the choice of $t \in \R^d$. Intuitively, bad choices of $t$ lead to situations where $ \big| \E[e^{\mathbf{i} t' Y_2}]\big|$ is significantly smaller than sampling uncertainty $ \big| (\E_n - \E) [e^{\mathbf{i} t' Y_2}] \big|$. This makes the empirical analog of the ratio a highly volatile function of $t \in \R^d$.

Our starting point towards a general quasi-Bayes representation of model (<ref>) is to consider characteristic function based implications of the representation in (<ref>). By applying the Fourier transform to both sides of (<ref>) and taking the logarithm, we obtain
\begin{align}
\label{koltarski2}   \log \varphi_{X}(t) = \int_{\mathcal{P}_t}  \frac{\E[\mathbf{i} Y_1 e^{\mathbf{i} \zeta' Y_2}  ]}{\E[ e^{\mathbf{i} \zeta' Y_2}  ]} d \zeta  \; \; \; \; \; \; \; \; \forall \: t \in \R^d.
\end{align}

By differentiating both sides of $(\ref{koltarski2})$ and rearranging, we obtain
\begin{align}
\label{koltarski3}  \varphi_{Y_2} (t) \nabla \log \varphi_{X}(t) = \E[\mathbf{i} Y_1 e^{\mathbf{i} \zeta' Y_2}  ]  \; \; \; \; \; \; \; \; \forall\: t \in \R^d.
\end{align}

We view (<ref>) as a collection of identifying restrictions on the model. Given the observed data $\{ (Y_{1,i} , Y_{2,i}) \}_{i=1}^n$, we estimate the empirical counterpart to (<ref>) using
\begin{align*}
& \widehat{\varphi}_{Y_2}(t) = \E_n[ e^{\mathbf{i} t' Y_2} ] = \frac{1}{n} \sum_{j=1}^n e^{\mathbf{i} t' Y_{2,j}} \; , \\ &  \widehat{\varphi}_{Y_1,Y_2}(t)  = \E_n [ \mathbf{i} Y_1 e^{\mathbf{i} \zeta' Y_2}  ]  = \frac{1}{n} \sum_{j=1}^n \mathbf{i} Y_{1,j} e^{\mathbf{i}t' Y_{2,j}}.
\end{align*}

If $\theta$ represents a distribution with characteristic function $\varphi_{\theta}$, the identifying restrictions in (<ref>) suggest a quasi-likelihood of the form

\begin{align} \label{quaslik2} L( \theta ) = -
\| \widehat{\varphi}_{Y_2} \nabla \log \varphi_{\theta}        - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2  .
\end{align}

Up to sampling uncertainty, the quasi-likelihood in (<ref>) is based on the objective function
\begin{align}  \label{quaslik2-obj} Q(\theta) = -\| \varphi_{Y_2} \big( \nabla \log \varphi_{\theta} - \nabla \log \varphi_{X}   \big)   \|_{ \mathbb{B}(T) }^2  .   \end{align}

A natural question, which we study in detail in Section <ref>, is whether the quasi-likelihood in (<ref>) is sufficient to identify the true latent distribution. From the objective function in (<ref>), this can be rephrased as to whether gradient based information is sufficient to determine the latent characteristic function. We show that this is indeed the case in dimension $d=1$ but that for higher dimensions, the quasi-likelihood in (<ref>) must be augmented with further information that accounts for the behavior of $f$ on the boundary $\partial \mathbb{B}(T)$ of $\mathbb{B}(T)$.[The boundary $\partial \Omega$ of a domain $\Omega \subset \R^d $ is the remainder of the closure of $\Omega$ after removing its interior. In the case of a ball $\mathbb{B}(T) = \{  t \in \R^d : \| t \| \leq T  \}$, this is just $\partial \mathbb{B}(T) = \{ t \in \R^d : \|t \| = T \}.$] Intuitively, the magnitude $\|f \|_{\mathbb{B}(T)}$ of a function $f : \mathbb{B}(T)  \rightarrow \mathbb{C} $  is  determined by the magnitude of its gradient $ \| \nabla f  \|_{\mathbb{B}(T)} $ and the behavior of  $f\vert_{\partial \mathbb{B}(T)}$, the function obtained by restricting $f$  to  $ \partial \mathbb{B}(T)$.[The necessity of considering more than just the gradient is clear as adding a large constant to $f$ modifies $\| f \|_{\mathbb{B}(T)}$ but leaves $\| \nabla f \|_{\mathbb{B}(T)}$ unchanged. In dimension $d=1$, the condition $f(0) = 0$ and knowledge of $\| \nabla  f \|_{\mathbb{B}(T)}$ is sufficient to  determine  the behavior of $f\vert_{\partial \mathbb{B}(T)}$. Unfortunately, this does not hold in higher dimensions. ]

Given a $\text{DP}_{\alpha}$ prior and an independent prior $G$ on covariance matrices, we consider (as defined in Section <ref>) a prior on Gaussian mixtures given by
\begin{align*}
\phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) d P(z) \;  , \; \; \;      (P, \Sigma) \sim \nu_{\alpha,G} =  \text{DP}_{\alpha} \otimes G.
\end{align*}

If $\varphi_{P,\Sigma}$ denotes the characteristic function of a Gaussian mixture $\phi_{P,\Sigma}$, we define the (preliminary) repeated measurements quasi-Bayes posterior by
\begin{align}
\label{qb-rep-0}  \nu_{\alpha,G}(  \varphi_{P,\Sigma} \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2        \big)  d \nu_{\alpha,G} (P,\Sigma)} .
\end{align}

We show in Section <ref> that the quasi-Bayes posterior in (<ref>) is valid for dimension $d=1$. For higher dimensions, as discussed above, we introduce an adjustment to account for the boundary $\partial \mathbb{B}(T)$. For a function $ g : \mathbb{B}(T) \rightarrow \mathbb{C}^d $, define the metric
\begin{align}
\label{metricold}  \| g \|_{\partial,\;\mathbb{B}(T)}^2 =  \int_{\mathbb{B}(T) } \| g(t) \|^2 dt +  \int_{\partial \mathbb{B}(T)} \int_{0}^1   \| g(t z) \|^2 dt d \mathcal{H}^{d-1}(z).
\end{align}

Here, $\mathcal{H}^{d-1}(.)$ denotes the $d-1$ dimensional Hausdorff measure.[Up to a known normalization constant $c$, integration w.r.t the Hausdorff measure is given by
$$ \int_{\partial \mathbb{B}(T)} h (z) d \mathcal{H}^{d-1}(z) = c \E \big[ h(V)  ] \; \; \; \text{where} \; \; \; V =  T \frac{Z}{\| Z \|} \; \; , \; \; Z \sim \mathcal{N}(0,I_d) $$

for every Borel function $ h : \partial \mathbb{B}(T) \rightarrow \R$.] Intuitively, for a function $h : \mathbb{B}(T) \rightarrow \R $ with $h(0) = 0$, we can write $ h(z) = \int_{0}^1 \langle \nabla h(tz) , z \rangle  dt $ for every $z \in \partial \mathbb{B}(T)$. The second term of (<ref>) therefore accounts for the boundary by specifically controlling the gradient along the straight line segment from the origin.

For $d > 1$, we define the repeated measurements quasi-Bayes posterior by
\begin{align}
\label{qb-rep2-0}  \nu_{\partial, \alpha,G}(  \varphi_{P,\Sigma} \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \big)  d \nu_{\alpha,G} (P,\Sigma)} .
\end{align}

If $(\epsilon_1,\epsilon_2,X)$ are mutually independent in model (<ref>), the roles of $Y_1$ and $Y_2$ can be interchanged in (<ref>). In particular, this provides us with two sets of identifying restrictions. It is straightforward to  modify the quasi-Bayes posteriors in (<ref>) and (<ref>) to include both sets of restrictions. One way to achieve this is to use the quasi-likelihood
\begin{align*} &  L^*( \theta ) = - \big(
\| \widehat{\varphi}_{Y_2} \nabla \log \varphi_{\theta}        - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2  +
\| \widehat{\varphi}_{Y_1} \nabla \log \varphi_{\theta}        - \widehat{\varphi}_{Y_2,Y_1}   \|_{ \mathbb{B}(T) }^2  \big) .  \end{align*}

In practice, the choice of quasi-likelihood used to induce a valid quasi-Bayes posterior is not unique. For example, one could base a quasi-likelihood on the identity (<ref>) directly. While such a choice is more straightforward for theoretical analysis, implementation requires evaluation of the path integral. As discussed previously, this is further complicated by the fact that the empirical analog of the ratio appearing inside the path integral is volatile  and highly sensitive to the choice of $t \in \R^d$.

§.§ Multi-Factor Models

In this section, we study the linear multi-factor model
\begin{align}
\label{mult-fac}\mathbf{Y}  = \mathbf{A} \mathbf{X}
\end{align}

where $\mathbf{Y} = (Y_1,\dots,Y_L)' \in \R^L$ is a vector of $L$ measurements, $\mathbf{X} = (X_1,\dots,X_K)'$ is a vector of $K$ latent and mutually independent factors and $\mathbf{A}$ is a known $L \times K$ matrix of factor loadings. We take for granted that measurements are demeaned so that $\E[\mathbf{X}] = \mathbf{0}$. The model in (<ref>) was initially introduced by <cit.>.[To ease the exposition, this section closely follows the notation introduced in <cit.>.]

In most cases, $ \mathbf{A} $ is not fully known but can be consistently estimated through means and covariances of the observed $ \mathbf{Y} = (Y_1,\dots,Y_L)$. In particular, the discrepancy from replacing $\mathbf{A}$ with an estimate $\widehat{\mathbf{A}}$ has stochastic order $O_{\mathbb{P}}(n^{-1/2})$. As the nonparametric rates of convergence are not faster than $O(n^{-1/2})$, all our main results go through with $\widehat{\mathbf{A}}$  replacing $\mathbf{A}$. For expositional simplicity, we continue with the case of known $\mathbf{A}$.

The repeated measurements model in (<ref>) can be realized as a special case of the multi-factor model in (<ref>). Indeed, consider univariate repeated measurements $(Y_1,Y_2)$ of a latent signal $X$ with measurement errors $(\epsilon_1, \epsilon_2)$. Then (<ref>) holds with
\begin{align}
\label{mult=rep}
\underbrace{
\begin{pmatrix}
Y_1 \\
\end{pmatrix}
=  \underbrace{
\begin{pmatrix}
1 & 1 & 0 \\
1 & 0 & 1
\end{pmatrix}
}_{\mathbf{A}}   \underbrace{  \begin{pmatrix}
X \\
\epsilon_1 \\
\epsilon_2
\end{pmatrix}}_{\mathbf{X}}
\end{align}

We are interested in recovering the distribution of $\mathbf{X}$ from a random sample of $\mathbf{Y}$. Denote the columns of $\mathbf{A}$ by $\mathbf{A} = (\mathbf{A}_1,\mathbf{A}_2,\dots,\mathbf{A}_K)$. In <cit.>, it is shown that the identifying restrictions of the model in (<ref>) may be expressed as
\begin{align}
\label{mult-fac-id1} \nabla \nabla'  \log \varphi_{\mathbf{Y}} (t)      = \sum_{k=1}^K \mathbf{A}_k \mathbf{A}_k'   (\log \varphi_{X_k})''(t' \mathbf{A}_k)     \; \; \; \; \; \;  \forall \; t \in \R^L \; ,
\end{align}

where $\nabla \nabla' \log \varphi_{\mathbf{Y}} (t) $ denotes the Hessian of the map $ t \rightarrow \log \varphi_{\mathbf{Y}} (t)$.

To simplify the exposition further, we introduce some additional notation. Let $\mathcal{V}_{ \mathbf{Y}   } (t)$ denote the vector of upper triangular elements of $\nabla \nabla' \log \varphi_{\mathbf{Y}} (t) $. Let $\mathcal{V}_{\mathbf{X}}(t) $ denote the vector with elements $  \{ (\log \varphi_{X_i})''(t' \mathbf{A}_i) \}_{i=1}^K  $. Let $V(\mathbf{A}_k)$  denote the vector of upper triangular elements of
$\mathbf{A}_k \mathbf{A}_k'$ and $\mathbf{Q} = [ V(\mathbf{A}_1) , \dots , V(\mathbf{A}_K) ] $ the matrix with columns composed of those vectors.

As the matrices in (<ref>) are symmetric, the identifying restrictions may be expressed as
\begin{align*}
\mathcal{V}_{\mathbf{Y}}(t) = \mathbf{Q} \mathcal{V}_{\mathbf{X}}(t) \; \; \; \; \forall \;  t \in \R^L.
\end{align*}

Equivalently, if $\mathbf{Q}$ has full column rank, the identifying restrictions are
\begin{align}
\label{mult-fac-rest0}   \mathbf{Q}^* \mathcal{V}_{\mathbf{Y}}(t) -   \mathcal{V}_{\mathbf{X}}(t)    = \mathbf{0}  \; \; \; \; \forall \;  t \in \R^L
\end{align}

where $\mathbf{Q}^* = (\mathbf{Q}' \mathbf{Q})^{-1} \mathbf{Q}'$. The empirical analog to $\mathcal{V}_{\mathbf{Y}}(t)$ is given below in (<ref>). As it involves the squared reciprocal of the characteristic function, the estimand is volatile and highly sensitive to the choice of $t \in \R^L$.[Intuitively, bad choices of $t$ lead to situations where $  \left| \varphi_{\mathbf{Y}} (t) \right| $ is significantly smaller than sampling uncertainty $ \big| (\E_n - \E) e^{\mathbf{i t' \mathbf{Y}}}  \big|$. This makes the function in (<ref>) a highly volatile function of $t \in \R^d$. Reweighting by $\widehat{\varphi}_{\mathbf{Y}}^2$ leads to a more stable quasi-likelihood.]  Similar to our strategy in the preceding section, we focus instead on a smoothed version of the identifying restriction. If the characteristic function of $\mathbf{Y}$ is non vanishing, the identifying restrictions can equivalently be expressed as
\begin{align*}
\varphi_{\mathbf{Y}}^2(t)  \big[ \mathbf{Q}^* \mathcal{V}_{\mathbf{Y}}(t) -   \mathcal{V}_{\mathbf{X}}(t) \big]    = \mathbf{0}  \; \; \; \; \forall \;  t \in \R^L
\end{align*}

Denote by $\mathbf{Q}_k^*$, the $k^{th}$ row of $\mathbf{Q}_k^*$. From the definition of $\mathcal{V}_{\mathbf{X}}(t)$, it follows that the identifying restrictions for the $k^{th}$ latent factor $X_k$ reduce to
\begin{align}
\label{mult-fac-rest}  \varphi_{\mathbf{Y}}^2(t) \big[ \mathbf{Q}_k^* \mathcal{V}_{\mathbf{Y}}(t) -  (\log \varphi_{X_k})''(t' \mathbf{A}_k)    \big] = 0 \; \; \; \; \forall \;  t \in \R^L.
\end{align}

Our starting point for a quasi-Bayes framework is to view (<ref>) as a collection of complex valued restrictions on the model. Given the observed data $\{ \mathbf{Y}_i  \}_{i=1}^n$, we estimate  the empirical counterpart to (<ref>) using

\begin{align}
& \nonumber \widehat{\varphi}_{\mathbf{Y}} = \E_n[ e^{\mathbf{i} t' \mathbf{Y} } ] =  \frac{1}{n} \sum_{j=1}^n e^{\mathbf{i} t' \mathbf{Y}_j} \; ,  \\ &    \big[ \nabla \nabla' \log \widehat{ \varphi}_{\mathbf{Y}} (t)          \big]_{l,k} = - \frac{\E_n[Y_l Y_k e^{\mathbf{i} t' \mathbf{Y}}  ] }{\widehat{\varphi}_{\mathbf{Y}}(t) } + \frac{\E_n[Y_l e^{\mathbf{i} t' \mathbf{Y}}  ] \E_n[ Y_k e^{\mathbf{i} t' \mathbf{Y}}  ]  }{\widehat{\varphi}_{\mathbf{Y}}^2(t)} .    \label{second-deriv}
\end{align}

Using these estimands, we obtain empirical analogs $\widehat{\varphi}_{\mathbf{Y}}$ and $ \widehat{\mathcal{V}}_{\mathbf{Y}} (t)$.

If $\theta$ represents a distribution with characteristic function $\varphi_{\theta}$, the identifying restrictions in (<ref>) suggest a quasi-likelihood of the form

\begin{align} \label{quaslik3} L( \theta ) = -
\| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{\theta})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2
\end{align}

Up to sampling uncertainty, the quasi-likelihood in (<ref>) is based on the objective function
\begin{align}  \label{quaslik3-obj} Q(\theta) = -\| \varphi_{\mathbf{Y}}^2 (t) \big[ (\log \varphi_{\theta})'' (t'\mathbf{A}_k) - (\log \varphi_{X_k})'' (t'\mathbf{A}_k)     \big]   \|_{ \mathbb{B}(T) }^2  .   \end{align}

A natural question is whether the quasi-likelihood in (<ref>) is sufficient to identify the true latent distribution. From the objective function in (<ref>), this can be rephrased as to whether gradient based information is sufficient to determine the latent characteristic function. Intuitively, this follows if $ \varphi_{\theta} = \varphi_{X_k}$ is the unique solution to the second order differential equation
\begin{align} \label{log-ode} (\log \varphi_{\theta})'' (t) = (\log \varphi_{X_k})'' (t) .    \end{align}

As all characteristic functions satisfy the boundary condition $ \log \varphi_{\theta} (0) = \log(1) = 0  $, (<ref>) has a unique solution if $\theta$ satisfies a second boundary condition. As the latent factor $X_k$ is demeaned, a natural boundary condition is given by $ (\log \varphi_{\theta})'(0) =   (\log \varphi_{X_k})'(0) = \mathbf{i} \E[X_k] = 0 $. This can be achieved by restricting $\theta$ to be a mean zero distribution.

Given a $\text{DP}_{\alpha}$ prior and an independent prior $G$ on covariance matrices, we consider (as defined in Section <ref>) a prior on Gaussian mixtures given by
\begin{align*}
\phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) d P(z) \;  , \; \; \;      (P, \Sigma) \sim \nu_{\alpha,G} =  \text{DP}_{\alpha} \otimes G.
\end{align*}

If $\varphi_{P,\Sigma}$ denotes the characteristic function of a Gaussian mixture $\phi_{P,\Sigma}$, we define the multi-factor      quasi-Bayes posterior for latent factor $X_k$ by
\begin{align}
\label{qb-mf-0}  \nu_{\alpha,G}( \varphi_{P, \sigma^2}   \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \big)  d \nu_{\alpha,G} ( P,\sigma^2 )}.
\end{align}

Without further modifications, the quasi-Bayes posterior in (<ref>) does not account for the mean zero boundary condition discussed above. To enforce this, we propose the demeaned quasi-posterior distribution obtained from demeaning samples from (<ref>). That is,
\begin{align} \label{qb-posterior-demean}  \overline{\nu}_{\alpha,G}(. \: | \:\mathcal{Z}_n) \sim   Z - \E[Z]  \; \; \; \; \; \; \text{where} \; \; \; \; \;   Z \sim \nu_{\alpha,G}(.|\mathcal{Z}_n) . \end{align}

From (<ref>), the identifying restrictions can be separated for each latent factor and this leads to a simple univariate quasi-Bayes posterior. However, as the latent factors  $(X_1,\dots,X_K)$ are mutually independent, the analysis is unchanged if we model them jointly using independent priors. Let $\mathbf{P} = (P_1,\dots, P_K)$ denote a vector of mixing distributions and $ \bm{\sigma}^2 = (\sigma_1^2 , \dots , \sigma_K^2) $ a vector of variance parameters. Let $\mathcal{V}_{\mathbf{P},\bm{\sigma}^2}(t) $ denote the vector with elements $  \{ (\log  \varphi_{P_i,\sigma_i^2} )'' (t' \mathbf{A}_i) \}_{i=1}^K  $. With some abuse of notation, we denote the vector of individual characteristic functions by  $  \varphi_{\mathbf{P}, \bm{\sigma}^2} =  (\varphi_{P_1,\sigma_1^2} , \dots , \varphi_{P_K , \sigma_K^2})$. The joint quasi-Bayes posterior induced from the multi-factor model in (<ref>) is then given by
\begin{align}
\label{qb-mf}  &  \nu_{\alpha,G}(  \varphi_{\mathbf{P}, \bm{\sigma}^2}   \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big( \widehat{\mathcal{V}}_{\mathbf{Y}} -  \mathbf{Q} \mathcal{V}_{\mathbf{P},\bm{\sigma}^2}    \big)      \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \widehat{\mathcal{V}}_{\mathbf{Y}} - \mathbf{Q} \mathcal{V}_{\mathbf{P},\bm{\sigma}^2}    \big)      \|_{ \mathbb{B}(T) }^2        \big)   d \nu_{\alpha,G} ( \mathbf{P}, \bm{\sigma}^2 )} \\ & \overline{\nu}_{\alpha,G}(. \: | \: \mathcal{Z}_n) \sim   Z - \E[Z]  \; \; , \; \;   Z \sim \nu_{\alpha,G}(.|\mathcal{Z}_n) .  \nonumber
\end{align}

Here $G$ is the common univariate prior on each variance parameter $ \{ \sigma_i^2 \}_{i=1}^K$. The prior $\nu_{\alpha,G}$ is the one obtained by placing mutually independent $(\text{DP}_{\alpha} \otimes G)$ priors on each of $(P_i, \sigma_i^2)_{i=1}^K$

Posterior Samples, ${\nu}_{\alpha,G}$

Posterior samples, $\overline{\nu}_{\alpha,G}$

Repeated Measurements (see Remark <ref>) multi-factor posterior with $X \sim  0.5 \mathcal{N}(-2,1) + 0.5 \mathcal{N}(2,1)$ and $\epsilon_1,\epsilon_2 \sim \mathcal{N}(0,1)$. Posterior draws for the latent distribution of $X$ relative to true latent histogram.

As Figure <ref> illustrates, in this specific example, the quasi-Bayes posterior $\nu_{\alpha,G}$ appears to concentrate on bimodal Gaussians that differ only by shifts in central tendency (mean). This is consistent with our discussion above in that they concentrate around solutions to the second order differential equation in (<ref>). These solutions identify the distribution up to the location parameter. The demeaned quasi-Bayes posterior $\overline{\nu}_{\alpha,G}$ produces posterior samples with a fixed mean at zero. As such, the boundary conditions for the differential equation are satisfied and posterior samples concentrate around the true mean-zero latent distribution.

§ THEORY

In this section, we develop the quasi-Bayes limit theory for the models in Section <ref>. First, we outline conditions on the prior in Section <ref>. In Sections <ref>-<ref>, we present the main results for each model.

§.§ Priors

The models and the associated quasi-Bayes posteriors in Section <ref> are all based on a common family of prior distributions. In all cases, we model latent distributions using a prior on Gaussian mixtures given by
\begin{align}
\label{dpgmm2} \phi_{P,\Sigma} (x)     =\int_{\R^d} \phi_{\Sigma}(x-z) d P(z) \;  , \; \; \; \; \; \; \;     (P, \Sigma) \sim \text{DP}_{\alpha} \otimes G.
\end{align}

We first state and then discuss the main assumptions that we impose on the Dirichlet process $\text{DP}_{\alpha}$ and covariance prior $G$.

[Dirichlet Prior]   The Dirichlet process $\text{DP}_{\alpha}$, in the sense of Definition <ref>, arises from a Gaussian base measure $\alpha$. That is,  $\alpha = \mathcal{N}(\mu^*, \Sigma^*)$ for some mean vector $\mu^* \in \R^d$   and positive definite matrix $\Sigma^* \in \mathbf{S}_+^d $.

[Covariance Prior]  $(i)$ $G$ is a probability measure with support contained in the space of positive
semi-definite matrices on $\R^{d \times d}$. $(ii)$ There exists $ \kappa \in (0,1   ], v_1  \geq  0 , v_2 \geq 0$ and universal constants $C,C',D,D' > 0 $ that satisfy
\begin{align*} & (a) \; \; \; \; \;  G \big(  \Sigma :  \lambda_{d} ( \Sigma^{-1})  > t_d    \big) \leq  C \exp(- C' t_d^{\kappa}) \; , \\ & (b) \; \; \; \; \; G  \bigg(   \Sigma : \bigcap_{1 \leq j \leq d} \bigg \{   t_j \leq  \lambda_j(\Sigma^{-1}) \leq  t_j(1+\delta)    \bigg\}     \bigg) \geq D t_1^{v_1} \delta^{v_2}  \exp(-D' t_d^{\kappa} )
\end{align*}

for every $  \delta \in (0,1) $ and  $0 < t_1 \leq t_2 \leq \dots \leq  t_d < \infty  $.

We note that, as the researcher chooses the prior, Assumption <ref> and <ref> can always be satisfied. For the Dirichlet process prior, by varying $(\mu^*,\Sigma^*)$, a researcher can impose varying degrees of prior information. In practice, setting $\mu^* = 0$ and $\Sigma^* = C I_{d}$ for a large $C > 0$ leads to a diffuse noninformative prior.

Assumption <ref> is frequently used in the literature on density estimation (; ). It allows for a general class of covariance priors. In dimension $d=1$, it holds with $\kappa=1/2$ if $G$ is the distribution of the square of an inverse-gamma distribution. In dimension $d > 1$, it holds with $\kappa = 1$ if $G$ is the Inverse-Wishart distribution. Various other choices may be used as well, with some leading to more straightforward computation in higher dimensions. For instance, we can always write $\Sigma = D C D$ where $C$ is a correlation matrix and $D = \sqrt{\text{Diag}(\Sigma)}$ is the scale matrix. In higher dimensions, it is often computationally simpler to place priors on $C$ and $D$ separately. For $D$, the diagonal can consist of independent univariate priors (e.g. inverse-gamma). For the correlation matrix $C$, a common choice is the class of Lewandowski-Kurowicka-Joe (LKJ) priors <cit.>.

For the base measure $\alpha$ in Assumption <ref>, one may choose $(\mu^*,\Sigma^*)$ using an empirical Bayes approach. All our main results go through with a data dependent prior $ \widehat{\alpha} = \mathcal{N}( \widehat{\mu}, \widehat{\Sigma} ) $, provided that asymptotically with probability approaching one, $\|  \widehat\mu \|$ is bounded and  the eigenvalues of $\widehat{\Sigma}$ are bounded away from zero and infinity.  For example, in the classical measurement error model (<ref>), one can choose $(\mu^*,\Sigma^*)$ to be the empirical mean and covariance matrix of the observed vector $Y$.

§.§ Main Results (Multivariate Deconvolution)

In this section, we present the main results for the deconvolution-based quasi-Bayes posterior of Section <ref>. Recall that in this case, we observe a sample $\{ Y_i \}_{i=1}^n$ and an auxiliary sample $\{ \epsilon_i  \}_{i=1}^m$, from the model in (<ref>). The main conditions that we impose on the observations is summarized in the following condition.
[Data]   $(i)$ $(Y_i)_{i=1}^n \in \R^d$ is a sequence of independent and identically distributed random vectors. $(ii)$ $(\epsilon_1, \dots , \epsilon_m) \stackrel{i.i.d}{\sim} F_{\epsilon} $ where $F_{\epsilon}$  is the distribution of the error $\epsilon$ in model (<ref>). $(iii)$ The sample size of the auxiliary sample grows at rate $m = m_n \asymp n$. $(iv)$ The distributions have finite second moments: $\E\big( \| Y \|^2 \big) < \infty $ and $\E\big( \| \epsilon \|^2 \big) < \infty $.

The rate requirement on the auxiliary sample size in Condition <ref>$(iii)$ is made for simplicity and could be relaxed further.[Lower sample sizes only influence the analysis through weaker statistical guarantees on the estimated characteristic function $\widehat{\varphi}_{\epsilon}(.)$. It is straightforward to incorporate this into the analysis.] As is common in the literature, we characterize the ill-posedness in the model through the decay rate of the characteristic function of the error $\varphi_{\epsilon}(.)$. This is summarized in the following definition.   We say a model is either mildly or severely ill-posed if

\begin{align*}
\inf_{t \in \R^d : \|t \|_{\infty} \leq T} \left| \varphi_{\epsilon}(t) \right| \asymp \begin{cases} T^{- \zeta} & \text{mildly ill-posed}  \\  \exp(-R T^{\zeta})   & \text{severely ill-posed}        \end{cases}    \end{align*}

for some $R, \zeta \geq 0$.

Recall that the  deconvolution-based quasi-Bayes posterior is given by
\begin{align}
\label{qb-dec-2}  \nu_{\alpha,G}(  \varphi_{P,\Sigma} \: | \: \mathcal{Z}_n) =  \frac{\exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T)  }^2        \big)}{\int \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T)  }^2        \big) d \nu_{\alpha,G}(P,\Sigma) }.
\end{align}

Intuitively, our general strategy to study (<ref>) proceeds as follows. The quantity $\| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T)  }^2 $ represents a smoothed version of the distance $\|  \phi_{P,\Sigma} - f_{X}  \|_{L^2}^2$, where $f_X$ denotes the latent density of $X$. We first obtain contraction rates with respect to this smoothed, weaker metric. Its implications towards distance in the stronger metric $ \|  \phi_{P,\Sigma} - f_X  \|_{ L^2  }^2 $ is then determined by the complexity of $\mathbb{B}(T)$ and the model ill-posedness.

Before considering the general case, we first study the case where the true latent density $f_X$ admits a representation as a (possibly infinite) Gaussian mixture:
\begin{align}
\label{d=gaussm} f_X(x) = \phi_{\Sigma_0} \star F_0(x) = \int_{\R^d} \phi_{\Sigma_0} (x-z) d F_0(z).
\end{align}

Here, $\Sigma_0 \in \mathbf{S}_+^d$  is a positive definite matrix and $F_0$ is a mixing distribution. Note that, we do not assume knowledge of $(\Sigma_0,F_0)$, only that the representation in (<ref>) is true for some $(\Sigma_0,F_0)$. Importantly, if $F_0$ is continuously distributed or has unbounded support, the specification in (<ref>) allows for a Gaussian mixture with infinite components. The model specification in (<ref>) is commonly used to study  Gaussian mixture based approaches to density estimation.[Convergence rates (for univariate density estimation) for the model in (<ref>) using a Gaussian mixture sieve maximum likelihood estimator were initially studied by <cit.>. Contraction rates (and improved rates on the MLE) for pure Bayes density estimation were subsequently obtained in <cit.>.]

Finite Gaussian mixture representations are widely used in empirical settings.[As in, for example, <cit.>; <cit.>; <cit.>.] Hence, the model in (<ref>) serves as a natural starting point. Intuitively, as the prior concentrates on Gaussian mixtures, the expectation is that the model should exhibit negligible bias in approximating a latent density with Gaussian mixture structure as in (<ref>). We formalize this intuition below by showing that nearly parametric rates of convergence can be obtained. To that end, our main assumption on the Gaussian mixture specification is the following.

[Exact Gaussian Mixtures]    $f_X = \phi_{\Sigma_0} \star F_0$ where $\Sigma_0 \in \R^{d \times d}$ is a positive definite matrix and   $F_0$ is a mixing distribution that satisfies $ F_0 \big(  t \in \R^d : \| t \| > z  \big) \leq C \exp(- C' z ^{\chi}) $ for all sufficiently large $z$, where $C,C',\chi \geq 0$ are universal constants.

Condition (<ref>) imposes an exponential tail on the mixing distribution $F_0$. Importantly, this covers all Gaussian mixtures that have modes contained in a compact subset of $\R^d$. As the mixing distribution may be continuously distributed, the corresponding Gaussian mixture may be infinite as well. The following result shows that, under suitable scaling of the covariance prior, the quasi-Bayes posterior contracts towards the true latent density $f_X$ at a nearly parametric rate.

Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref> and $\sigma_n^2$  is as specified below. Furthermore, suppose $\Sigma_0$ is in the support of $G$. Let $\kappa, \chi  > 0$ be as in Assumption <ref> and  Condition <ref>, respectively.

$(a)$  Let  $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1    \} $. Suppose the model is mildly ill-posed with  $\zeta \geq 0$ as in Definition <ref>. If $T_n \asymp  \sqrt{\log n} \sqrt{\log \log n}  $ and $\sigma_n^2 \asymp (\log n)^{- \lambda/\kappa} $, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_X  -  \phi_{P,\Sigma}      \|_{L^2} >  C \frac{(\log n)^{(\lambda + \zeta)/2} }{\sqrt{n}} (\log \log n)^{\zeta/2}  \:     \bigg| \mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

$(b)$ Let  $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1  , d/ \zeta + 1/2   \} $. Suppose the model is severely ill-posed with  $\zeta \in (0,2]$ and $R > 0$ as in Definition <ref>. If $T_n =  (c_0 \log n)^{1/ \zeta} $ for any $c_0$ satisfying $c_0 R  = \gamma < 1/2$ and $\sigma_n^2$ is defined by
\begin{align*}
\sigma_n^2 \asymp \begin{cases} (\log n)^{2 / \zeta -1 - \lambda/\kappa  } (\log \log n)^{- 2/\kappa}  & \zeta \in (0,2)  \\  (\log n)^{- \lambda/\kappa}  & \zeta =2. \end{cases}
\end{align*}

Then, there exists a universal constant $C > 0$ and $V \in (0,1/2)  $ such that
\begin{align*}
\begin{cases}   \nu_{\alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{-1/2 + \gamma} (\log n)^{\lambda/2}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1)  & \zeta \in (0,2) \\ \nu_{\alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{-V}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1) & \zeta=2.    \end{cases}
\end{align*}

Under a mildly ill-posed regime, the quasi-Bayes posterior attains nearly (up to a logarithmic factors) parametric rates of convergence. If the model is severely ill-posed at a rate $\zeta$ that is lower than Gaussian type ill-posedness, nearly parametric rates in the form  $n^{-1/2 + \epsilon}$ for any $\epsilon > 0$ can be obtained. In the extreme case where the model exhibits Gaussian ill-posedness, the model bias (which is determined by the decay rate of a Gaussian mixture characteristic function) has similar order to the variance. In this case, the rate is still polynomial but the exponent $V$ depends on second order factors such as the constant $R$ in Definition <ref> and the eigenvalues of the matrix $\Sigma_0$ in representation (<ref>). In the case of univariate frequentist kernel deconvolution, a similar finding can be found in <cit.>.

The strategy of scaling the covariance prior can be traced back to <cit.> where it was employed for univariate pure Bayesian density estimation. This requirement was eventually removed in <cit.>. We conjecture that the scaling may be an artifact of the proof and can possibly be relaxed. As our quasi-Bayesian setup is quite different to a  pure Bayesian model, the analysis in <cit.> is not directly applicable here. We leave this investigation for future work. In our simulations and empirical illustrations, we do not apply any scaling to the covariance prior.

An interesting aspect of the quasi-Bayes framework is that all formal analysis takes place directly in the Fourier domain. As the Fourier transform $\mathcal{F}$ is a distance preserving (up to a constant) isomorphism between $L^2$ spaces, $ \| .\|_{L^2}  $ contraction rates in the Fourier domain translate directly to $ \| . \|_{L^2}$ rates for the density. However, this can also be used to derive contraction rates in stronger metrics such as $\| . \|_{L^{\infty}}$. In particular, as the Fourier transform satisfies $\| f  \|_{\infty} \leq \| \mathcal{F}[f]   \|_{L^1}$, we have the following variant of Theorem <ref>.

Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref> and $\sigma_n^2$  is as specified below. Furthermore, suppose $\Sigma_0$ is in the support of $G$. Let $\kappa, \chi  > 0$ be as in Assumption <ref> and  Condition <ref>, respectively.

$(a)$  Let  $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1    \} $. Suppose the model is mildly ill-posed with  $\zeta \geq 0$ as in Definition <ref>. If $T_n \asymp  \sqrt{\log n} \sqrt{\log \log n}  $ and $\sigma_n^2 \asymp (\log n)^{- \lambda/\kappa} $, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_X  -  \phi_{P,\Sigma}      \|_{L^\infty} >  C \frac{(\log n)^{(\lambda + d/2 +  \zeta)/2} }{\sqrt{n}} (\log \log n)^{(\zeta+d/2)/2}  \:     \bigg|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

$(b)$ Let  $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1  , d/ \zeta   \} $. Suppose the model is severely ill-posed with  $\zeta \in (0,2]$ and $R > 0$ as in Definition <ref>. If $T_n =  (c_0 \log n)^{1/ \zeta} $ for any $c_0$ satisfying $c_0 R  = \gamma < 1/2$ and $\sigma_n^2$ is defined by
\begin{align*}
\sigma_n^2 \asymp \begin{cases} (\log n)^{2 / \zeta -1 - \lambda/\kappa  } (\log \log n)^{- 2/\kappa}  & \zeta \in (0,2)  \\  (\log n)^{- \lambda/\kappa}  & \zeta =2. \end{cases}
\end{align*}

Then, there exists a universal constant $C > 0$ and $V \in (0,1/2)  $ such that
\begin{align*}
\begin{cases}   \nu_{\alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^\infty} >  C n^{-1/2 + \gamma} (\log n)^{(\lambda + d/\zeta)/2}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1)  & \zeta \in (0,2) \\ \nu_{\alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^\infty} >  C n^{-V}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1) & \zeta=2.    \end{cases}
\end{align*}

The  $L^2$ and $L^{\infty}$ rates in Theorem <ref> and Corollary <ref> appear to be novel even for the special case of density estimation.[Rates of convergence with Dirichlet Process priors in $L^1$ or Hellinger risk can be found in <cit.>] As discussed in Remark <ref>, this is an interesting consequence of our characteristic function based quasi-Bayesian approach. While it may be possible for pure-Bayesian approaches to achieve similar results, the analysis to verify this appears to be  significantly more complicated. For ease of exposition, all the remaining results in Section <ref> are stated only for the $\| . \|_{L^2}$ metric. By analogous arguments to that of Corollary <ref>  and Remark <ref>, it is straightforward to extend these results to stronger metrics such as $\| . \|_{L^\infty}$.

As Gaussian mixtures can approximate any density $f_X \in L^2$ arbitrarily well, it is straightforward to modify the preceding results to obtain consistency for a general density.[Here, consistency means that
$$ \nu_{\alpha,G_n} \big(  \| f_X - \phi_{P,\Sigma}  \|_{L^2} > \epsilon  \big| \mathcal{Z}_n \big) = o_{\mathbb{P}}(1)  \; \; \; \; \; \; \forall \; \; \epsilon > 0. $$
] We focus instead on the more difficult task of establishing a rate of convergence. To that end, consider the case where the true latent density $f_X$ is not an exact Gaussian mixture and thus, the prior model exhibits non-negligible bias. In this model, the Gaussian mixture bias is closely related to the quantity
\begin{align} \label{gm-bias} \mathcal{B}(\sigma) =   \min_{P} \|f_X \star f_{\epsilon}  - \phi_{P, \sigma^2 I}   \star f_{\epsilon}     \|_{L^2} \asymp \min_{P} \| \varphi_{\epsilon}\big(  \varphi_{X} - \varphi_{P,\sigma^2 I}  \big)   \|_{L^2} ,  \end{align}

where $\sigma^2 I$ denotes the identity matrix with each diagonal entry $\sigma^2$ and the minimum is over all mixing distributions $P$.

For a conservative upper bound on the rate at which $\mathcal{B}(\sigma) $ tends to $0$ as $\sigma \rightarrow 0$, observe that by picking $P = f_X$, we obtain

\begin{align*}
\| \varphi_{\epsilon}\big(  \varphi_{X} - \varphi_{P,\sigma^2 I}  \big)   \|_{L^2} =  \| (e^{- \| t \|^2 \sigma^2 /2} - 1 ) \varphi_{X} (t) \varphi_{\epsilon}(t)  \|_{L^2} \lessapprox \sigma^2 .
\end{align*}

The preceding bound follows from $ \left| e^{-\| t \|^2 \sigma^2 / 2} - 1 \right| \leq \|t \|^2 \sigma^2 / 2 $. As such, the concluding estimate is valid provided that $ \int_{\R^d} \| t \|^4 \left|  \varphi_{X} (t) \right|^2 \left| \varphi_{\epsilon}(t)  \right|^2 < \infty  $ or equivalently $f_{X} \star f_{\epsilon} \in \mathbf{H}^2 $. For $f_{X} \star f_{\epsilon}$ with smoothness order greater than $2$, this argument does not provide better rates. In particular, with higher smoothness, the choice $ P = f_X$  is not optimal in (<ref>): one can typically do  better by allowing $ P = P_{X,\epsilon,\sigma}$ to depend implicitly on all the features $(f_{\epsilon},f_{X},\sigma)$.

We consider the following characterization of the Gaussian mixture bias.
[Gaussian Mixture Bias]
$(i)$ $f_X \in \mathbf{H}^p(R)$ for some $p > 1/2$ and $R < \infty$. $(ii)$ There exists universal constants $  C, M    < \infty$ and $\chi > 0$  such that for all $\sigma > 0 $ sufficiently small, there exists a mixing distribution $F_{\sigma} = F_{X,\epsilon,\sigma}$ supported on  the cube $ I_{\sigma} = [-C  (\log \sigma^{-1})^{1/\chi},C  (\log \sigma^{-1})^{1/\chi} ]^d$ that satisfies $ \| f_{X} \star f_{\epsilon}  - \phi_{F_{\sigma}, \sigma^2 I } \star f_{\epsilon}  \|_{L^2} \leq  M \sigma^{p  + \zeta}  $ .

Variants of Condition <ref> are commonly used in density estimation (; ). A stronger version of Condition <ref> is also used in <cit.> within the context of univariate pure Bayes density deconvolution. For examples of explicit constructions of $F_{\sigma}$ in Condition <ref>, see <cit.> and <cit.> for density estimation and <cit.> for deconvolution. We note that, at least in our quasi-Bayes setup, Condition <ref> can be weakened further. In particular, it suffices that the approximation holds with respect to the weaker norm $\| . \|_{\mathbb{B}(T_n)}$ and for a sequence $\sigma_n$, where $(\sigma_n^{-1},T_n)$ grow at a suitable rate. Finally, it is worth noting that Bayesian procedures do not require knowledge of the optimal mixing distribution $F_{\sigma}$ in Condition <ref> for implementation. The existence is purely a proof device towards obtaining theoretical guarantees.

The following result derives the quasi-Bayes posterior contraction rate when the Gaussian mixture bias is as in Condition <ref>.

Suppose the model is mildly ill-posed with  $\zeta \geq 0$ as in Definition <ref>. Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Let $\kappa, \chi  > 0$ be as in Assumption <ref> and Condition <ref>, respectively. Define $\lambda = \max \{ \chi^{-1}(d+2), \chi^{-1} d + 1   \}$. If $T_n \asymp n^{1/[2(p + \zeta) + d]} \sqrt{\log n} $ and $\sigma_n^2 \asymp T_n^{2-d/\kappa}/(\log n)^{\lambda/\kappa + 1} \log \log n $, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_X -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{-p/[2(p + \zeta) + d]} (\log n)^{(\lambda + \zeta)/2 + d/4}  \:     \bigg|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

Theorem <ref> provides contraction rates for the mildly ill-posed case. As the theorem illustrates, rates are determined by the Gaussian mixture bias. From our discussion above, this  implicitly depends on the underlying smoothness of the density. Rates for the severely ill-posed case can be obtained in a similar manner. In this case, the Gaussian mixture bias decays exponentially in $\sigma$ (see e.g. ).

§.§ Main Results (Repeated Measurements)

In this section, we present the main results for the repeated measurements quasi-Bayes posterior of Section <ref>. Recall that in this case, we observe a sample $\{ Y_{1,i}, Y_{2,i} \}_{i=1}^n$  from the model in (<ref>). The main conditions that we impose on the observations is summarized in the following condition.
[Data]   $(i)$ $(Y_{1,i},Y_{2,i})_{i=1}^n $ is a sequence of independent and identically distributed random vectors. $(ii)$ Finite second moments: $\E\big( \| Y_1 \|^2 \big) < \infty $ and $\E\big( \| Y_2 \|^2 \big) < \infty $.

To begin, we consider the repeated measurements quasi-Bayes posterior without explicit boundary correction. From Section <ref>, this takes the form
\begin{align}
\label{qb-rep-2}  \nu_{\alpha,G}(  \varphi_{P,\Sigma} \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T) }^2        \big)  d \nu_{\alpha,G} (P,\Sigma)} .
\end{align}

Up to sampling uncertainty, the quasi-Bayes posterior in (<ref>) is based on a transformation of the objective function
\begin{align}
\label{rep-obj} Q(  \varphi_{P,\Sigma} ) =  \| \varphi_{Y_2} \big( \nabla \log \varphi_{P,\Sigma}  - \nabla \log \varphi_{X}  \big)  \|_{ \mathbb{B}(T) }^2 .
\end{align}

From this formulation, some natural questions arise. What are the statistical guarantees of such a posterior? Does $L^2$ gradient behavior over a domain like $\mathbb{B}(T)$  provide enough identifying information to recover the true latent density? What is the difference between this posterior and one where boundary behavior is explicitly accounted for? In the remainder of this section, we investigate these questions in detail.

As in Section <ref> (see (<ref>) and the discussion following it), first we consider the case where the true latent density $f_X$ admits a representation as a (possibly infinite) Gaussian mixture:
\begin{align}
\label{d=gaussm2} f_X(x) = \phi_{\Sigma_0} \star F_0(x) = \int_{\R^d} \phi_{\Sigma_0} (x-z) d F_0(z).
\end{align}

Our main assumption on the Gaussian mixture specification is as follows.
[Exact Gaussian Mixtures in Repeated Measurements]  $(i)$ $f_X = \phi_{\Sigma_0} \star F_0$ where $\Sigma_0 \in \R^{d \times d}$ is a positive definite matrix. $(ii)$ The mixing distribution $F_0$ satisfies $ F_0 \big(  t \in \R^d : \| t \| > z  \big) \leq  C \exp(- C' z ^{\chi}) $, $\left| \varphi_{F_0} (t)  \right| \geq c \exp(- c' \| t \|^2) $ and $ \|  \nabla \log \varphi_{F_0} (t) \| \leq C \| t \| $ for all sufficiently large $z$ and $\|t \|$, where $\chi > 0$ and $c,c',C,C' > 0$ are universal constants.

By integrating the differential inequality, the Condition $ \|  \nabla \log \varphi_{F_0} (t) \| \leq C \| t \|  $ essentially states that $\varphi_{F_0}(.)$ has at most Gaussian decay. As a first result, we show that quasi-Bayes posterior in (<ref>) contracts rapidly around the logarithmic gradient of the true characteristic function.

Suppose the characteristic function of $Y_2$ has at most Gaussian decay, that is $  \inf_{\| t \|_{\infty} \leq T }  \left| \varphi_{Y_2} (t)  \right| \geq \exp(-R T^2) $ for some $R > 0$. Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G$ where $G$ satisfies Assumption <ref>. Furthermore, suppose $\Sigma_0$ is in the support of $G$. Let $ \chi  > 0$ be as in Condition <ref> and   $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1    \} $. If $T_n = (c_0 \log n)^{1/2}$ for any $c_0$ satisfying $c_0 R = \gamma < 1/2$, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G} \bigg(  \| \nabla \log  \varphi_{X}  - \nabla \log \varphi_{P,\Sigma}      \|_{\mathbb{B}(T_n)} >  C n^{-1/2 + \gamma}  (\log n)^{\lambda /2}   \:     \bigg|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

By picking $c_0$ sufficiently small, Theorem <ref> shows that the quasi-Bayes posterior contracts around the gradient of $\log \varphi_X$ at nearly parametric rates of the form $n^{-1/2 + \epsilon}$ for any $\epsilon > 0$.

While the results of Theorem <ref> are encouraging, the question remains as to whether this can be improved to convergence in  a stronger metric such as $\|  f_X - \phi_{P,\Sigma}  \|_{L^2}$. As $ \varphi_{X}  $ and $\varphi_{P,\Sigma}$ are the characteristic functions of random variables, they necessarily satisfy the initial value condition  $ \nabla \log \varphi_X (0) = \nabla \log \varphi_{P,\Sigma}(0) = 0  $. From this initial value condition, in dimension $d=1$, the fundemental theorem of calculus and Cauchy-Schwarz imply
\begin{align}   \label{schwarz-trick}
\sup_{t \in \mathbb{B}(T)}  \left|\log \varphi_X (t) - \log \varphi_{P,\Sigma}(t)           \right| & = \sup_{t \in \mathbb{B}(T)}\left| \int_0^t  \big[  \nabla \log \varphi_X (s) - \nabla \log \varphi_{P,\Sigma}(s)     ]  ds  \right| \\ & \nonumber \leq \sqrt{T} \| \nabla \log  \varphi_{X}  - \nabla \log \varphi_{P,\Sigma}      \|_{\mathbb{B}(T)} .
\end{align}

In particular, at least in dimension $d=1$, contraction of gradients is informative towards recovering the true latent distribution. This is formalized in the next result.

Suppose $d=1$ and the characteristic function of $Y_2$ has at most Gaussian decay, that is $  \inf_{\| t \|_{\infty} \leq T }  \left| \varphi_{Y_2} (t)  \right| \geq \exp(-R T^2) $ for some $R > 0$. Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Furthermore, suppose $\Sigma_0$ is in the support of $G$. Let $\kappa,\chi > 0$ be as in Assumption <ref> and Condition <ref> respectively and define $ \lambda = \max \{ 1/2 + 3/\chi , 2 \}  $. If $T_n = (c_0 \log n)^{1/2}$ for any $c_0$ satisfying $c_0 R = \gamma < 1/2$ and $\sigma_n^2 \asymp (\log n)^{-\lambda/\kappa}$, then there exists a universal constant $C > 0$ and $V \in (0,1/2)  $ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{-V}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1) .
\end{align*}

With Gaussian decay on $\varphi_{Y_2}$, the model bias (which is determined by the decay rate of a Gaussian mixture characteristic function) has similar order to the variance. In this case, as in the discussion following Theorem <ref>, the precise value of the exponent $V$ depends on second order factors such as the constant $R$  and the eigenvalues of the matrix $\Sigma_0$ in the representation (<ref>).

The strategy employed to prove Theorem <ref> is based on (<ref>). Unfortunately, for dimension $d >1$ and a function $f : \mathbb{B}(T_n) \rightarrow \mathbb{C}$, there is no direct relationship between $  \| f \|_{\mathbb{B}(T_n)} $ and $\| \nabla f  \|_{\mathbb{B}(T_n)}$, even with initial value condition $f(0) = 0$. However, for sufficiently nice domains like $\mathbb{B}(T_n)$, it can be shown that[We show this formally in Lemma <ref>. Our argument essentially follows from the general Poincaré trace results in <cit.>]
\begin{align}
\label{poincare} \| f \|_{\mathbb{B}(T_n)} \leq  C_n \big( \| \nabla f \|_{\mathbb{B}(T_n)} + \| f  \|_{\partial \mathbb{B}(T_n) }      \big) \;,
\end{align}

holds, where $C_n$ is a constant that depends only on $T_n$ and $\| . \|_{ \partial \mathbb{B}(T_n)}$ is the $L^2$ norm on the boundary $\partial \mathbb{B}(T_n) $ of $\mathbb{B}(T_n)$.[The $L^2$ norm on the boundary is taken with respect to the $d-1$ dimensional Hausdorff measure $\mathcal{H}^{d-1}$. Specifically,
$$ \| f  \|_{\partial \mathbb{B}(T_n)}^2 = \int_{\partial \mathbb{B} (T_n) } \left| f(t) \right|^2 d \mathcal{H}^{d-1}(t).  $$
] In the theory of Sobolev spaces, inequality (<ref>)  is commonly referred to   as a  Poincaré inequality <cit.>.

The discussion above suggests that a suitable modification of the quasi-Bayes posterior to account for the second term in $(\ref{poincare})$ may resolve the problem. The starting point in our analysis towards this goal is to observe that a function $f : \mathbb{B}(T_n) \rightarrow \mathbb{C}$ with  initial value condition $f(0) = 0$ satisfies
\begin{align}
\label{lineintegral} f(z) = \int_{0}^1 \langle \nabla f(tz) , z    \rangle dt \; \; \; \; \; \; \;  \forall \: z \in \partial \mathbb{B}(T_n).
\end{align}

Specifically, (<ref>) says that the value of $f(.)$ at a boundary point $z \in \partial \mathbb{B}(T_n)$ can be determined by a scaled average of $\nabla f$ on the line segment connecting the origin to $z$. In particular, by accounting for values of $\nabla f$ on this line segment, we can induce a boundary corrected quasi-Bayes posterior. This motivates the following distance metric. For a function $ g : \mathbb{B}(T_n) \rightarrow \mathbb{C}^d $, define the metric
\begin{align}
\label{metricnew1}   \| g \|_{\partial,\;\mathbb{B}(T)}^2 =  \int_{\mathbb{B}(T_n) } \| g(t) \|^2 dt +  \int_{\partial \mathbb{B}(T_n)} \int_{0}^1   \| g(t z) \|^2 dt d \mathcal{H}^{d-1}(z).
\end{align}

As in Section <ref>, $d \mathcal{H}^{d-1}$ denotes the $d-1$ dimensional Hausdorff measure  on $\partial \mathbb{B}(T_n)$. For $d  > 1$, we define the (boundary corrected) repeated measurements quasi-Bayes posterior by
\begin{align}
\label{qb-rep2}  \nu_{\partial, \alpha,G}(  \varphi_{P,\Sigma} \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \big)  d \nu_{\alpha,G} (P,\Sigma)} .
\end{align}

The following result verifies that quasi-Bayes posterior in (<ref>) contracts towards the true latent distribution, for all dimensions.

Suppose the characteristic function of $Y_2$ has at most Gaussian decay, that is $  \inf_{\| t \|_{\infty} \leq T }  \left| \varphi_{Y_2} (t)  \right| \geq \exp(-R T^2) $ for some $R > 0$. Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Furthermore, suppose $\Sigma_0$ is in the support of $G$. Let $\kappa,\chi > 0$ be as in Assumption <ref> and Condition <ref> respectively and define $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1 \}  $. If $T_n = (c_0 \log n)^{1/2}$ for any $c_0$ satisfying $c_0 R = \gamma < 1/2$ and $\sigma_n^2 \asymp (\log n)^{-\lambda/\kappa}$, then there exists a universal constant $C > 0$ and $V \in (0,1/2)  $ such that
\begin{align*}
\nu_{\partial, \alpha,G_n} \bigg(  \| f_{X}  -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{-V}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1) .
\end{align*}

Next, analogous to our discussion in Section <ref>, we consider the case where the true latent density $f_X$ is not an exact Gaussian mixture and thus, the prior model exhibits non-negligible bias. In this model, the Gaussian mixture bias is closely related to the quantity
\begin{align} \label{gm-bias2}
\mathcal{B}(\sigma,T) = \min_{P}   \|  \varphi_{Y_2} \big( \nabla \log \varphi_{P, \sigma^2 I} - \nabla \log \varphi_{X}      \big)   \|_{\partial,\;\mathbb{B}(T)}
\end{align}

where $\sigma^2 I$ denotes the identity matrix with each diagonal entry $\sigma^2$ and the minimum is over all mixing distributions $P$. For a conservative upper bound on the rate at which $\mathcal{B}(\sigma)$ tends to $0$ as $\sigma \rightarrow 0$, observe that by picking $P = f_X$ we obtain
\begin{align*}
\|  \varphi_{Y_2} \big( \nabla \log \varphi_{P, \sigma^2 I} - \nabla \log \varphi_{X}      \big)   \|_{\partial,\;\mathbb{B}(T)} & =  \|  \varphi_{Y_2} \big( \nabla \log \big( \varphi_X \varphi_{\sigma^2 I}     \big) - \nabla \log \varphi_{X}      \big)   \|_{\partial,\;\mathbb{B}(T)} \\ & =  \|  \varphi_{Y_2} \nabla \log \varphi_{\sigma^2 I}         \|_{\partial,\;\mathbb{B}(T)}.
\end{align*}

Since the logarithmic gradient of the Gaussian characteristic function is $\nabla \log \varphi_{\sigma^2 I} (t) =  - \sigma^2  t $, the preceding equality reduces to
\begin{align}
\|  \varphi_{Y_2} \big( \nabla \log \varphi_{P, \sigma^2 I} - \nabla \log \varphi_{X}      \big)   \|_{\partial,\;\mathbb{B}(T)} \leq \sigma^2 \|  \varphi_{Y_2}(t)   t  \|_{\partial,\;\mathbb{B}(T)} \lessapprox \sigma^2 (1+T^{d/2-1}). \label{bias-final}
\end{align}

The preceding bound is valid provided that[By change of variables on $ (\partial \mathbb{B}(T), d \mathcal{H}^{d-1}) $ and the coarea formula <cit.>
$$ \int_{0}^1 \int_{\partial \mathbb{B}(T)} t^2 \|z \|^2 \left| \varphi_{Y_2}(tz)   \right|^2 d \mathcal{H}^{d-1}(z) dt \lessapprox T^{d-1} \int_{0}^1 \int_{\partial \mathbb{B}(t \times T)}  \frac{\left| \varphi_{Y_2} (y) \right|^2}{\| y \|^{d-3}} d \mathcal{H}^{d-1}(y) dt      = T^{d-2} \int_{\mathbb{B}(T)  } \frac{\left| \varphi_{Y_2}(y) \right|^2}{\| y \|^{d-3}} dy \lessapprox T^{d-2}.$$
] $ \int_{\R^d} \| t \|^2 \left|  \varphi_{Y_2}(t)  \right|^2 dt < \infty   $ or equivalently  that $ f_{Y_2} = f_X \star f_{\epsilon_2} \in \mathbf{H}^1$. For $ f_X \star f_{\epsilon_2}$ with smoothness order greater than $1$, this argument does not provide better rates. Analogous to the discussion in Section <ref>, the choice $P = f_X$ in (<ref>) may not be optimal. One way to see this directly is to express $\varphi_{Y_2} = \varphi_{X} \varphi_{\epsilon_2}$ and write
\begin{align}
\varphi_{Y_2} \big( \nabla \log \varphi_{P, \sigma^2 I} - \nabla \log \varphi_{X} \big) =  \nabla \log \varphi_{P, \sigma^2 I}   \big[\varphi_{\epsilon_2} \big( \varphi_{X} -  \varphi_{P, \sigma^2 I}  \big) \big] + \varphi_{\epsilon_2}  \big( \nabla \varphi_{P, \sigma^2 I} - \nabla \varphi_{X}   \big). \label{rep-alt}
\end{align}

If we consider mixing distributions $P$ with characteristic function having at most Gaussian decay, then $  \sup_{t \in \mathbb{B}(T)} \| \nabla \log \varphi_{P, \sigma^2 I} (t)  \| \lessapprox T  $. In particular, this implies that the Gaussian mixture bias is bounded above by
\begin{align}
\mathcal{B}(\sigma,T) \lessapprox   \min_{P} \bigg[ T \|  \varphi_{\epsilon_2} \big(  \varphi_{X} -  \varphi_{P, \sigma^2 I}  \big)    \|_{\partial,\;\mathbb{B}(T)} +   \|  \varphi_{\epsilon_2} \big(  \nabla \varphi_{X} -   \nabla \varphi_{P, \sigma^2 I}  \big)    \|_{\partial,\;\mathbb{B}(T)}      \bigg].    \label{gm-bias-explain}
\end{align}

The quantity on the right closely resembles the Gaussian mixture bias that arises from a deconvolution model as in (<ref>). While this illustrates some of factors that determine the bias, the preceding argument is however quite loose as it treats the characteristic function and its gradient separately. We generally expect the bias in (<ref>) to have smaller order than the right side of (<ref>). Indeed, the simple choice $P = f_X$ on the right side of $(\ref{gm-bias-explain})$ leads to a worse bound than (<ref>) and requires more stringent[To be specific, it requires $f_{\epsilon_2} \in  \mathbf{H}^2, f_{X} \in \mathbf{H}^1 , \int_{\R^d} \| t \|^4 \|  \nabla \varphi_{X}(t) \|^2 \left| \varphi_{\epsilon}(t)  \right|^2 dt < \infty    $  instead of the much weaker requirement that $f_X \star f_{\epsilon} \in \mathbf{H}^1$ in (<ref>).] smoothness conditions imposed on $(f_X,f_{\epsilon})$. Motivated by this discussion, we consider the following characterization of the Gaussian mixture bias.

[Gaussian Mixture Bias]
$(i)$ $f_X \in \mathbf{H}^p$ for some $p > 0$ and $ \sup_{t \in \R^d} \| \nabla \log \varphi_{X} (t) \| \leq D  $ for some $D < \infty$. $(ii)$ There exists universal constants $  C, M    < \infty$ and $\chi > 0$  such that for all $\sigma > 0 $ sufficiently small and $T > 0$ sufficiently large, there exists a mixing distribution $F_{\sigma} = F_{X,\epsilon,\sigma}$ supported on  the cube $ I_{\sigma} = [-C  (\log \sigma^{-1})^{1/\chi},C  (\log \sigma^{-1})^{1/\chi} ]^d$ that satisfies $(a)$ $ \|  \varphi_{Y_2} \big(  \nabla \log \varphi_{F_{\sigma}, \sigma^2 I} - \nabla \log \varphi_{X}     \big)  \|_{\partial,\;\mathbb{B}(T)}^2  \leq  M (1+T^{  d-2 }) \sigma^{2(s + \zeta )}  $ for some  $s > \max \{3/2 , d/2 + 1/2 \}  $, $(b)$ $\inf_{\|t \| \leq T} \left| \varphi_{F_{\sigma}} (t)  \right| \geq T^{- \gamma_1} \sigma^{- \gamma_2}$  for some $\gamma_1,\gamma_2 < \infty $ and $(c)$ $\| \nabla \log \varphi_{F_{\sigma}}   \| \leq M $.

The following result derives the quasi-Bayes posterior contraction rate when the Gaussian mixture bias is as in Condition <ref>. This condition is stated to cover the mildly ill-posed setting. The requirement in the severely ill-posed case is similar except that the Gaussian mixture bias is expected to be exponential in $\sigma$.

Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Let $\kappa, \chi  > 0$ be as in Assumption <ref> and <ref>, respectively. Define $\lambda = \max \{ \chi^{-1}(d+2), \chi^{-1} d + 1   \}$ and $\beta = \max \{ 0 , d-2 \}$. Suppose there exists $\zeta  > 0$ such that the characteristic function of $Y_2$ satisfies $\inf_{\|t \| \leq T } \left| \varphi_{Y_2} (t)   \right| \geq R T^{- \zeta} $ for some $R > 0$. If $T_n \asymp n^{1/[2(s + \zeta) +(d- \beta)   ]} \sqrt{\log n} $ and $\sigma_n^2 \asymp T_n^{2-d/\kappa}/(\log n)^{\lambda/\kappa + 1} \log \log n $, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_X -  \phi_{P,\Sigma}      \|_{L^2} >  C n^{- \min \{ p,s - \beta/2 -1.5 \} /[2(s+ \zeta) + (d- \beta) ]} (\log n)^{ \lambda/2 }  \:     \bigg|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

As in Theorem <ref>, rates are determined by the Gaussian mixture bias. An interesting avenue for future research would be explicitly characterize the Gaussian mixture bias based on lower level smoothness assumptions on the latent density and unobserved errors. As the analysis in <cit.> and <cit.> shows, the precise construction of the optimal mixing distribution can be quite involved. We leave this investigation for future work.

§.§ Main Results (Multi-Factor Models)

In this section, we present the main results for the multi-factor quasi-Bayes posterior of Section <ref>. Recall that in this case, we observe a sample $\{ \mathbf{Y}_i \}_{i=1}^n$  from the model in (<ref>). The main conditions that we impose on the observations are summarized in the following condition.
[Data]   $(i)$ $( \mathbf{Y}_i )_{i=1}^n $ is a sequence of independent and identically distributed random vectors. $(ii)$ Finite second moment: $\E\big( \| \mathbf{Y} \|^2 \big) < \infty $. $(iii)$ The factors $X_1,\dots,X_K$ are  mutually independent and demeaned, i.e $\E[X_k] = 0$ for $k=1,\dots,K$.

The multi-factor quasi-Bayes posterior for latent factor $X_k$ is given by
\begin{align}
\label{qb-mf-2}  \nu_{\alpha,G}( \varphi_{P, \sigma^2}   \: | \:  \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \big)  d \nu_{\alpha,G} ( P,\sigma^2 )}.
\end{align}

Up to sampling uncertainty, the quasi-Bayes posterior in (<ref>) is based on a transformation of the objective function
\begin{align}
\label{rep-obj-2} Q(  \varphi_{P,\sigma^2} ) =  \| \varphi_{\mathbf{Y}}^2(t) \big[  \big(\log \varphi_{P,\sigma^2} \big)''(t' \mathbf{A}_k)  - \big(\log \varphi_{X_k} \big)'' (t' \mathbf{A}_k)  \big]  \|_{ \mathbb{B}(T) }^2 .
\end{align}

As a first observation, we note that near minimizers of (<ref>) are not uniquely identified without further normalization restrictions. Intuitively, such minimizers are (approximately) solutions to the  second order differential equation
\begin{align}
\label{ode} \big(\log \varphi_{P,\sigma^2} \big)'' (t) = \big(\log \varphi_{X_k} \big)'' (t).
\end{align}

For a unique solution of (<ref>) to exist, we require two boundary conditions. As all characteristic functions equal 1 at zero, all solutions must necessarily satisfy the first boundary condition $(\log \varphi_{P,\sigma^2})(0) = 0$. For the second condition, observe that by Condition <ref>$(iii)$, the true factors are demeaned and hence their characteristic functions satisfy   $(\log \varphi_{X_k})'(0) = \mathbf{i} \E[X_k] = 0$. Unfortunately, solutions (or near solutions) to (<ref>) do not necessarily satisfy this boundary condition. One way to enforce this constraint is to force the normalization $ (\log \varphi_{P,\sigma^2})'(0) = 0  $ directly on the prior. In essence, this would entail replacing the prior with a restricted prior that is supported on mean zero Gaussian mixtures. While this is theoretically possible, in practice a restricted prior entails significant computational challenges. A second option, that we use in the remainder of this section, is to use the original quasi-Bayes posterior in (<ref>) but to demean the posterior samples. To that end, consider the demeaned posterior measure
\begin{align} \label{zdemean}   \overline{\nu}_{\alpha,G} (.|\mathcal{Z}_n) \sim  Z - \E[Z]  \; \;  \; \; \; \; \text{where} \; \; \;   Z \sim \nu_{\alpha,G}(.|\mathcal{Z}_n).  \end{align}

Each realization of (<ref>) is a demeaned Gaussian mixture. If $(P,\Sigma)$ is the mixing distribution and covariance matrix associated to the original Gaussian mixture sample from (<ref>), we denote the density and characteristic function of the demeaned mixture by $ \overline{\phi}_{P,\Sigma}  $ and $ \overline{\varphi}_{P,\Sigma} $, respectively.

First, we consider the case where the latent density $f_{X_k}$ admits a representation as a (possibly infinite) Gaussian mixture:
\begin{align}
\label{d=gaussm3} f_{X_k}(x) = \phi_{ \sigma_{0}^2} \star F_{0}(x) = \int_{\R} \phi_{ \sigma_{0}^2} (x-z) d F_{0}(z). \; \; \; \; \; \; \;  \; \; \; \;
\end{align}

Our main assumption on the Gaussian mixture specification is as follows.
[Exact Gaussian Mixtures in Multi-Factor Models]  $(i)$ $f_{X_k} = \phi_{ \sigma_{0}^2 } \star F_{0}$  for some constant $\sigma_0^2  > 0$. $(ii)$ The mixing distributions $F_{0,k}$ satisfy $ F_{0} \big(  t \in \R : \left|t \right| > z  \big) \leq  C \exp(- C' z ^{\chi}) $, $\left| \varphi_{F_{0}} (t)  \right| \geq c \exp(- c' t^2) $ and $ \left|   \partial_{t}^2 \log \varphi_{F_0} (t) \right| \leq C  $ for all sufficiently large $z$ and $\left| t \right|$, where $\chi > 0$ and $c,c',C,C' > 0$ are universal constants.

By integrating the differential inequality, the Condition $ \left|   \partial_{t}^2 \log \varphi_{F_0} (t) \right| \leq C   $ essentially states that $\varphi_{F_0}(.)$ has at most Gaussian decay. As a first result, we show that quasi-Bayes posterior in (<ref>) contracts rapidly around the logarithmic gradient of the true characteristic function.

Suppose the characteristic function of $\mathbf{Y}$ has at most Gaussian decay, that is $  \inf_{\| t \|_{\infty} \leq T }  \left| \varphi_{\mathbf{Y}} (t)  \right| \geq \exp(-R T^2) $ for some $R > 0$. Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Furthermore, suppose $\sigma_0^2$ is in the support of $G$. Let $\kappa,\chi > 0$ be as in Assumption <ref> and Condition <ref> respectively and define $ \lambda = \max \{ \chi^{-1}(d+2) + d/2 , d+1 \}  $. If $T_n = (c_0 \log n)^{1/2}$ for any $c_0$ satisfying $2 c_0 R = \gamma < 1/2$ and $\sigma_n^2 \asymp (\log n)^{-\lambda/\kappa}$, then there exists a universal constant $C > 0$ and $V \in (0,1/2)  $ such that
\begin{align}
\overline{\nu}_{\alpha,G_n} \bigg(  \| f_{X_k}  -  \phi_{P,\sigma^2}      \|_{L^2} >  C n^{-V}  \:     \big|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1) .
\end{align}

Similar to the preceding sections, with Gaussian decay on $\varphi_{\mathbf{Y}}$, the model bias (which is determined by the decay rate of a Gaussian mixture characteristic function) has similar order to the variance. In this case, as in the discussion following Theorem <ref>, the precise value of the exponent $V$ depends on second order factors such as the constant $R$  and $\sigma_0^2$ in representation (<ref>).

Next, analogous to our discussion in Section <ref> and <ref>, we consider the case where the true latent density $f_{X_k}$ is not an exact Gaussian mixture and thus, the prior model exhibits non-negligible bias. In this model, the Gaussian mixture bias is closely related to the quantity
\begin{align} \label{gm-bias3}
\mathcal{B}(\sigma,T) = \min_{P}   \| \varphi_{\mathbf{Y}}^2 (t) \big[  \big(\log \varphi_{P,\sigma^2} \big)''(t' \mathbf{A}_k)  - \big(\log \varphi_{X_k} \big)''(t' \mathbf{A}_k)  \big]  \|_{ \mathbb{B}(T) }.
\end{align}

where  the minimum is over all mixing distributions $P$.[For a conservative upper bound on the rate at which $\mathcal{B}(\sigma,T) $ tends to $0$ as $\sigma \rightarrow 0$, observe that by picking $P = f_X$, we obtain $
\mathcal{B}(\sigma,T) \leq   \sigma^2 \| \varphi_{\mathbf{Y}}^2 (t)  \|_{ \mathbb{B}(T) } \lessapprox \sigma^2.$ The final bound is valid  if  $\| \varphi_{\mathbf{Y}}^2 (t)  \|_{ L^2 } < \infty $ which ,by the Hausdorff–Young inequality, is true whenever $ \int_{\R^L} \left|f_{\mathbf{Y}} (y) \right|^{4/3} dy < \infty $. Analogous to our discussions in the preceding sections, one can typically do better by allowing $ P = P_{X,\epsilon,\sigma}$ to depend implicitly on all the features $(f_{\mathbf{Y}},f_{X_k},\sigma)$. ]

By similar reasoning to the preceding sections, we consider the following characterization of the Gaussian mixture bias.

[Gaussian Mixture Bias]
$(i)$ $f_{X_k} \in \mathbf{H}^p$ for some $p > 0$ and $ \sup_{t \in \R} \left| (\log \varphi_{X_k})'' (t)   \right| \leq D $ for some $D < \infty$. $(ii)$ There exists universal constants $  C, M    < \infty$ and $\chi > 0$  such that for all $\sigma > 0 $ sufficiently small and $T > 0$ sufficiently large, there exists a mixing distribution $F_{\sigma} = F_{X_k,\mathbf{Y},\sigma}$ supported on  the cube $ I_{\sigma} = [-C  (\log \sigma^{-1})^{1/\chi},C  (\log \sigma^{-1})^{1/\chi} ]$ that satisfies $(a)$ $ \| \varphi_{\mathbf{Y}}^2 (t) \big[  \big(\log \varphi_{F_{\sigma},\sigma^2} \big)''(t' \mathbf{A}_k)  - \big(\log \varphi_{X_k} \big)''(t' \mathbf{A}_k)  \big]  \|_{ \mathbb{B}(T) }  \leq  M \sigma^{2 \zeta +s}  $ for some  $s >  \max \{(5-d) / 2 , 0 \}$, $(b)$ $\inf_{\|t \| \leq T} \left| \varphi_{F_{\sigma}} (t)  \right| \geq T^{- \gamma_1} \sigma^{- \gamma_2}$  for some $\gamma_1,\gamma_2 < \infty $,  $(c)$ $ \left| (\log \varphi_{F_{\sigma}})''  (t)   \right| \leq M $ and $\sup_{r=1,2} \left| \partial_{t}^r  \varphi_{F_{\sigma}} (t)  \right| \leq M $.

The following result derives the quasi-Bayes posterior contraction rate when the Gaussian mixture bias is as in Condition <ref>.

Suppose Conditions <ref>, <ref> hold and the covariance prior is taken as $G_n \sim G / \sigma_n^2$ where $G$ satisfies Assumption <ref>. Let $\kappa, \chi  > 0$ be as in Assumption <ref> and Condition <ref>, respectively. Define $\lambda = \max \{ \chi^{-1}(d+2), \chi^{-1} d + 1   \}$. Suppose there exists $\zeta  > 0$ such that the characteristic function of $\mathbf{Y}$ satisfies $\inf_{\|t \|_{\infty} \leq T } \left| \varphi_{\mathbf{Y}} (t)   \right| \geq R T^{- \zeta} $ for some $R > 0$. If $T_n \asymp n^{1/[2(s + 2\zeta) +d   ]} \sqrt{\log n} $ and $\sigma_n^2 \asymp T_n^{2-d/\kappa}/(\log n)^{\lambda/\kappa + 1} \log \log n $, then there exists a universal constant $C > 0$ such that
\begin{align*}
\nu_{\alpha,G_n} \bigg(  \| f_{X_k} -  \phi_{P,\sigma^2}      \|_{L^2} >  C n^{- \min \{ p,s + (d-5)/2 \} /[2(s+ 2\zeta) + d ]} (\log n)^{\lambda/2 + \zeta + 5/4}  \:     \bigg|\mathcal{Z}_n  \bigg) = o_{\mathbb{P}}(1).
\end{align*}

Let $\mathbf{P} = (P_1,\dots, P_K)$ denote a vector of mixing distributions and $ \bm{\sigma}^2 = (\sigma_1^2 , \dots , \sigma_K^2) $ a vector of variance parameters. Let $\mathcal{V}_{\mathbf{P},\bm{\sigma}^2}(t) $ denote the vector with elements $  \{ (\log  \varphi_{P_i,\sigma_i^2} )'' (t' \mathbf{A}_i) \}_{i=1}^K  $. With some abuse of notation, we denote the vector of individual characteristic functions by  $  \varphi_{\mathbf{P}, \bm{\sigma}^2} =  (\varphi_{P_1,\sigma_1^2} , \dots , \varphi_{P_K , \sigma_K^2})$. The joint multi-factor quasi-Bayes posterior is given by
\begin{align}
\label{qb-mf-theory}  &  \nu_{\alpha,G}(  \varphi_{\mathbf{P}, \bm{\sigma}^2}   \: | \: \mathcal{Z}_n) =  \frac{  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big( \widehat{\mathcal{V}}_{\mathbf{Y}} -  \mathbf{Q} \mathcal{V}_{\mathbf{P},\bm{\sigma}^2}    \big)      \|_{ \mathbb{B}(T) }^2        \big)  }{\int  \exp \big( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \widehat{\mathcal{V}}_{\mathbf{Y}} - \mathbf{Q} \mathcal{V}_{\mathbf{P},\bm{\sigma}^2}    \big)      \|_{ \mathbb{B}(T) }^2        \big)   d \nu_{\alpha,G} ( \mathbf{P}, \bm{\sigma}^2 )} \\ & \overline{\nu}_{\alpha,G}(. \: | \: \mathcal{Z}_n) \sim   Z - \E[Z]  \; \; , \; \;   Z \sim \nu_{\alpha,G}(.|\mathcal{Z}_n) .  \nonumber
\end{align}

Here $G$ is the common univariate prior on each variance parameter $ \{ \sigma_i^2 \}_{i=1}^K$. The prior $\nu_{\alpha,G}$ is the one obtained by placing mutually independent $(\text{DP}_{\alpha} \otimes G)$ priors on each of $(P_i, \sigma_i^2)_{i=1}^K$. An analogous argument to the preceding cases provides contraction rates for the joint quasi-Bayes posterior. The main idea being that, as the priors are independent among the factors and the factors are mutually independent, analysis of the quasi-Bayes posterior in (<ref>) reduces to studying univariate quasi-posteriors. In principle, one could obtain rates where certain factors are modelled as  exact (infinite) Gaussian mixtures and others with  non-neglegible Gaussian mixture bias.

§ CONCLUSION

This paper developed a quasi-Bayes framework for three classes of latent variable models. In each case, we used the models  identifying restrictions to construct a characteristic function based quasi-likelihood. We then combined this with a prior to induce a quasi-Bayes posterior. As the prior is supported on infinite Gaussian mixtures, our modeling framework connects with a large empirical literature (e.g. ; ; ) that utilizes finite Gaussian mixtures to model rich forms of heterogeneity. Simulation and empirical exercises demonstrate that our quasi-Bayes procedures are viable and perform favorably relative to existing alternatives. We end this section by highlighting some future directions for research.

In this paper, we provided first steps towards a characteristic function based quasi-Bayes framework for latent variable models. The results could be extended to several strands of literature that make use of identification arguments in characteristic space. Possible extensions include nonlinear regression with measurement error (; ), instrumental variable based approaches to measurement error (e.g. ) and nonlinear earnings dynamics (). These extensions are currently in progress.

The theory and implementation of our quasi-Bayes procedures could be expanded upon in a variety of ways. Empirical Bayes selection of priors and tuning parameters that our procedures depend on is an important topic to be investigated. Another possible avenue is to consider a hierarchical Bayes setup where the prior hyperparameters are themselves modelled using a prior. A more challenging task would be to establish the frequentist validity of quasi-Bayes credible intervals. This would, at the very least, require the quasi-Bayes objective function to be optimally-weighted. As the quasi-Bayes objective functions is an $L^2$ norm over characteristic function based moment restrictions, optimal weighting in this context closely resembles the definition proposed in <cit.>. We hope to address these issues in future work.

§ APPENDIX : PROOFS

The following notation is frequently referred to in the proofs and so is listed here for convenience. The imaginary unit is denoted by $\mathbf{i}$. Let $\lambda$ denote the Lebesgue measure on $\R^d$. We denote the diameter of a set $A \subset \R^d$ by $\mathcal{D}(A) = \sup_{x,y \in A} \|   x-y\| $. Given a positive definite matrix $\Sigma \in \mathbf{S}_+^d$, we denote the ordered eigenvalues by $\lambda_1(\Sigma) \leq \dots \leq \lambda_d(\Sigma)$. To indicate the dependence of the quasi-posterior on the data $\mathcal{Z}_n$ and  $T$ (the radius of the ball $\mathbb{B}(T)$), we will often use the notation $\nu(.|\mathcal{Z}_n,T)$.

Given a random vector $Z \in \R^d$ with $\E\big( \| Z \|^2  \big) < \infty$ and $t \in \R^d$, define
\begin{align*}
\chi_t(Z)  = e^{\mathbf{i} t' Z} = \cos(t'Z) + \mathbf{i} \sin(t'Z).
\end{align*}

Then, there exists a universal constant $D > 0$ such that for every $T > 0$ we have

\begin{align*}
\E  \bigg( \sup_{ \| t \|_{\infty} \leq T} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| \bigg) \leq D \frac{ \max \big \{ \sqrt{\log T} , 1  \big \}  }{\sqrt{n}}.
\end{align*}

It suffices to show the result for the real and imaginary part separately. We verify it for the real part, the imaginary part is completely analogous. For the remainder of this proof, we continue assuming $\chi_t(Z) = \cos(t'Z)$.  Let $\mathcal{F} = \{   \chi_t(Z) : \left| t \right| \leq T  \} $. By an application of <cit.>,  there exists a universal constant $ L > 0 $ such that
\begin{align*}
\E  \bigg( \sup_{   \|t \|_{\infty} \leq T} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| \bigg) \leq \frac{L}{\sqrt{n}} \int_{0}^{8    }  \sqrt{\log N_{[]}(\mathcal{F}, \| . \|_{L^2(\mathbb{P})},  \epsilon  ) } d \epsilon.
\end{align*}

Let $\{  t_i \}_{i=1}^M  $  denote a minimal $\delta > 0$ covering of  $[-T,T]^d$. Define the functions
$$ e_{i}(Z) = \sup_{ t \in \R^d :   \| t \|_{\infty} \leq T , \|  t- t_i \|_{\infty} < \delta  } \ \big| \:   \chi_t(Z)  -   \chi_{t_i}(Z)   \:   \big| \; \; \; \; \; i=1,\dots,M . $$

It follows that $ \big \{   \chi_{t_i}(Z)   - e_i \; ,  \;  \chi_{t_i}(Z)  + e_i   \big \}_{i=1}^M  $ is a bracket covering for $\mathcal{F}$. Since the mapping $t \rightarrow \chi_t(Z)$ has Lipschitz constant bounded by $ \| Z \| $, we obtain $
\| e_i \|_{L^2(\mathbb{P})}^2    \leq  \E\big( \| Z \|^2  \big) \delta^2$. Since $M \leq (3 T \delta^{-1})^d$, it follows that there exists a universal constant $L >0$ such that
\begin{align*}
\int_{0}^{8    }  \sqrt{\log N_{[]}(\mathcal{F}, \| . \|_{L^2(\mathbb{P})},  \epsilon  ) } d \epsilon \leq L \max \big \{ \sqrt{\log T} ,1   \big  \}.
\end{align*}

Given a random variable $Z \in \R^d$ with $\E\big( \| Z \|^2  \big) < \infty$ and $t \in \R$, define
\begin{align*}
\chi_t(Z)  = e^{\mathbf{i} t'Z} = \cos(t'Z) + \mathbf{i} \sin(t'Z).
\end{align*}

Then, there exists a universal constant $L > 0$ such that for any sequence $T= T_n \uparrow \infty$ with $\log(T_n) \lessapprox n$, we have that

\begin{align*}
\mathbb{P}  \bigg( \sup_{ \left|t \right| \leq T_n} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| \leq D \frac{\sqrt{\log T_n}}{\sqrt{n}} \bigg)  \rightarrow 1.
\end{align*}

It suffices to show the result for the real and imaginary part separately. We verify it for the real part, the imaginary part is completely analogous. For the remainder of this proof, we continue assuming $\chi_t(Z) = \cos(t'Z)$.  Let $\mathcal{F} = \{   \chi_t(Z) : \left| t \right| \leq T_n  \} $. Since $\mathcal{F}$ is uniformly bounded,  <cit.> and Lemma <ref> imply that there exists universal constants $C,D > 0$ which satisfy
\begin{align*}
\mathbb{P}  \bigg( \sup_{ \left|t \right| \leq T_n} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| >   D \frac{\sqrt{\log T_n}}{\sqrt{n}}  + x  \bigg) \leq \exp \bigg(    - \frac{x^2}{C \big[ 1 + \sqrt{\log T_n}/ \sqrt{n} \:  ] n^{-1} }  \bigg)
\end{align*}

for all $x > 0$. With $x =  \sqrt{\log T_n} / \sqrt{n}  $  and observing that $\sqrt{\log T_n} \lessapprox \sqrt{n} $, we obtain
$$ \mathbb{P}  \bigg( \sup_{ \left|t \right| \leq T_n} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| \leq L \frac{\sqrt{\log T_n}}{\sqrt{n}} \bigg)  \rightarrow 1 $$

for some universal constant $ L > 0$.

Consider a measurable partition $\R^d = \bigcup_{j=0}^N  V_j $ and points $z_j \in V_j$ for $j=1,\dots,N$. Let $F^* = \sum_{j=1}^N w_j \delta_{z_j}  $ denote the discrete probability measure with weight $w_j$ at $z_j$. Then, for any probability measure $F$ on $\R^d$ with $\int_{\R^d} \| x \|^2 d F(x) < \infty  $, we have that

$$ \left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t) \right| \leq \| t \| \sup_{j=1,\dots,N} \mathcal{D}(V_j) + 2 \sum_{j=1}^N \left|  F(V_j) - w_j  \right|. $$

\begin{align*}
\| \nabla \mathcal{F}[F](t) - \nabla \mathcal{F}[F^*] (t) \| & \leq \bigg( \int_{\R^d} \| x \|^2 dF (x)   \bigg)^{1/2}  \bigg \{  \sum_{j=1}^N \left| F(V_j) - w_j  \right|   \bigg \}^{1/2}    \\ &    + \bigg( 1+ \sup_{ v \in \bigcup_{j=1}^N V_j  } \| v \|  \| t \| \bigg) \sup_{j=1,\dots,N} \mathcal{D}(V_j)     + \sup_{i=1,\dots,N} \| z_i \| \sum_{j=1}^N \left| F(V_j) - w_j  \right|.
\end{align*}

\begin{align*}
\sup_{k=1,\dots,d} \left|  \partial_{t_k}^2 \F[F](t)   - \partial_{t_k}^2 \F[F^*](t)   \right| &  \leq \bigg( \int_{\R^d} \| x \|_{\infty}^4 dF (x)   \bigg)^{1/2}  \bigg \{  \sum_{j=1}^N \left| F(V_j) - w_j  \right|   \bigg \}^{1/2}    \\ &   + \bigg( 2   \sup_{ v \in \bigcup_{j=1}^N V_j  } \| v \|   + \sup_{ v \in \bigcup_{j=1}^N V_j  }   \| v \|^2  \| t \| \bigg) \sup_{j=1,\dots,N} \mathcal{D}(V_j)   \\ &    + \sup_{i=1,\dots,N} \| z_i \|^2 \sum_{j=1}^N \left| F(V_j) - w_j  \right| .
\end{align*}

Here, $\mathcal{D}(A)$ denotes the diameter of a set $A$.

Observe that

\begin{align*}
& \mathcal{F}[F](t) - \mathcal{F}[F^*] (t ) \\ & =  \int_{\R^d} e^{ \mathbf{i} t'x} dF(x) - \int_{\R^d} e^{ \mathbf{i} t'x} dF^*(x)   \\ &  =  \int_{\R^d} e^{ \mathbf{i} t'x} dF(x) - \sum_{j=1}^N w_j e^{\mathbf{i} t' z_j}  \\ & =  \int_{V_0} e^{\mathbf{i} t'x} dF(x) + \sum_{j=1}^N \int_{V_j} \big( e^{\mathbf{i} t'x} - e^{\mathbf{i} t' z_j}    \big) dF(x) + \sum_{j=1}^N e^{\mathbf{i} t' z_j} \big[ F(V_j) - w_j     ].
\end{align*}

Since the mapping $\mu \rightarrow e^{\mathbf{i} t'  \mu } $ has Lipschitz constant at most $\| t \|$ and $ \sum_{j=1}^N F(V_j) = 1  $, we obtain
\begin{align*}
\left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t)   \right| & \leq F(V_0) + \| t \|  \sup_{j=1,\dots,N}  \mathcal{D}(V_j) \sum_{j=1}^N F(V_j)    + \sum_{j=1}^N \left|  F(V_j) - w_j  \right| \\ & \leq F(V_0) + \| t \|  \sup_{j=1,\dots,N} \mathcal{D}(V_j)   + \sum_{j=1}^N \left|  F(V_j) - w_j  \right|.
\end{align*}

Since $ V_0 = \R^d \setminus \cup_{j \geq 1} V_j  $ and $F(\R^d) = 1 = \sum_{j=1}^N w_j$, we obtain
$$  F(V_0) = \sum_{j=1}^N w_j - \sum_{j=1}^N F(V_j) \leq  \sum_{j=1}^N \left| F(V_j) - w_j  \right| . $$

For the second claim, first observe that the moment condition on $F$ ensures that the gradient exists. By differentiating the preceding expression for $\mathcal{F}[F](t) - \mathcal{F}[F^*] (t )$, we obtain
\begin{align*}
&  \nabla\mathcal{F}[F](t) - \nabla \mathcal{F}[F^*] (t ) \\ & = \mathbf{i} \int_{V_0} x e^{\mathbf{i} t'x} dF(x) + \mathbf{i}  \sum_{j=1}^N \int_{V_j} \big(  x e^{\mathbf{i} t'x} -  z_j e^{\mathbf{i} t' z_j}    \big) dF(x) + \mathbf{i}\sum_{j=1}^N z_j e^{\mathbf{i} t' z_j} \big[ F(V_j) - w_j     ] .
\end{align*}

For $\mu \in V_j$, the mapping $\mu \rightarrow  \mu e^{\mathbf{i} t'  \mu } $ has Lipschitz constant at most $ 1+  \sup_{v \in V_j }\| v \|  \| t \| $ and by Cauchy-Schwarz, $ \int_{V_0} \| x \| d F(x) \leq \big( \int_{\R^d} \| x \|^2 dF (x)   \big)^{1/2} \big\{  F(V_0)   \big \}^{1/2}$. By using the same bound for $F(V_0)$ as above, we obtain
\begin{align*}
& \|  \nabla\mathcal{F}[F](t) - \nabla \mathcal{F}[F^*] (t ) \| \\ & \leq \bigg( \int_{\R^d} \| x \|^2 dF (x)   \bigg)^{1/2}  \bigg \{  \sum_{j=1}^N \left| F(V_j) - w_j  \right|   \bigg \}^{1/2} +  \bigg( 1+  \sup_{ v \in \cup_{j=1}^N V_j  } \| v \|    \| t \|  \bigg) \sup_{j=1,\dots,N} \mathcal{D}(V_j) \\ & \;  + \sup_{i=1,\dots,N} \| z_i \| \sum_{j=1}^N \left| F(V_j) - w_j  \right|.
\end{align*}

For the third claim, let $z_{j,k}$ denote the $k^{th}$ element of $z_j$. Twice differentiating (with respect to $t_k$) the expression for $\mathcal{F}[F](t) - \mathcal{F}[F^*] (t )$  yields
\begin{align*}
&   \partial_{t_k}^2 \mathcal{F}[F](t) - \partial_{t_k}^2 \mathcal{F}[F^*] (t ) \\ & = - \int_{V_0} x_k^2 e^{\mathbf{i} t'x} dF(x) -  \sum_{j=1}^N \int_{V_j} \big(  x_k^2 e^{\mathbf{i} t'x} -  z_{j,k}^2 e^{\mathbf{i} t' z_j}    \big) dF(x) - \sum_{j=1}^N z_{j,k}^2 e^{\mathbf{i} t' z_j} \big[ F(V_j) - w_j     ] .
\end{align*}

For $\mu = (\mu_1,\dots,\mu_d) \in V_j$ and a fixed component $\mu_k$  the mapping $\mu \rightarrow \mu_k^2 e^{\mathbf{i} t' \mu} $ has Lipschitz constant at most $2 \sup_{v \in V_j} \| v \| + \sup_{v \in V_j} \| v \|^2 \| t \|  $ and by Cauchy-Schwarz, we have the bound $ \int_{V_0}  x_k^2 d F(x) \leq \big( \int_{\R^d}  \| x \|_{\infty}^4  dF (x)   \big)^{1/2} \big\{  F(V_0)   \big \}^{1/2}$. The claim follows from an analogous bound to the preceding case.

Consider a measurable partition $\R^d = \bigcup_{j=0}^N  V_j $ and points $z_j \in V_j$ for $j=1,\dots,N$. Let $F^* = \sum_{j=1}^N w_j \delta_{z_j}  $ denote the discrete probability measure with weight $w_j$ at $z_j$. Then, there exists a universal constant $D > 0 $ such that for any positive definite matrix $\Sigma \in \R^{d \times d}$ with minimum eigenvalue $\underline{\sigma}^2 > 0$ and probability measure $F$ on $\R^d$, we have that
\begin{align*}
& \| \varphi_{F,\Sigma}  - \varphi_{F^*,\Sigma}  \|_{L^2} \leq D \bigg[ \underline{\sigma}^{-(d+2)/2} \sup_{j=1,\dots,N} \mathcal{D}(V_j) + \underline{\sigma}^{-d/2}  \sum_{j=1}^N \left|  F(V_j) - w_j  \right|    \bigg] \; ,  \\ & \| \varphi_{F,\Sigma}  - \varphi_{F^*,\Sigma}  \|_{\mathbb{B}(T)} \leq D  \bigg[ T^{(d+2)/2} \sup_{j=1,\dots,N} \mathcal{D}(V_j) + T^{d/2}  \sum_{j=1}^N \left|  F(V_j) - w_j  \right|   \bigg].
\end{align*}

By Lemma <ref>, we have
\begin{align*}
\left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t) \right| \leq \| t \| \sup_{j=1,\dots,N} \mathcal{D}(V_j) + 2 \sum_{j=1}^N \left|  F(V_j) - w_j  \right| .
\end{align*}

Since $  e^{- t' \Sigma t} \leq e^{- \| t \|^2 \underline{\sigma}^2} $, it follows that
\begin{align*}
& \|  \varphi_{F,\Sigma} - \varphi_{F^*,\sigma}  \|_{L^2}^2 \\ & = \int_{\R^d} \left|  \varphi_{\Sigma} (t)  \right|^2  \left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t) \right| ^2 dt \\ & = \int_{\R^d} e^{- t' \Sigma t}  \left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t) \right| ^2 dt  \\ & \leq \int_{\R^d} e^{- \|t \|^2 \underline{\sigma}^2}  \left| \mathcal{F}[F](t) - \mathcal{F}[F^*] (t) \right| ^2 dt \\ & \leq  2 \sup_{j=1,\dots,N} \{ \mathcal{D}(V_j) \}^2 \int_{\R^d} e^{- \|t \|^2 \underline{\sigma}^2} \| t \|^2 dt +  8 \bigg \{ \sum_{j=1}^N \left|  F(V_j) - w_j  \right| \bigg \}^2 \int_{\R^d} e^{- \|t \|^2 \underline{\sigma}^2} dt.
\end{align*}

The first claim follows from observing that the two integrals scale with rate at most $\underline{\sigma}^{-(d+2)}$ and $\underline{\sigma}^{-d}$, respectively. The second claim follows by an analogous argument from truncating the integral to the set $\{ \|t\|_{\infty} \leq T  \}$ and using the trivial bound $e^{-t' \Sigma t} \leq 1$.

Suppose $F$ is a probability measure supported on $[-R,R]^d$ for some $ R > 0$. Then, given any $k \geq 1$, there exists a discrete probability measure $ F' $ with at most $(k+1)^d+1$ support points in $[-R,R]^d$ such that for every $t \in \R^d$, we have
\begin{align*}
& (i) \; \; \; \;  \left|  \F[F](t) - \F[F'] (t)  \right| \leq  2 \frac{\| t \|^{k+1}  (e \sqrt{d} R)^{k+1}  }{(k+1)^{k+1}} \; , \\ & (ii) \; \; \; \;  \|  \nabla \F[F](t) - \nabla \F[F'] (t)  \| \leq 2 \sqrt{d}R \frac{\| t \|^{k}  (e \sqrt{d} R)^{k}  }{k^{k}} \; , \\ & (iii) \; \;  \sup_{l=1,\dots,d}  \left| \partial_{t_l}^2 \F[F](t) - \partial_{t_l}^2  \F[F'] (t)  \right| \leq 2 dR^2\frac{\| t \|^{k-1}  (e \sqrt{d} R)^{k-1}  }{(k-1)^{k-1}} .
\end{align*}

In particular, there exists universal constants $C,D > 0$ such that for all $T,R $ sufficiently large and $\epsilon \in (0,1)$, the choice $ k = \lceil C \max \{ \log(\epsilon^{-1}) , R T   \} \rceil     $  satisfies
\begin{align*}
& (i) \; \; \; \;  \sup_{\| t \|_{\infty} \leq T}  \left|  \F[F](t) - \F[F'] (t)  \right| \leq  D \epsilon \; , \\ & (ii) \; \; \; \; \sup_{\| t \|_{\infty} \leq T}   \|  \nabla \F[F](t) - \nabla \F[F'] (t)  \| \leq D \epsilon \; , \\ & (iii)  \; \; \; \sup_{\| t \|_{\infty} \leq T} \sup_{l=1,\dots,d}  \left| \partial_{t_l}^2 \F[F](t) - \partial_{t_l}^2  \F[F'] (t)  \right| \leq D \epsilon.
\end{align*}

Furthermore, the support points of $F'$ can be chosen on the grid $ \mathcal{Z} = \{ T^{-1} \epsilon   (z_1,\dots,z_d) : z_i \in \mathbb{Z} \; , \; \left| z_i \right| \leq \lceil  R / ( T^{-1} \epsilon  )   \rceil     \} $, with a multiplictive penalty of at most $R$ and $R^2$ in cases $(ii)$ and $(iii)$, respectively.

Given any $k \geq 1$, by <cit.>, there exists a discrete measure $F'$ with at most $(k+1)^d + 1$ support points on $[-R,R]^d$ such that
\begin{align*}
\int_{\R^d}  z_1^{l_1} \dots z_d^{l_d} dF(z) =  \int_{\R^d}  z_1^{l_1} \dots z_d^{l_d} dF'(z) \; \; \; \; \; \; \; \; \forall \; \; \;  0  \leq l_1, \dots , l_d \leq k.
\end{align*}

For any $t,z \in \R^d$ and $j \in \mathbb{Z}$, a multinomial expansion yields
$$ (t'z)^j = \sum_{k_1 + \dots +  k_d = j, k_1 \geq 0 , \dots , k_d \geq 0} \frac{j!}{k_1! \dots k_d!} \prod_{i=1}^d t_i^{k_i} z_i^{k_i}.  $$

It follows that $F$ and $F'$ assign the same expectation to $(t'z)^j$ for every $t \in \R^d$, provided that $j \leq k$. This yields

\begin{align*}
\F[F](t) -   \F[F'] (t)   =      \int_{\R^d}  e^{\mathbf{i} t' z} d(F-F')(z)     & = \int_{\R^d} \sum_{j=0}^{\infty} \frac{(\mathbf{i} t'z)^j}{j!}    d(F-F')(z)    \\ & = \int_{\R^d}   \sum_{j=k+1}^{\infty}   \frac{(\mathbf{i} t'z)^j}{j!}     d(F-F')(z)  .
\end{align*}

Observe that for every $x \in \R$, we have the bound $ \left| \sum_{j=k+1}^{\infty} (\mathbf{i}x)^j / j!  \right| =    \left|  e^{\mathbf{i}x} - \sum_{j=0}^{k} (\mathbf{i}x)^j / j!   \right| \leq \left|x \right|^{k+1} / (k+1)! \leq \left|x \right|^{k+1} e^{k+1} / (k+1)^{k+1} $. For any $z \in [-R,R]^d$, we have that $ \left| t'z \right| \leq  \| t \| \| z \| \leq \| t \| \sqrt{d} R$. Since $F,F'$ are probability measures and have support contained in $[-R,R]^d$, it follows that
\begin{align*}
\left|  \F[F](t) - \F[F'] (t)  \right|   &  \leq \int_{\R^d}   \left| \sum_{j=k+1}^{\infty}    \frac{(\mathbf{i} t'z)^j}{j!}  \right|    dF + \int_{\R^d} \left| \sum_{j=k+1}^{\infty}    \frac{(\mathbf{i} t'z)^j}{j!}  \right|    dF' \\ & \leq   2 \frac{\| t \|^{k+1}  (e \sqrt{d} R)^{k+1}  }{(k+1)^{k+1}}.
\end{align*}

For the second claim, the same reasoning as above implies that $F$ and $F'$ assign the same expectation to the vector $z (t'z)^j$ for every $t \in \R^d$, provided that $j \leq k-1$. This yields

\begin{align*}
\nabla \F[F](t) - \nabla \F[F'] (t)   =       \mathbf{i} \int_{\R^d}  z e^{\mathbf{i} t' z} d(F-F')(z)     & = \mathbf{i} \int_{\R^d} z \sum_{j=0}^{\infty}  \frac{(\mathbf{i} t'z)^j}{j!}    d(F-F')(z)    \\ & =\mathbf{i} \int_{\R^d}  z  \sum_{j=k}^{\infty}   \frac{(\mathbf{i} t'z)^j}{j!}     d(F-F')(z)  .
\end{align*}

We have $ \| z \| \leq \sqrt{d} R  $ for every $z$ in the support of $F,F'$. From using the bound in the preceding case with $k$ replacing $k+1$, it follows that
\begin{align*}
\| \nabla \F[F](t) - \nabla \F[F'] (t) \|  & \leq  \sqrt{d}R  \bigg(  \int_{\R^d}   \left| \sum_{j=k}^{\infty}    \frac{(\mathbf{i} t'z)^j}{j!}  \right|    dF + \int_{\R^d} \left| \sum_{j=k}^{\infty}    \frac{(\mathbf{i} t'z)^j}{j!}  \right|    dF'   \bigg) \\ & \leq  2 \sqrt{d}R \frac{\| t \|^{k}  (e \sqrt{d} R)^{k}  }{k^{k}}.
\end{align*}

For the third claim, let $z_l$ denote the $l^{th}$ coordinate of $z = (z_1,\dots,z_d)$. The same reasoning as above implies that $F$ and $F'$ assign the same expectation to the vector $z_l^2 (t'z)^j$ for every $t \in \R^d$, provided that $j \leq k-2$. This yields
\begin{align*}
\partial_{t_l}^2  \F[F](t) - \partial_{t_l}^2  \F[F'] (t)   =       - \int_{\R^d}  z_l^2 e^{\mathbf{i} t' z} d(F-F')(z)     & = - \int_{\R^d} z_l^2 \sum_{j=0}^{\infty}  \frac{(\mathbf{i} t'z)^j}{j!}    d(F-F')(z)    \\ & = - \int_{\R^d}  z_l^2  \sum_{j=k-1}^{\infty}   \frac{(\mathbf{i} t'z)^j}{j!}     d(F-F')(z)  .
\end{align*}

Since $ z_l^2 \leq \| z  \|^2 \leq d R^2  $, the claim follows from an analogous bound to the preceding case.

For the final claim regarding the location of the support points, suppose $F' = \sum_{i=1}^N p_i \delta_{\mu_i}$ is a discrete probability measure that satisfies the requirements of the first part of the Lemma. Let $F^* = \sum_{i=1}^N p_i \delta_{\mu_i^*} $ denote the probability measure obtained by replacing each $\mu_i$ with $\mu_i^* \in \arg \min \limits_{t \in \mathcal{Z}} \| \mu_i - t   \| $. From the definition of $\mathcal{Z}$, it follows that $  \| \mu_i  - \mu_i^* \| \leq D T^{-1}   \epsilon $. We claim $F'$ satisfies all the same bounds. For the first bound, observe that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T} \left| \mathcal{F}[F'] - \mathcal{F}[F^*]   \right|   & =  \sup_{\| t \|_{\infty} \leq T} \left|  \sum_{j=1}^N p_j \big[  e^{\mathbf{i}t \mu_j} - e^{\mathbf{i}t \mu_j^*}   ]      \right| \leq  \sup_{\| t \|_{\infty} \leq T}  \sup_{j=1,\dots,N} \left|    e^{\mathbf{i}t \mu_j} - e^{\mathbf{i}t \mu_j^*}   \right|.
\end{align*}

Since the mapping $\mu \rightarrow e^{\mathbf{i}t \mu}$ has Lipschitz constant bounded by $ \| t \|$, it follows that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T} \left| \mathcal{F}[F'] - \mathcal{F}[F^*]   \right|  \leq \sqrt{d} T \sup_{j=1,\dots,N}  \| \mu_j - \mu_j^*  \| \leq D \epsilon.
\end{align*}

For the second bound, observe that for $\mu \in [-R ,R]^d$, the mapping $\mu \rightarrow \mu e^{\mathbf{i}t \mu}$ has Lipschitz constant at most $1 + \sqrt{d} R \| t \| $. Hence

\begin{align*}
\sup_{\| t \|_{\infty} \leq T}  \| \nabla \F[F'](t) - \nabla \F[F^*] (t) \| & \leq \sup_{\| t \|_{\infty} \leq T}  \sup_{j=1,\dots,N} \| \mu_j e^{\mathbf{i}t' \mu_j} - \mu_j^* e^{\mathbf{i}t' \mu_j^*}   \| \\ &  \leq D (RT+1) \sup_{j=1,\dots,N}  \| \mu_j - \mu_j^*  \| \\ & \leq D R \epsilon.
\end{align*}

For the third bound, let $\mu_{j,l}$ denote the $l^{th}$ component of $\mu_j$. On $[-R,R]^d$, the mapping  $\mu_j \rightarrow \mu_{j,l}^2 e^{\mathbf{i}t' \mu_j}$ has Lipschitz constant at most $ 2 \sqrt{d} R + d R^2 \| t \|$. Hence
\begin{align*}
\sup_{\| t \|_{\infty} \leq T} \sup_{l=1,\dots,d}  \left| \partial_{t_l}^2 \F[F](t) - \partial_{t_l}^2  \F[F'] (t)  \right| & \leq  \sup_{\| t \|_{\infty} \leq T} \sup_{l=1,\dots,d} \sup_{j=1,\dots,N} \left|  \mu_{j,l}^2 e^{\mathbf{i}t' \mu_j} - (\mu_{j,l}^*)^2 e^{\mathbf{i}t' \mu_j^*}    \right| \\ & \leq D (R + R^2 T) \sup_{j=1,\dots,N}  \| \mu_j - \mu_j^*  \| \\ & \leq D R^2 \epsilon.
\end{align*}