# Asymptotic Normality of the Posterior Distributions in a Class of Hidden
Markov Models

Chunlei Wang<EMAIL_ADDRESS>Department of Statistics and Actuarial
Science, The University of Iowa  Sanvesh Srivastava sanvesh-
<EMAIL_ADDRESS>Department of Statistics and Actuarial Science, The
University of Iowa

###### Abstract

We show that the posterior distribution of parameters in a hidden Markov model
with parametric emission distributions and discrete and known state space is
asymptotically normal. The main novelty of our proof is that it is based on a
testing condition and the sequence of test functions is obtained using an
optimal transportation inequality.

Keywords: Bayesian inference; Bernstein-von Mises theorem; hidden Markov
model; transportation inequality.

##  1 Introduction

Hidden Markov models with parametric emission distributions and discrete and
known state space, which we shorten as HMMs, have been widely used in diverse
areas for analyzing dependent discrete-time data [11, 10, 26]. Our focus is on
studying asymptotic properties of the posterior distribution of parameters in
an HMM [4, 5]. We prove a Bernstein-von Mises theorem for HMMs using a testing
condition. Informally, this theorem says that the posterior distribution of
parameters in an HMM converges in distribution to a normal distribution that
centers at the maximum likelihood estimator of parameters, and its covariance
matrix equals inverse of the Fisher information matrix.

Consider the setup of a general hidden Markov model as a discrete-time
stochastic process $(X_{t},Y_{t})_{t\geq 1}$, where $(Y_{t})_{t\geq 1}$
process is observed and $(X_{t})_{t\geq 1}$ process is unobserved and Markov.
Its state space includes all possible values of $(X_{t})_{t\geq 1}$ and
emission distribution is defined as the conditional distribution of $Y_{t}$
given $X_{t}$. Our HMMs are a special case of this setup in that the emission
distributions are parametric and state space is discrete and known. There are
many theoretical results about frequentist estimation in hidden Markov models
with discrete, continuous, or unknown state spaces and parametric or non-
parametric emission distributions [1, 22, 2, 18, 7, 8]. The equivalent
asymptotic Bayesian results have started to emerge recently [5, 12, 25, 13,
9].

Our theoretical results are closest to those of De Gunst and Shcherbakova [5].
They prove a Bernstein-von Mises theorem for HMMs using Taylor expansion of
the log likelihood function. Our proof differs from theirs in that we adopt
the setup of Ghosal et al. [14] for deriving posterior contraction rates in
general Bayesian models. Using this setup, we adapt the proof of Theorem 10.1
in [23] to show that the posterior distribution of parameters in an HMM is
asymptotically normal. Compared to [5], our assumption on the log likelihood
function is milder in that our proof assumes only the first order smoothness
of the log likelihood function. More importantly, our proof relies on the
local asymptotic normality condition for HMMs, which is satisfied under the
assumptions of Bickel et al. [2], De Gunst and Shcherbakova [5]. Our proof
techniques are similar to those used for investigating asymptotic properties
of the posterior distribution in extensions of HMMs with non-parametric
emission distributions [12, 25]. A key step in our proof is to establish a
testing condition based on a sequence of test functions. We are not aware of
any existing test function that is tuned for such applications in HMMs.

Our main contribution is to establish the testing condition using a sequence
of test functions obtained from an optimal transportation cost inequality. The
testing condition measures the complexity of the HMM family and establishing
it is a key step in showing the consistency of posterior distribution [14].
Our construction of the sequence of test functions is based on the
$L^{1}$-transportation cost information inequality for HMMs [6, 19] and
Lipschitz concentration of log likelihood function of HMMs [20]. Hu [17] has
constructed the likelihood ratio test with exponentially decaying error
probabilities for testing simple hypotheses. Extending this work, we construct
the sequence of test functions with exponentially decaying error probabilities
for testing general hypotheses using the $L^{1}$-transportation cost
information inequality. The sequence of test functions satisfies the testing
condition in [14, 23] and is used to prove the Bernstein-von Mises theorem for
HMMs.

This paper is structured in four sections. In Section 2, we state the HMM
setup, define the prediction filter, and restructure the results of
Kontorovich et al. [19] and Hu [17] in the context of this paper. The
assumptions and main results about the testing condition are presented in
Section 3. The contraction rate and asymptotic normality of the posterior
distribution of parameters in an HMM follow immediately from the testing
condition. Finally, Section 4 contains proofs of the main results.

##  2 Setup and Background

Consider the HMM setup in greater detail. Recall that the HMM is a discrete-
time stochastic process $(X_{t},Y_{t})_{t\geq 1}$ and that $(Y_{t})_{t\geq 1}$
process is observed and $(X_{t})_{t\geq 1}$ process is unobserved and Markov.
The hidden Markov chain $(X_{t})_{t\geq 1}$ has state space
$\mathcal{X}=\\{1,\ldots,S\\}$ with a known $S\in\mathbb{N}$. The
$(X_{t})_{t\geq 1}$ process has an initial stationary distribution
$r(a)=\mathbb{P}(X_{1}=a)$ for $a\in\mathcal{X}$ and is time homogeneous with
transition kernel

$\displaystyle Q_{ab}=q(a,b)=\mathbb{P}(X_{t+1}=b|X_{t}=a),\quad t\geq
1,\;a,b\in\mathcal{X}.$ (2.1)

Given $(X_{t})_{t\geq 1}$, $(Y_{t})_{t\geq 1}$ is a sequence of independent
random variables on the metric space $(\mathcal{Y},d_{\mathcal{Y}})$ and the
conditional distribution of $Y_{t}$ depends only on $X_{t}$. Furthermore, the
conditional distribution of $Y_{t}$ given $X_{t}=x_{t}$ does not depend on $t$
and has a density function $g(\cdot\mid x_{t})$ with respect to the
$\sigma$-finite measure $\mu$ on $\mathcal{Y}$. Let
$\Theta\subseteq\mathbb{R}^{d}$ denote the paramter space with fixed dimension
$d\in\mathbb{N}$. The density functions $r(\cdot)$, $q(\cdot,\cdot)$, and
$g(\cdot\mid\cdot)$ belong to a parametric family indexed by $\theta$, which
is denoted as
$\\{r_{\theta}(\cdot),q_{\theta}(\cdot,\cdot),g_{\theta}(\cdot\mid\cdot),\theta\in\Theta\subseteq\mathbb{R}^{d}\\}$.
Our focus is Bayesian inference on $\theta$ given the observed data.

Consider the joint and marginal distributions of the augmented and observed
data. Let $n$ be the number of observations, $(Y_{1},\ldots,Y_{n})$ be the
observed data, $(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n})$ be the augmented
data, and $c_{i}^{j}$ be the shorthand for sequence $c_{i},\ldots,c_{j}$ for
$i\leq j$. Then, the joint and marginal densities of the augmented and
observed data under $(\text{counting measure})^{n}\otimes\mu^{n}$ and
$\mu^{n}$, respectively, are

$\displaystyle p_{\theta}(x_{1}^{n},y_{1}^{n})$
$\displaystyle=r_{\theta}(x_{1})\prod_{k=1}^{n-1}q_{\theta}(x_{k},x_{k+1})g_{\theta}(y_{k}|x_{k})g_{\theta}(y_{n}|x_{n}),$
$\displaystyle p_{\theta}(y_{1}^{n})$
$\displaystyle=\sum_{(x_{1}^{n})\in\mathcal{X}^{n}}p_{\theta}(x_{1}^{n},y_{1}^{n}),$
(2.2)

where $x_{1}^{n}\in\mathcal{X}^{n}$, $y_{1}^{n}\in\mathcal{Y}^{n}$, and
$\theta\in\Theta$. Denote the distribution of $Y_{1}^{n}$ as
$\mathbb{P}_{\theta}^{(n)}$, density $p_{\theta}(y_{1}^{n})$ as
$p_{\theta}^{(n)}$, and the expectation under
$\operatorname{\mathbb{P}}^{(n)}_{\theta}$ as $\mathbb{E}_{\theta}^{(n)}$. If
$\theta_{0}\in\Theta$ is the true parameter, then
$\operatorname{\mathbb{P}}^{(n)}_{\theta_{0}}$ is the true distribution of
$Y_{1}^{n}$.

We now define related conditional distributions and the _extended_ HMM
sequence. For $2\leq t\leq n$ and $\theta\in\Theta$, the conditional
distribution of $X_{t}$ given $Y_{1}^{t-1}$ is called the _prediction filter_
at time $t$ and is defined as

$\displaystyle
p_{t}^{\theta}\vcentcolon=\left\\{\mathbb{P}_{\theta}(X_{t}=1\mid
Y_{1}^{t-1}),\ldots,\mathbb{P}_{\theta}(X_{t}=S\mid
Y_{1}^{t-1})\right\\}^{\mathsf{T}}\in\mathcal{E},$ (2.3)

where $\mathcal{E}$ is the space of all probability distributions on
$\mathcal{X}$ equipped with total variation distance $\|\cdot\|_{\text{TV}}$.
Baum’s forward equation further implies that for $t=1,\ldots,n-1$,

$\displaystyle p_{t+1}^{\theta}(\cdot)$
$\displaystyle=\frac{\operatorname{\mathbb{P}}_{\theta}(X_{t+1}=\cdot,Y_{t}\mid
Y_{1}^{t-1})}{\operatorname{\mathbb{P}}_{\theta}(Y_{t}\mid
Y_{1}^{t-1})}=\frac{\sum_{x_{t}}q_{\theta}(x_{t},\cdot)g_{\theta}(Y_{t}\mid
X_{t}=x_{t})p_{t}^{\theta}(x_{t})}{\sum_{x_{t}}g_{\theta}(Y_{t}\mid
X_{t}=x_{t})p_{t}^{\theta}(x_{t})}$
$\displaystyle\vcentcolon=f^{\theta}(Y_{t},p_{t}^{\theta}),$ (2.4)

where $f^{\theta}$ is a measurable function on
$\mathcal{Z}=\mathcal{Y}\times\mathcal{E}$ equipped with metric
$d_{\mathcal{Z}}=d_{\mathcal{Y}}+\|\cdot\|_{\text{TV}}$. Extending the
definition of $f^{\theta}(Y_{t},p_{t}^{\theta})$ for lags $s=0,1,\ldots,t-1$,
we recursively define $p_{t+1}^{\theta}$ for $s=0,\ldots,t-1$ as

$\displaystyle
p_{t+1}^{\theta}=f_{0}^{\theta}(Y_{t}^{t},p^{\theta}_{t})=f_{1}^{\theta}(Y_{t-1}^{t},p^{\theta}_{t-1})=\cdots=f_{s}^{\theta}(Y_{t-s}^{t},p^{\theta}_{t-s})=\cdots=f_{t-1}^{\theta}(Y_{1}^{t},p^{\theta}_{1}),$
(2.5)

where $f_{0}^{\theta}=f^{\theta}$ and $p_{1}^{\theta}=r_{\theta}$. The
_extended HMM_ sequence is defined as
$(Y_{t},p^{\theta}_{t})_{t=1}^{n}\in(\mathcal{Y}\times\mathcal{E})^{n}=\mathcal{Z}^{n}$,
where $(p^{\theta}_{t})_{t=1}^{n}$ is defined in (2.5). Let
$l_{n}(\theta,Y_{1}^{n})=\log p_{\theta}(Y_{1}^{n})$ be the log likelihood
function of $\theta$, where $p_{\theta}(y_{1}^{n})$ is defined in (2.2). Using
the extended HMM sequence, $l_{n}(\theta,Y_{1}^{n})$ is expressed as

$\displaystyle l_{n}(\theta,Y_{1}^{n})$ $\displaystyle=\log
p_{\theta}(Y_{1})+\log\prod_{t=2}^{n}p_{\theta}(Y_{t}|Y_{t-1},\ldots,Y_{1})$
$\displaystyle=\log\left\\{\sum_{x_{1}}g_{\theta}(Y_{1}|x_{1})r_{\theta}(x_{1})\right\\}+\sum_{t=2}^{n}\log\left\\{\sum_{x_{t}}g_{\theta}(Y_{t}|x_{t})\operatorname{\mathbb{P}}_{\theta}(X_{t}=x_{t}|Y_{t-1},\ldots,Y_{1})\right\\}$
$\displaystyle=\sum_{t=1}^{n}\log\sum_{x_{t}}p_{t}^{\theta}(x_{t})g_{\theta}(Y_{t}|x_{t}).$
(2.6)

The log likelihood function in (2.6) depends on
$(Y_{t},p^{\theta}_{t})_{t=1}^{n}$. The main advantage of (2.6) is that the
log likelihood function $l_{n}(\theta,Y_{1}^{n})$ can be expressed in an
additive form and easier to manipulate. Unfortunately, this is not possible if
we use $p_{\theta}(y_{1}^{n})$ in (2.2) to define $l_{n}(\theta,Y_{1}^{n})$.

We now state the optimal transportation inequality and related results that
are required for constructing the sequence of test functions.

######  Definition 2.1.

Let $(E,d)$ be a metric space. A probability measure $\mu$ on $(E,d)$
satisfies $L^{1}$-transportation cost information inequality if for any
probability measure $\nu$ on $(E,d)$, there exists a constant $C>0$ such that

$\displaystyle W_{1}^{d}(\mu,\nu)\leq\sqrt{2CH(\nu\mid\mu)},$ (2.7)

where $W_{1}^{d}$ is the optimal transportation cost with cost function
$d(\cdot,\cdot)$ and $H(\cdot\mid\cdot)$ denotes the Kullback-Leibler
divergence. If a measure $\mu$ satisfies (2.7), we denote this relation as
$\mu\in T_{1}(C)$.

The $L^{1}$-transportation cost information inequality is important because it
is an equivalent characterization of sub-Gaussian measure concentration.

######  Theorem 2.2.

[3, Theorem 3.1] A measure $\mu$ defined on a metric space $(E,d)$ satisfies
$T_{1}(C)$-inequality if and only if for any $\mu$-integrable Lipschitz
function $F:E\rightarrow\mathbb{R}$ and any $\lambda\in\mathbb{R}$,

$\int_{E}e^{\lambda(F-\langle F\rangle_{\mu})}d\mu\leq
e^{\tfrac{\lambda^{2}}{2}C\|F\|^{2}_{\text{\text{Lip}}}},$

where $\langle F\rangle_{\mu}=\int_{E}Fd\mu$ and $\|F\|_{\text{\text{Lip}}}$
is the Lipschitz constant of $F$. In this case, for any $r>0$,

$\displaystyle\mu(F-\langle F\rangle_{\mu}>r)\leq
e^{-\frac{r^{2}}{2C\|F\|^{2}_{\text{Lip}}}}.$ (2.8)

An important property of $L^{1}$-transportation cost information inequality is
called the _tensorization principle_. For $i=1,\dots,n$, let $\mu_{i}$ be a
probability measure on metric space $(E_{i},d_{i})$ such that $\mu_{i}\in
T_{1}(C)$. Consider the product measure $\mu_{1}\otimes\cdots\otimes\mu_{n}$
on $(E_{1}\times\cdots\times E_{n},d_{l_{1}})$, where the $l_{1}$-metric is
defined as

$\displaystyle
d_{l_{1}}(u,v)=\sum_{i=1}^{n}d_{i}(u_{i},v_{i}),\;u=(u_{1},\ldots,u_{n}),\;v=(v_{1},\ldots,v_{n}),\;u_{i},v_{i}\in
E_{i}.$ (2.9)

Then, the tensorization principle for $L^{1}$-transportation cost information
inequality states that

$\displaystyle\mu_{1}\otimes\cdots\otimes\mu_{n}\in T_{1}(nC);$ (2.10)

see [24, Chapter 2] for greater details. For a product measure, this property
is called _independent tensorization_. Under mild conditions, the probability
measure $\mathbb{P}^{n}_{\theta}$ induced by $(Y_{t})_{t=1}^{n}$ satisfies the
$T_{1}(nC_{H}^{\theta})$-inequality on the metric space
$(\mathcal{Y}^{n},d_{l_{1}})$ for some constant $C_{H}^{\theta}$. If
$\mathcal{Y}$ is countable, then the constant $C_{H}^{\theta}$ is determined
by the mixing property of the hidden Markov chain $(X_{t})_{t\geq 1}$ [19].
Furthermore, $C_{H}^{\theta}$ is also related to the measure concentration of
emission distribution if $\mathcal{Y}$ is uncountable [17]. We end this
section by extending the former two results in the following theorem.

######  Theorem 2.3.

If the $r$-step transition matrix $Q^{r}_{\theta}$ of $(X_{t})_{t\geq 1}$ is
positive entrywise for some $r>0$ and $\theta\in\Theta$, then

$\displaystyle
D_{\theta}=\frac{1}{2}\sum_{t=1}^{\infty}\sup_{a,a^{\prime}}\sum_{b\in\mathcal{X}}|Q_{\theta}^{t}(a,b)-Q_{\theta}^{t}(a^{\prime},b)|<\infty,$
(2.11)

and $\mathbb{P}_{\theta}^{(n)}(Y_{1}^{n}\in\cdot)\in T_{1}(nC_{H}^{\theta})$
with respect to $(\mathcal{Y}^{n},d_{l_{1}})$, where $C_{H}^{\theta}$ is given
as:

  1. 1.

In the case $(\mathcal{Y},d_{\mathcal{Y}})$ is countable,
$C_{H}^{\theta}=(D_{\theta}+1)^{2};$

  2. 2.

In the case $(\mathcal{Y},d_{\mathcal{Y}})$ is uncountable, if for any
$a\in\mathcal{X}$, the emission distribution
$\mathbb{P}_{\theta}(Y_{1}\in\cdot|X_{1}=a)\in T_{1}(C_{\mathcal{Y}})$ for
some constant $C_{\mathcal{Y}}$, and there exists a constant $L>0$ such that
for any $a,b\in\mathcal{X}$,

$W_{1}^{d_{\mathcal{Y}}}\\{\mathbb{P}_{\theta}(Y_{1}\in\cdot\mid
X_{1}=a),\mathbb{P}_{\theta}(Y_{1}\in\cdot\mid X_{1}=b)\\}\leq L1_{a\neq b},$

then $C_{H}^{\theta}=C_{\mathcal{Y}}+L^{2}(D_{\theta}+1)^{2}.$

##  3 Main results

Consider the setup for studying asymptotic properties of the posterior
distribution of $\theta$. Assume that the observe data $Y_{1}^{n}$ has
$\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$ as its distribution. Given a
prior distribution $\Pi_{n}$ on the parameter space $\Theta$ with respect to
Borel $\sigma$-field $\mathcal{F}$, the posterior distribution of $\theta$
conditional on $Y_{1}^{n}$ is

$\displaystyle\Pi_{n}(B\mid
Y_{1}^{n})=\frac{\int_{B}d\operatorname{\mathbb{P}}^{(n)}_{\theta}(Y_{1}^{n})\,d\Pi_{n}(\theta)}{\int_{\Theta}d\operatorname{\mathbb{P}}^{(n)}_{\theta}(Y_{1}^{n})\,d\Pi_{n}(\theta)},\quad
B\in\mathcal{F}.$ (3.1)

The posterior distribution is consistent if it concentrates on arbitrary small
neighborhoods of $\theta_{0}$ with
$\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$ probability tending to 1 as $n$
tends to infinity. Let $\|\cdot\|_{2}$ be the Euclidean distance. Then, the
contraction rate in metric space $(\Theta,\|\cdot\|_{2})$ is at least
$\epsilon_{n}$ if

$\displaystyle\Pi_{n}\\{\theta\in\Theta:\|\theta-\theta_{0}\|_{2}>M_{n}\epsilon_{n}\mid
Y_{1}^{n}\\}\rightarrow 0\text{ in
$\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$-probability for every
$M_{n}\rightarrow\infty$}.$

The contraction rate measures the size of small neighborhoods of $\theta_{0}$
on which the posterior distribution puts almost all mass.

A general technique for deriving posterior contraction rates is based on
proving the following testing condition [14, Theorem 7.3]. Let $\epsilon_{n}$
be a sequence satisfying $\epsilon_{n}\rightarrow 0$ and
$(n\epsilon_{n}^{2})^{-1}=O(1)$, the testing condition says that for a
universal constants $K>0$ and every sufficiently large $j>0$, there exists a
sequence test functions $\phi_{n}$ such that

$\displaystyle\operatorname{\mathbb{E}}_{\theta_{0}}^{(n)}(\phi_{n})\leq
e^{-Kj^{2}n\epsilon_{n}^{2}},\quad\sup_{\theta\in\Theta:j\epsilon_{n}<\|\theta-\theta_{0}\|_{2}\leq
2j\epsilon_{n}}\operatorname{\mathbb{E}}_{\theta}^{(n)}(1-\phi_{n})\leq
e^{-Kj^{2}n\epsilon_{n}^{2}}.$ (3.2)

Once the testing condition with rate $\epsilon_{n}$ in (3.2) is established,
the posterior distribution is consistent with the same rate $\epsilon_{n}$ for
any prior distributions $\Pi_{n}$ putting enough amount of mass near the true
parameter $\theta_{0}$ [14, 15]. Furthermore, if the local asymptotic
normality property (LAN) of
$\\{\operatorname{\mathbb{P}}_{\theta}^{(n)}:\theta\in\Theta\\}$ also holds,
then the Bernstein-von Mises theorem follows immediately [23].

Our main contribution is to construct $\phi_{n}$ in an HMM and prove that it
satisfies the testing condition in (3.2). Starting from the simple hypothesis,
we construct the likelihood ratio test with exponentially decaying error based
on [17]. Applying the $L^{1}$-transportation cost information inequality for
HMMs, we show that the log likelihood function is a Lipschitz function of the
extended HMM sequence $(Y_{t},p^{\theta}_{t})_{t=1}^{n}$, and it concentrates
on its mean with sub-Gaussian tail behavior. Then, for any $\epsilon>0$, some
$0<\xi<1$, and any $\theta_{1}$ with $\|\theta_{1}-\theta_{0}\|_{2}>\epsilon$,
we consider hypothesis of $\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$
versus the complement
$\\{\operatorname{\mathbb{P}}_{\theta}^{(n)}:\theta\in\Theta,\|\theta-\theta_{1}\|_{2}\leq\xi\epsilon\\}$.
By taking the liklihood ratio tests of
$\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$ versus
$\operatorname{\mathbb{P}}_{\theta^{\prime}}^{n}$ where $\theta^{\prime}$ is
the center of
$\\{\theta\in\Theta,\|\theta-\theta_{1}\|_{2}\leq\xi\epsilon\\}$, the
constructed test has an exponential decaying error. Finally, we prove the
testing condition (3.2) by an entropy bound of
$\\{\operatorname{\mathbb{P}}_{\theta}^{(n)}:\theta\in\Theta,j\epsilon_{n}<\|\theta-\theta_{0}\|_{2}\leq
2j\epsilon_{n}\\}$.

We require the following assumptions to ensure that a sequence of test
functions $\phi_{n}$ exists that satisfies (3.2).

  1. ($A_{1}$)

For any $\theta\in\Theta$, $\mathbb{P}_{\theta}^{(n)}(Y_{1}^{n}\in\cdot)\in
T_{1}(nC_{H})$ on metric space $(\mathcal{Y}^{n},d_{l^{1}})$ for some constant
$C_{H}>0$.

  2. ($A_{2}$)

For any $\theta\in\Theta$, there exists constants $\delta_{1},\delta_{2}>0$
such that the hidden Markov chain $(X_{t})_{t\geq 1}$ is ergodic,
$\|\partial_{y}f^{\theta}_{t}\|_{\infty}<\delta_{1}$, and
$\sum_{t=0}^{\infty}\|\partial_{p}f^{\theta}_{t}\|_{\infty}<\delta_{2}$, where

$\displaystyle\|\partial_{y}f^{\theta}_{t}\|_{\infty}=\sup_{y\neq
y^{\prime},p\in\mathcal{E}}\frac{\|f_{t}^{\theta}(y,p)-f_{t}^{\theta}(y^{\prime},p)\|_{\text{TV}}}{d_{\mathcal{Y}}(y,y^{\prime})},$
$\displaystyle\|\partial_{p}f_{t}^{\theta}\|_{\infty}=\sup_{p_{1}\neq
p_{1}^{\prime}\in\mathcal{E},y_{1}^{t+1}\in\mathcal{Y}^{t+1}}\frac{\|f^{\theta}_{t}(y_{1}^{t+1},p_{1})-f_{t}^{\theta}(y_{1}^{t+1},p_{1}^{\prime})\|_{\text{TV}}}{\|p_{1}-p_{1}^{\prime}\|_{\text{TV}}}.$
(3.3)

  3. ($A_{3}$)

For any $\theta\in\Theta$, the function
$\log\left\\{\sum_{x_{1}}p_{1}^{\theta}(x_{1})g_{\theta}(Y_{1}|x_{1})\right\\}$
is Lipschitz with norm $L/2$ on metric space
$(\mathcal{Z}=\mathcal{Y}\times\mathcal{E},d_{\mathcal{Z}}=d_{\mathcal{Y}}+\|\cdot\|_{\text{TV}})$.

  4. ($A_{4}$)

$\Theta\subset\mathbb{R}^{d}$ is compact and $\theta_{0}$ lies in its
interior.

  5. ($A_{5}$)

As $n\rightarrow\infty$,
$\frac{1}{n}\mathbb{E}^{(n)}_{\theta_{1}}\\{l_{n}(\theta_{1},Y_{1}^{n})-l_{n}(\theta_{2},Y_{1}^{n})\\}\rightarrow
J(\theta_{1}\mid\theta_{2})$ uniformly for $\theta_{1},\theta_{2}\in\Theta$,
where

$\displaystyle
J(\theta_{1}\mid\theta_{2})=\int_{\mathcal{Y}\times\mathcal{E}}\log\frac{\sum_{x\in\mathcal{X}}p(x)g_{\theta_{1}}(y\mid
x)}{\sum_{x\in\mathcal{X}}p(x)g_{\theta_{2}}(y\mid x)}R_{\theta_{1}}(dy,dp),$

and $R_{\theta_{1}}$ is the stationary distribution of $(Y_{t},p_{t})$ under
$\theta_{1}$.

  6. ($A_{6}$)

There exist constants $\kappa_{1},\kappa_{2}$ such that
$0<\kappa_{1}\leq\kappa_{2}\leq 2\kappa_{1}$ and for any
$\theta_{1},\theta_{2}\in\Theta$,

$\kappa_{1}\|\theta_{1}-\theta_{2}\|_{2}\leq
J(\theta_{1}|\theta_{2})\leq\kappa_{2}\|\theta_{1}-\theta_{2}\|_{2}.$

Assumptions ($A_{1}$)-($A_{3}$) are based on the conditions in Le Gland and
Mevel [20, Theorem 3.1] and are global over $\Theta$. Assumption ($A_{1}$)
states that the probability measure $\operatorname{\mathbb{P}}_{\theta}^{(n)}$
satisfies the $L^{1}$-transportation cost information inequality on $\Theta$
with $nC_{H}$. Theorem 2.3 shows that the constant $C_{H}$ is expressed in
terms of the emission distribution and mixing property of hidden Markov chain.
Assumption ($A_{2}$) implies that the extended HMM sequence
$(Y_{t},p_{t})_{t=1}^{n}$ is stationary and satisfies $L^{1}$-transportation
cost information inequality on $\Theta$ [17]. Assumption ($A_{3}$) states that
the $\log$ likelihood function is Lipschitz on metric space
$(\mathcal{Z},d_{\mathcal{Z}})$, which is mild and satisfied in many cases
[20, Example 3.1]. Assumption ($A_{5}$) implies the uniform convergence of
normalized log likelihood ratio function, which is slightly stronger than [22,
Proposition 4]. Assumption ($A_{6}$) specifies the equivalence of
$J(\theta_{1}\mid\theta_{2})$ and $\|\theta_{1}-\theta_{2}\|_{2}$ which is a
regular condition in parametric models [14, Example 7.1].

######  Theorem 3.1 (Testing Condition).

Let $\epsilon_{n}>0$ with $\epsilon_{n}\rightarrow 0$ such that
$(n\epsilon_{n}^{2})^{-1}=O(1)$. Under Assumptions ($A_{1}$)-($A_{6}$), there
exist positive constants $C^{\prime},K$ and a sequence of test functions
$\phi_{n}$ such that for every sufficiently large $j$,

$\displaystyle\mathbb{E}_{\theta_{0}}^{(n)}(\phi_{n})\leq
C^{\prime}e^{-Kn\epsilon_{n}^{2}j^{2}},\quad\sup_{\theta\in\Theta:j\epsilon_{n}<\|\theta-\theta_{0}\|_{2}\leq
2j\epsilon_{n}}\mathbb{E}_{\theta}^{(n)}(1-\phi_{n})\leq
C^{\prime}e^{-Kn\epsilon_{n}^{2}j^{2}}.$ (3.4)

Theorem 3.1 shows that under certain regularity conditions, there exists a
sequence of test functions $\phi_{n}$ satisfying the testing condition in
(3.2) for HMMs. The existsnce of such tests plays a crucial role in
establishing the convergence of posterior distributions [14, 15, 16]. We can
directly apply the testing condition in (3.4) to establish the contraction
rate and asymptotic normality of posterior distribution.

######  Corollary 3.2 (Posterior Convergence).

Let $\epsilon_{n}>0$ with $\epsilon_{n}\rightarrow 0$ such that
$(n\epsilon_{n}^{2})^{-1}=O(1)$. For $k>1$, define

$\displaystyle K(p_{\theta_{0}}^{(n)},p_{\theta}^{(n)})$ $\displaystyle=\int
p_{\theta_{0}}^{(n)}\log(p_{\theta_{0}}^{(n)}/p_{\theta}^{(n)})d\mu^{n},$
$\displaystyle V_{k,0}(p_{\theta_{0}}^{(n)},p_{\theta}^{(n)})$
$\displaystyle=\int
p_{\theta_{0}}^{(n)}|\log(p_{\theta_{0}}^{(n)}/p_{\theta}^{(n)})-K(p_{\theta_{0}}^{(n)},p_{\theta}^{(n)})|^{k}d\mu^{n},$
$\displaystyle B_{n}(\theta_{0},\epsilon;k)$
$\displaystyle=\\{\theta\in\Theta:K(p_{\theta_{0}}^{(n)},p_{\theta}^{(n)})\leq
n\epsilon^{2},V_{k,0}(p_{\theta_{0}}^{(n)},p_{\theta}^{(n)})\leq
n^{k/2}\epsilon^{k}\\}.$ (3.5)

Assume for some $k>1$, the prior distribution $\Pi_{n}$ of $\theta$ satisfies

$\displaystyle\frac{\Pi_{n}(\theta\in\Theta:j\epsilon_{n}<\|\theta-\theta_{0}\|_{2}\leq
2j\epsilon_{n})}{\Pi_{n}(B_{n}(\theta_{0},\epsilon_{n};k))}\leq
e^{Kn\epsilon_{n}^{2}j^{2}/2}.$ (3.6)

Then under the condition of Theorem 3.1, for every $M_{n}\rightarrow\infty$,
we have that

$\displaystyle\mathbb{E}_{\theta_{0}}^{(n)}\Pi_{n}(\theta\in\Theta:\|\theta-\theta_{0}\|_{2}\geq
M_{n}\epsilon_{n}\mid Y_{1}^{n})\rightarrow 0.$ (3.7)

######  Corollary 3.3 (Bernstein-von Mises theorem).

Given observation $Y_{1}^{n}$, denote the log likelihood as $l_{n}(\theta)$
and $l_{n}^{\prime}(\theta)$ as the derivative of $l_{n}$ with respect to
$\theta$. Assume the HMM model
$\\{\mathbb{P}_{\theta}^{(n)}:\theta\in\Theta\\}$ satisfies the local
asymptotic normality condition (LAN):

$\displaystyle
l_{n}\left(\theta_{0}+\frac{h}{\sqrt{n}}\right)-l_{n}(\theta_{0})=\frac{1}{\sqrt{n}}h^{\top}l^{\prime}_{n}(\theta_{0})-\frac{1}{2}h^{\top}I_{\theta_{0}}h+o_{\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}}(1),$
(3.8)

where $h\in\\{h\in\mathbb{R}^{d}:\theta_{0}+\frac{h}{\sqrt{n}}\in\Theta\\}$
and $l_{n}^{\prime}(\theta_{0})$ weakly converges to $N_{d}(0,I_{\theta_{0}})$
under $\operatorname{\mathbb{P}}_{\theta_{0}}^{(n)}$ probability. Assume the
prior distribution has continuous density $\pi(\theta)>0$ on $\Theta$ and the
Fisher information matrix $I_{\theta_{0}}$ at $\theta_{0}$ is nonsingular.
Then under the condition of Theorem 3.1,

$\displaystyle\mathbb{E}_{\theta_{0}}^{(n)}\|\Pi_{n}\left\\{\sqrt{n}(\theta-\theta_{0})\mid
Y_{1}^{n}\right\\}-N_{d}(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{TV}\rightarrow
0,$ (3.9)

where
$\Delta_{n,0}=\frac{1}{\sqrt{n}}I_{\theta_{0}}^{-1}l^{\prime}_{n}(\theta_{0}).$

Bickel et al. [2] have developed the first set of results showing the
consistency and asymptotic normality of maximum likelihood estimators of the
parameters in an HMM. Corollary 3.3 provides similar theoretical gaurantees
about Bayesian estimation in HMMs. Specifically, it states that the posterior
distribution of $\theta$ given the observations $Y_{1}^{n}$ in an HMM is
asymptotically normal with random mean vector
$\theta_{0}+\frac{1}{\sqrt{n}}\Delta_{n,0}$ and covariance matrix
$(nI_{\theta_{0}})^{-1}$. Compared to the similar results in [5], which are
based on Taylor expansion of the log likelihood function, Corollary 3.3
provides an alternative approach to prove the Bernstein-von Mises theorem
using the LAN condition (3.8) and testing condition (3.4). The LAN condition
is a general regularity assumption on the class of parametric models
$\\{\operatorname{\mathbb{P}}_{\theta}^{(n)}:\theta\in\Theta\\}$, which is
satisfied under the theoretical setup in [2, 5]. In our proof, we construct
the sequence of test functions based on a likelihood ratio test and prove the
testing condition using the optimal transportation inequality for HMMs.

##  4 Proofs

In this section, for simplicity, we denote $\mathbb{P}_{\theta}^{(n)}$ and
$\mathbb{E}_{\theta}^{(n)}$ as $\mathbb{P}_{\theta}$ and
$\mathbb{E}_{\theta}$, respectively.

### Proof of Theorem 2.3

In the case $\mathcal{Y}$ is countable, the proof is based on Kontorovich et
al. [19, Theorem 1.3].

######  Theorem 4.1.

[Kontorovich et al. [19, Theorem 1.3]] Suppose $\mathcal{X}$ is a countable
space, $\mathcal{F}$ is the collection of all subsets of $\mathcal{X}^{n}$,
$\mathbb{P}$ is the probability measure on $(\mathcal{X}^{n},\mathcal{F})$,
and $\mathbb{E}$ is the expectation under $\mathbb{P}$. Suppose
$\phi:\mathcal{X}^{n}\mapsto\mathbb{R}$ is a $c$-Lipschitz function (with
respect to Hamming metric) for some $c>0$, then for any $r>0$,

$\mathbb{P}\\{|\phi-\mathbb{E}\phi|>r\\}\leq
2\exp(-\frac{r^{2}}{2nc^{2}\|\Delta_{n}\|_{\infty}^{2}}).$

Under $\operatorname{\mathbb{P}}_{\theta}$, $(X_{t})_{t\geq 1}$ is a Markov
chain with transition matrix $Q_{\theta}$, and the mixing coefficient
$\|\Delta_{n}\|_{\infty}$ is bounded by the Markov contraction coefficients
[19, Lemma 7.1]:

$\displaystyle\|\Delta_{n}\|_{\infty}\leq\frac{1}{2}\sum_{t=1}^{\infty}\sup_{a,a^{\prime}}\sum_{b\in\mathcal{X}}|Q_{\theta}^{t}(a,b)-Q_{\theta}^{t}(a^{\prime},b)|+1=D_{\theta}+1.$
(4.1)

The transportation inequality of HMM with countable $\mathcal{Y}$ is given as
an example in [19, Section 7.2] and $C_{H}^{\theta}=(D_{\theta}+1)^{2}$.

In the case $\mathcal{Y}$ is uncountable, Theorem 2.2 implies that proving the
second part of Theorem 2.3 is equivalent to showing that for any Lipschitz
function $F:\mathcal{Y}^{n}\mapsto\mathbb{R}$ with
$\|F\|_{\text{Lip}}\leq\alpha$ (with respect to $d_{l^{1}}$),

$\mathbb{E_{\theta}}e^{\lambda(F-\mathbb{E_{\theta}}F)}\leq\exp\left\\{\frac{(\lambda\alpha)^{2}}{2}n(C_{\mathcal{Y}}+L^{2}(D_{\theta}+1)^{2})\right\\},\quad\forall\lambda\in\mathbb{R}.$

Let $G_{n}(X_{1}^{n})=\mathbb{E_{\theta}}[F\mid X_{1}^{n}]$ for $n\geq 1$.
Then, for $x_{1}^{n}\neq\tilde{x}_{1}^{n}$,

$\displaystyle|G_{n}(x_{1}^{n})-G_{n}(\tilde{x}_{1}^{n})|$
$\displaystyle=|\int F(y_{1}^{n})d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=x_{1}^{n})-\int
F(y_{1}^{n})d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=\tilde{x}_{1}^{n})|$ $\displaystyle=|\int
F(y_{1}^{n})\left\\{d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=x_{1}^{n})-d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=\tilde{x}_{1}^{n})\right\\}|$
$\displaystyle\overset{(i)}{\leq}\sup_{\|\Phi\|_{\text{Lip}}\leq\alpha}\int\Phi(y_{1}^{n})\left\\{d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=x_{1}^{n})-d\mathbb{P}_{\theta}(Y_{1}^{n}=y_{1}^{n}\mid
X_{1}^{n}=\tilde{x}_{1}^{n})\right\\}$ $\displaystyle\overset{(ii)}{=}\alpha
W_{1}^{d_{l_{1}}}\\{\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot\mid
X_{1}^{n}=x_{1}^{n}),\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot\mid
X_{1}^{n}=\tilde{x}_{1}^{n})\\}$
$\displaystyle\overset{(iii)}{=}\alpha\inf_{\Pi}\int\sum_{t=1}^{n}d_{\mathcal{Y}}(y_{t},\tilde{y}_{t})d\Pi(y_{1}^{n},\tilde{y}_{1}^{n})$
$\displaystyle=\alpha\sum_{t=1}^{n}1_{x_{t}\neq\tilde{x}_{t}}W_{1}^{d_{\mathcal{Y}}}\\{\mathbb{P}_{\theta}(Y_{t}\in\cdot|X_{t}=x_{t}),\mathbb{P}_{\theta}(Y_{k}\in\cdot|X_{k}=\tilde{x}_{t})\\}$
$\displaystyle\overset{(iv)}{\leq}\alpha
L\sum_{k=1}^{n}1_{x_{k}\neq\tilde{x}_{k}}=\alpha Ld_{H}(x,\tilde{x}),$

where $\Phi$ on the RHS of $(i)$ is an $\alpha$-Lipschitz function on
$(\mathcal{Y}^{n},d_{l_{1}})$, $(ii)$ follows from the duality form of $W_{1}$
distance, $(iii)$ follows from the definition of $W_{1}^{d_{l_{1}}}$ where
$\Pi$ is the joint distribution with marginal distributions
$\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot|x_{1}^{n})$ and
$\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot|\tilde{x}_{1}^{n})$, and $(iv)$ follows
from assumption that
$W_{1}^{d_{\mathcal{Y}}}\\{\mathbb{P}_{\theta}(Y_{1}\in\cdot\mid
X_{1}=a),\mathbb{P}_{\theta}(Y_{1}\in\cdot\mid X_{1}=b)\\}\leq L1_{a\neq b}$;
therefore, $G_{n}$ is a Lipschitz function on $\mathcal{X}^{n}$ with respect
to Hamming metric $d_{H}$. Using (4.1), we have that
$\mathbb{P}_{\theta}(X_{1}^{n}\in\cdot)\in T_{1}(n(D_{\theta}+1)^{2})$ on
$(\mathcal{X}^{n},d_{l_{1}})$. By Theorem 2.2, for any $\lambda>0$,

$\mathbb{E}_{\theta}e^{\lambda(G_{n}-\mathbb{E}_{\theta}G_{n})}\leq\exp\left(\frac{(\lambda\alpha
L)^{2}}{2}n(D_{\theta}+1)^{2}\right).$

Using independent tensorization, we have
$\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot|X_{1}^{n})\in T_{1}(nC_{\mathcal{Y}})$
on $(\mathcal{Y}^{n},d_{H})$, which implies that

$\mathbb{E}_{\theta}\\{e^{\lambda(F-E_{\theta}[F\mid X_{1}^{n}])}\mid
X_{1}^{n}\\}=\mathbb{E}_{\theta}\\{e^{\lambda(F-G_{n})}\mid
X_{1}^{n}\\}\leq\exp\left(\frac{(\lambda\alpha)^{2}}{2}nC_{\mathcal{Y}}\right).$

Therefore,

$\displaystyle\mathbb{E}_{\theta}e^{\lambda(F-\mathbb{E}_{\theta}F})$
$\displaystyle=\mathbb{E}_{\theta}\left(\mathbb{E}_{\theta}\\{e^{\lambda(F-G_{n})}\mid
X_{1}^{n}\\}e^{\lambda(G_{n}-\mathbb{E}_{\theta}F)}\right)$
$\displaystyle\leq\mathbb{E}_{\theta}\left(\mathbb{E}_{\theta}\left\\{e^{\lambda(F-E_{\theta}[F\mid
X_{1}^{n}])}\mid X_{1}^{n}\right\\}\right)\exp\left\\{\frac{(\lambda\alpha
L)^{2}}{2}n(D_{\theta}+1)^{2}\right\\}$
$\displaystyle\leq\exp\left\\{\frac{(\lambda\alpha)^{2}}{2}n(C_{\mathcal{Y}}+L^{2}(D_{\theta}+1)^{2})\right\\}.$

$\square$

## Proof of Theorem 3.1

We start by proving the following lemma that is based on Hu [17, Theorem 3.3].

######  Lemma 4.1.

If Assumptions ($A_{1}$)-($A_{3}$) hold, then for
$\theta_{1}\neq\theta_{2}\in\Theta$,

$F_{n}(\theta_{1},\theta_{2})=l_{n}(\theta_{1},Y_{1}^{n})-l_{n}(\theta_{2},Y_{1}^{n})$

is a Lipschitz function on $(\mathcal{Z}^{n},d_{l^{1}})$ with
$\|F_{n}(\theta_{1},\theta_{2})\|_{\text{Lip}}\leq L$ and for any
$\theta\in\Theta$,

$\mathbb{E}_{\theta}e^{\lambda\left\\{F_{n}(\theta_{1},\theta_{2})-\mathbb{E}_{\theta}F_{n}(\theta_{1},\theta_{2})\right\\}}\leq\exp\left\\{\frac{\lambda^{2}}{2}L^{2}nC_{H}(1+\delta)^{4}\right\\},$

where $\delta=\max(\delta_{1},\delta_{2})$ in Assumption ($A_{2}$).

### Proof of Lemma 4.1

First we show that for any $\theta\in\Theta$, there exists a constant
$C_{E}>0$ such that the probability measure induced by the extended HMM
$\\{Y_{t},p_{t}^{\theta}\\}_{t=1}^{n}$ satisfies $T_{1}(nC_{E})$-inequality on
metric space $(\mathcal{Z}^{n}=(\mathcal{Y}\times\mathcal{E})^{n},d_{l_{1}})$,
where the $l_{1}$-metric on $\mathcal{Z}^{n}$ is defined as:
$d_{l_{1}}(z_{1}^{n},\tilde{z}_{1}^{n})=\sum_{k=1}^{n}d_{\mathcal{Z}}(z_{k},\tilde{z}_{k})$
and

$d_{\mathcal{Z}}(z_{k},\tilde{z}_{k})=d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})+\|p_{k}^{\theta}-p_{k}^{\theta}\|_{\text{TV}},\quad
z_{k}=(y_{k},p_{k}^{\theta}),\tilde{z}_{k}=(\tilde{y}_{k},\tilde{p}_{k}^{\theta})\in\mathcal{Z}.$

That is, given $p_{1}^{\theta}=r_{\theta}\in\mathcal{E}$ and for any
$\theta\in\Theta$, we want to show the probability measure
$\mathbb{P}_{\theta}((Y_{i},p_{i}^{\theta})_{i=1}^{n}\in\cdot)\in
T_{1}(nC_{E})$ on $(\mathcal{Z}^{n},d_{l^{1}})$, and $C_{E}$ is given as

$C_{E}=C_{H}(1+\delta)^{4},\quad\delta=\max(\delta_{1},\delta_{2}).$

Let $F$ be a Lipschitz function on $(\mathcal{Z}^{n},d_{l_{1}})$ with
$\|F\|_{\text{Lip}}\leq 1$. By the Lipschitz property and definition of
$d_{l_{1}}$, for $1\leq k\leq n$ we have that,

$\displaystyle\|\partial_{y_{k}}F\|_{\infty}:=\sup_{z_{1}^{k-1},y_{k}\neq\tilde{y}_{k},p_{k},z_{k+1}^{n}}\frac{|F(z_{1}^{k-1},(y_{k},p_{k}),z_{k+1}^{n})-F(z_{1}^{k-1},(\tilde{y}_{k},p_{k}),z_{k+1}^{n})|}{d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})}\leq
1,$
$\displaystyle\|\partial_{p_{k}}F\|_{\infty}:=\sup_{z_{1}^{k-1},p_{k}\neq\tilde{p}_{k},y_{k},z_{k+1}^{n}}\frac{|F(z_{1}^{k-1},(y_{k},p_{k}),z_{k+1}^{n})-F(z_{1}^{k-1},(y_{k},\tilde{p}_{k}),z_{k+1}^{n})|}{\|p_{k}-\tilde{p}_{k}\|_{\text{TV}}}\leq
1.$ (4.2)

For $k=2,\ldots,n$, (2.5) implies that the prediction filters $p_{k}^{\theta}$
only depend on $y_{1}^{k-1}$, so $F(z_{1}^{n})$ can be written as a function
$G$ on $\mathcal{Y}^{n}$.

Next, we get a bound for $\|\partial_{y_{k}}G\|_{\infty}$. Suppose
$y_{1}^{n}\neq\tilde{y}_{1}^{n}$ only differ at the $k$th coordinates
$y_{k}\neq\tilde{y}_{k}$. Then,

$G(y_{1}^{n})-G(\tilde{y}_{1}^{n})=F(y_{1}^{n},p_{1}^{n})-F(\tilde{y}_{1}^{n},\tilde{p}_{1}^{n}).$

For $j\leq k$, $p_{j}=\tilde{p}_{j}$ because of
$y_{1}^{k-1}=\tilde{y}_{1}^{k-1}$. When $j=k+1$, Assumption ($A_{2}$) implies
that

$\displaystyle\|p_{k+1}-\tilde{p}_{k+1}\|_{\text{TV}}$
$\displaystyle=\|f^{\theta}(y_{k},p_{1}^{k})-f^{\theta}(\tilde{y_{k}},p_{1}^{k})\|_{\text{TV}}$
$\displaystyle\leq\delta_{1}d_{\mathcal{Y}}(y_{k},\tilde{y}_{k});$ (4.3)

Finally, for $j\geq k+2$,

$\displaystyle\|p_{j}-\tilde{p}_{j}\|_{\text{TV}}$
$\displaystyle=\|f^{\theta}_{j-k-2}(y_{k+1}^{j-1},p_{k+1})-f^{\theta}_{j-k-2}(y_{k+1}^{j-1},\tilde{p}_{k+1})\|_{\text{TV}}$
$\displaystyle\leq\|\partial_{p_{0}}f^{\theta}_{j-k-2}\|_{\infty}\cdot\|p_{k+1}-\tilde{p}_{k+1}\|_{\text{TV}};$
(4.4)

therefore, if $y_{1}^{n},\tilde{y}_{1}^{n}$ differ only at the $k$th
coordinates, then

$\displaystyle|G(y_{1}^{n})-G(\tilde{y}_{1}^{n})|$
$\displaystyle=|F(y_{1}^{n},p_{1}^{n})-F(\tilde{y}_{1}^{n},\tilde{p}_{1}^{n})|$
$\displaystyle\leq\|\partial_{y_{k}}F\|_{\infty}d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})+\sum_{j=k+1}^{n}\|\partial_{p_{j}}F\|_{\infty}\|p_{j}-\tilde{p}_{j}\|_{\text{TV}}$
$\displaystyle\overset{(i)}{\leq}d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})+\left(\|\partial_{p_{k+1}}F\|_{\infty}+\sum_{j=k+2}^{n}\|\partial_{p_{0}}f^{\theta}_{j-k-2}\|_{\infty}\right)\|p_{k+1}-\tilde{p}_{k+1}\|_{\text{TV}}$
$\displaystyle\overset{(ii)}{\leq}d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})\left(1+\delta_{1}+\delta_{1}\delta_{2}\right)<d_{\mathcal{Y}}(y_{k},\tilde{y}_{k})\left(1+\delta\right)^{2},$
(4.5)

where $(i)$ follows from (Proof of Lemma 4.1) and $(ii)$ follows from (Proof
of Lemma 4.1), (Proof of Lemma 4.1), and Assumption ($A_{2}$). Under
$\operatorname{\mathbb{P}}_{\theta}$, $G(y_{1}^{n})$ is a Lipschitz function
on $(\mathcal{Y}^{n},d_{l_{1}})$ with $\|G\|_{\text{Lip}}\leq(1+\delta)^{2}$.
Using Assumption ($A_{1}$), $\mathbb{P}_{\theta}(Y_{1}^{n}\in\cdot)\in
T_{1}(nC_{H})$; Theorem 2.2 implies that for any $\lambda\in\mathbb{R}$,

$\displaystyle\mathbb{E}_{\theta}e^{\lambda\left\\{F(Y_{1}^{n},p_{1}^{n})-\mathbb{E}_{\theta}F(Y_{1}^{n},p_{1}^{n})\right\\}}$
$\displaystyle=\mathbb{E}_{\theta}e^{\lambda\\{G(Y_{1}^{n})-\mathbb{E}_{\theta}G(Y_{1}^{n})\\}}$
$\displaystyle\leq\exp\left\\{\frac{\lambda^{2}}{2}nC_{H}\left(1+\delta\right)^{4}\right\\}.$
(4.6)

Hence, $\mathbb{P}_{\theta}((Y_{i},p_{i}^{\theta})_{i=1}^{n}\in\cdot)\in
T_{1}(nC_{E})$ with $C_{E}=C_{H}(1+\delta)^{4}.$

Concluding the proof of the lemma, we show that $F_{n}(\theta_{1},\theta_{2})$
is a Lipscthiz function on $(\mathcal{E}^{n},d_{l_{1}})$. Expressing the log-
likelihood additively, we have

$\displaystyle F_{n}(\theta_{1},\theta_{2})$
$\displaystyle=l_{n}(\theta_{1},Y_{1}^{n})-l_{n}(\theta_{2},Y_{1}^{n})$
$\displaystyle=\sum_{t=1}^{n}\left(\log\sum_{x_{t}}p_{t}^{\theta_{1}}(x_{t})g_{\theta_{1}}(Y_{t}|x_{t})-\log\sum_{x_{t}}p_{t}^{\theta_{2}}(x_{t})g_{\theta_{2}}(Y_{t}|x_{t})\right).$
(4.7)

By Assumption ($A_{3}$), for $i=1,2$, the function
$\log\sum_{x_{t}}p_{t}^{\theta_{i}}(x_{t})g_{\theta_{i}}(Y_{t}|x_{t})$ is a
Lipschitz with norm $L/2$ on metric space
$(\mathcal{Z}=\mathcal{Y}\times\mathcal{E},d_{\mathcal{Z}}=d_{\mathcal{Y}}+\|\cdot\|_{\text{TV}})$.
So $F_{n}(\theta_{1},\theta_{2})$ is a Lipshitz function on
$(\mathcal{Z}^{n},d_{l_{1}})$ with $\|F_{n}\|_{\text{Lip}}\leq L$. Using
Theorem 2.2, we have that

$\mathbb{E}^{n}_{\theta}e^{\lambda\left\\{F_{n}(\theta_{1},\theta_{2})-\mathbb{E}_{\theta}^{n}F_{n}(\theta_{1},\theta_{2})\right\\}}\leq\exp\left\\{\frac{\lambda^{2}}{2}L^{2}nC_{H}(1+\delta)^{4}\right\\}.$

$\square$

As a special case of Lemma 4.1, we have

$\displaystyle\mathbb{E}^{n}_{\theta_{1}}e^{\lambda\left\\{F_{n}(\theta_{1},\theta_{2})-nH_{n}(\theta_{1}|\theta_{2})\right\\}}\leq\exp\left\\{\frac{\lambda^{2}}{2}L^{2}nC_{H}(1+\delta)^{4}\right\\}.$
(4.8)

Now we prove Theorem 3.1 in three steps. First, we show the existence of a
test for $\operatorname{\mathbb{P}}_{\theta_{0}}$ versus
$\operatorname{\mathbb{P}}_{\theta_{1}}$ with exponentially decaying error
probabilities. Second, we show the existence of a test for
$\operatorname{\mathbb{P}}_{\theta_{0}}$ versus the complement
$\\{\operatorname{\mathbb{P}}_{\theta}:\theta\in\Theta,\|\theta-\theta_{1}\|_{2}\leq\xi\epsilon\\}$
with exponentially decaying error probabilities for any $\epsilon>0$, some
$\xi<1$, and any $\theta_{1}\in\Theta$ satisfying
$\|\theta_{1}-\theta_{0}\|_{2}>\epsilon$. Finally, the proof is completed by
showing the existence of a test for $\operatorname{\mathbb{P}}_{\theta_{0}}$
versus the complement
$\\{\operatorname{\mathbb{P}}_{\theta}:\theta\in\Theta,\|\theta-\theta_{0}\|_{2}\geq
M\epsilon\\}$ with exponentially decaying error probabilities for any
$\epsilon>0$ and sufficiently large $M>0$.

_Step 1: Show the existence of a test with exponentially decaying error
probabilities for simple hypotheses._

For $\theta_{1}\neq\theta_{0}\in\Theta$, we consider the simple hypotheses

$\displaystyle H_{0}:\theta=\theta_{0}\quad\text{vs}\quad
H_{1}:\theta=\theta_{1}.$ (4.9)

Based on Lemma 4.1, we show the existence of a test with Type 1 and 2 error
probability decaying exponentially to $0$ as $n\rightarrow\infty$. For
$\theta\neq\theta^{\prime}\in\Theta$, define

$H_{n}(\theta|\theta^{\prime})=\frac{1}{n}\mathbb{E}_{\theta}\\{l_{n}(\theta,Y_{1}^{n})-l_{n}(\theta^{\prime},Y_{1}^{n})\\}.$

Using Jensen’s inequality, we have $H_{n}(\theta|\theta^{\prime})\geq 0$. By
Assumption ($A_{5}$), $H_{n}(\theta|\theta^{\prime})\rightarrow
J(\theta|\theta^{\prime})>0$ as $n\rightarrow\infty$; so there exists an
integer $N=N(\theta,\theta^{\prime})$ depending on $(\theta,\theta^{\prime})$
such that
$H_{n}(\theta|\theta^{\prime})\geq\frac{1}{2}J(\theta|\theta^{\prime})$ for
any $n>N(\theta,\theta^{\prime})$. Let the critical value
$r\in(0,\frac{1}{4}J(\theta_{0}|\theta_{1})).$ We construct the likelihood
ratio test for the hypotheses in (4.9) as

$\displaystyle\phi_{\theta_{1}}=1\\{l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{1},Y_{1}^{n})\leq
nr\\}\equiv 1\\{F_{n}(\theta_{0},\theta_{1})\leq nr\\},$ (4.10)

and reject $H_{0}$ if $\phi_{\theta_{1}}=1$. For any
$n>N(\theta_{0},\theta_{1})$, we have
$H_{n}(\theta_{0}|\theta_{1})-r>\frac{1}{4}J(\theta_{0}|\theta_{1})$ and the
probability of Type 1 error of the test in (4.10) is

$\displaystyle\mathbb{P}_{\theta_{0}}(\text{reject}\;H_{0})$
$\displaystyle=\mathbb{P}_{\theta_{0}}(\phi_{\theta_{1}}=1)=\operatorname{\mathbb{P}}_{\theta_{0}}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{1},Y_{1}^{n})\leq
nr]$
$\displaystyle=\operatorname{\mathbb{P}}_{\theta_{0}}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{1},Y_{1}^{n})-nH_{n}(\theta_{0}|\theta_{1})\leq
n\\{r-H_{n}(\theta_{0}|\theta_{1})\\}]$
$\displaystyle=\operatorname{\mathbb{P}}_{\theta_{0}}[F_{n}(\theta_{0},\theta_{1})-\mathbb{E}_{\theta_{0}}F_{n}(\theta_{0},\theta_{1})\leq
n\\{r-H_{n}(\theta_{0}|\theta_{1})\\}]$
$\displaystyle\overset{(i)}{\leq}\exp\left(-\frac{n(r-H_{n}(\theta_{0}|\theta_{1}))^{2}}{\tilde{C}}\right)\overset{(ii)}{\leq}\exp\left(-\frac{nJ(\theta_{0}|\theta_{1})^{2}}{16\tilde{C}}\right),$
(4.11)

where $(i)$ follows from (4.8) with $\tilde{C}=L^{2}C_{H}(1+\delta)^{4}$, and
$(ii)$ follows from
$H_{n}(\theta_{0}|\theta_{1})-r>\frac{1}{4}J(\theta_{0}|\theta_{1})>0$.

The exponentially decaying bound on the probability of Type 2 error follows
similarly by switching the roles of $\theta_{0}$ and $\theta_{1}$. There
exists $N(\theta_{1},\theta_{0})\in\mathbb{N}$ such that for any
$n>N(\theta_{1},\theta_{0})$,
$H_{n}(\theta_{1}|\theta_{0})\geq\frac{1}{2}J(\theta_{1}|\theta_{0})$, and the
probability of Type 2 error of $\phi_{\theta_{1}}$ in (4.10) is

$\displaystyle\mathbb{P}_{\theta_{1}}(\text{fail to reject}\;H_{0})$
$\displaystyle=\mathbb{P}_{\theta_{1}}(\phi_{\theta_{1}}=0)=\mathbb{P}_{\theta_{1}}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{1},Y_{1}^{n})>nr]$
$\displaystyle=\mathbb{P}_{\theta_{1}}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{1},Y_{1}^{n})+nH_{n}(\theta_{1}|\theta_{0})>n\\{H_{n}(\theta_{1}|\theta_{0})+r\\}]$
$\displaystyle=\mathbb{P}_{\theta_{1}}[F_{n}(\theta_{1},\theta_{0})-\mathbb{E}_{\theta_{1}}F_{n}(\theta_{1},\theta_{0})<-n\\{H_{n}(\theta_{1}|\theta_{0})+r\\}]$
$\displaystyle\leq\exp\left(-\frac{n(H_{n}(\theta_{1}|\theta_{0})+r)^{2}}{\tilde{C}}\right)\leq\exp\left(-\frac{nJ(\theta_{1}|\theta_{0})^{2}}{4\tilde{C}}\right).$
(4.12)

By choosing $n>\max\\{N(\theta_{1},\theta_{0}),N(\theta_{1},\theta_{0})\\},$
we can see that the probabilities of Type 1 and Type 2 error of test (4.10)
decay exponentially in $n$.

_Step 2: For any $\epsilon>0$, some $\xi<1$, any $\theta_{1}\in\Theta$
satisfying $\|\theta_{1}-\theta_{0}\|_{2}>\epsilon$, show the existence of a
test for $\operatorname{\mathbb{P}}_{\theta_{0}}$ versus the complement
$\\{\operatorname{\mathbb{P}}_{\theta}:\theta\in\Theta,\|\theta-\theta_{1}\|_{2}\leq\xi\epsilon\\}$
with exponentially decaying error probabilities._

For any $\epsilon>0$, let $\theta^{\prime}\in\Theta$ satisfy
$\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}<2\epsilon$,
$U\subset\\{\theta\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}<2\epsilon\\}$
be an open neighborhood with diameter $\frac{\epsilon}{2}$, and $\theta_{U}\in
U$ be a center such that $\|\theta-\theta_{U}\|_{2}<\frac{\epsilon}{4}$ for
any $\theta\in U$. We consider the following hypotheses:

$\displaystyle H_{0}:\theta=\theta_{0}\quad\text{vs}\quad H_{1}:\theta\in U.$
(4.13)

We want to show the existence of a test $\phi_{U}^{\epsilon}$ with
exponentially decaying Type 1 and 2 error probabilities; that is,

$\mathbb{E}_{\theta_{0}}(\phi^{\epsilon}_{U})\lesssim
e^{-n\tilde{c}\epsilon^{2}},\quad\sup_{\theta\in
U}\,\mathbb{E}_{\theta}(1-\phi_{U}^{\epsilon})\lesssim
e^{-n\tilde{c}\epsilon^{2}},$

where $\lesssim$ denotes inequality up to a fixed constant. Let $r>0$ be the
critical value to be chosen later. Then, define

$\displaystyle\phi_{U}^{\epsilon}=1\\{l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})\leq
nr\\},$ (4.14)

and reject $H_{0}$ in (4.13) if $\phi_{U}^{\epsilon}=1$.

The corresponding probability of Type 1 error of $\phi_{U}^{\epsilon}$ is

$\displaystyle\mathbb{P}_{\theta_{0}}(\text{reject $H_{0}$})$
$\displaystyle=\mathbb{E}_{\theta_{0}}(\phi_{U}^{\epsilon})$
$\displaystyle=\mathbb{P}_{\theta_{0}}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})\leq
nr]$
$\displaystyle\overset{(i)}{\leq}\exp\left(-\frac{n(r-H_{n}(\theta_{0}|\theta_{U}))^{2}}{\tilde{C}}\right).$
(4.15)

where $(i)$ follows from Step 1 provided $r-H_{n}(\theta_{0}|\theta_{U})<0$.

The corresponding probability of Type 2 error of $\phi_{U}^{\epsilon}$ is

$\displaystyle\sup_{\theta\in U}\,$
$\displaystyle\mathbb{P}_{\theta}(\text{fail to reject
$H_{0}$})=\sup_{\theta\in U}\mathbb{E}_{\theta}(1-\phi_{U}^{\epsilon})$
$\displaystyle=\sup_{\theta\in
U}\mathbb{P}_{\theta}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})>nr]$
$\displaystyle=\sup_{\theta\in
U}\mathbb{P}_{\theta}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta,Y_{1}^{n})+l_{n}(\theta,Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})>nr]$
$\displaystyle=\sup_{\theta\in
U}\mathbb{P}_{\theta}[l_{n}(\theta_{0},Y_{1}^{n})-l_{n}(\theta,Y_{1}^{n})+nH_{n}(\theta|\theta_{0})$
$\displaystyle\quad+l_{n}(\theta,Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})-nH_{n}(\theta|\theta_{U})>n(r+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U}))]$
$\displaystyle\leq\sup_{\theta\in
U}\\{\mathbb{P}_{\theta}[l_{n}(\theta,Y_{1}^{n})-l_{n}(\theta_{0},Y_{1}^{n})-nH_{n}(\theta|\theta_{0})<-\frac{n}{2}(r+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U}))]$
$\displaystyle\quad+\mathbb{P}_{\theta}[l_{n}(\theta,Y_{1}^{n})-l_{n}(\theta_{U},Y_{1}^{n})-nH_{n}(\theta|\theta_{U})>\frac{n}{2}(r+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U}))]\\}$
$\displaystyle\overset{(i)}{\leq}2\sup_{\theta\in
U}\left\\{\exp\left(-\frac{n(r+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U}))^{2}}{4\tilde{C}}\right)\right\\}.$
(4.16)

where $(i)$ follows from Lemma 4.1 provided
$r+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U})>0$.

Now we choose the critical value $r$ such that

$\frac{1}{4}J(\theta_{0}|\theta_{U})<r<\frac{1}{2}J(\theta_{0}|\theta_{U}).$

From Step 1, there exists $N(\theta_{0},\theta_{U})\in\mathbb{N}$ such that
for any $n>N(\theta_{0},\theta_{U})$,
$H_{n}(\theta_{0}|\theta_{U})\geq\frac{3}{4}J(\theta_{0}|\theta_{U})$; so we
have that

$\displaystyle
r-H_{n}(\theta_{0}|\theta_{U})\leq-\frac{1}{4}J(\theta_{0}|\theta_{U})\overset{(i)}{\leq}-\frac{\kappa_{2}}{4}\|\theta_{0}-\theta_{U}\|_{2}\leq-\frac{9\kappa_{2}}{16}\epsilon,$
(4.17)

where $(i)$ follows from Assumption ($A_{6}$). Combining the results in (Proof
of Lemma 4.1) and (4.17) gives the exponentially decaying bounds for the Type
1 error:

$\mathbb{E}_{\theta_{0}}(\phi_{U}^{\epsilon})\leq\exp\left[-n\epsilon^{2}\frac{81\kappa_{2}^{2}}{256\tilde{C}}\right].$

Assumption ($A_{5}$) implies that $H_{n}(\theta|\theta_{0})\rightarrow
J(\theta|\theta_{0})$ uniformly for any $\theta\in U$; so there exists
$N^{\prime}(\theta_{0})\in\mathbb{N}$ such that for any
$n>N^{\prime}(\theta_{0})$,
$H_{n}(\theta|\theta_{0})\geq\frac{15}{16}J(\theta|\theta_{0})$ for any
$\theta\in U$. Furthermore, by choosing
$n>\max\\{N^{\prime}(\theta_{0}),N^{\prime}(\theta_{U})\\}$, we also have
$\frac{15}{16}J(\theta|\theta_{U})\leq
H_{n}(\theta|\theta_{U})\leq\frac{17}{16}J(\theta|\theta_{U})$ for any
$\theta\in U$. Therefore,

$\displaystyle\inf_{\theta\in U}\\{r$
$\displaystyle+H_{n}(\theta|\theta_{0})-H_{n}(\theta|\theta_{U})\\}\geq\frac{1}{4}J(\theta_{0}|\theta_{U})+\inf_{\theta\in
U}\\{\frac{3}{4}J(\theta|\theta_{0})-\frac{4}{5}J(\theta|\theta_{U})\\}$
$\displaystyle\overset{(i)}{\geq}\frac{\kappa_{1}}{4}\|\theta_{0}-\theta_{U}\|_{2}+\frac{3\kappa_{1}}{4}\inf_{\theta\in
U}\|\theta-\theta_{0}\|_{2}-\frac{4\kappa_{2}}{5}\sup_{\theta\in
U}\|\theta-\theta_{U}\|_{2}$
$\displaystyle\overset{(ii)}{\geq}(\frac{9\kappa_{1}}{16}-\frac{\kappa_{2}}{5})\epsilon>0,$
(4.18)

where $(i)$ follows from Assumption ($A_{6}$) and $(ii)$ follows from for
$\theta\in U$,
$\|\theta_{0}-\theta_{U}\|_{2}\geq\frac{3\epsilon}{4},\|\theta_{0}-\theta\|_{2}\geq\frac{\epsilon}{2}$,
and $\|\theta-\theta_{U}\|_{2}\leq\frac{\epsilon}{4}.$ Combining the results
in (Proof of Lemma 4.1) and (Proof of Lemma 4.1) gives the exponentially
decaying bounds for the Type 2 error:

$\sup_{\theta\in U}\mathbb{E}_{\theta}(1-\phi_{U}^{\epsilon})\leq
2\exp\left[-n\epsilon^{2}\frac{(\frac{9\kappa_{1}}{16}-\frac{\kappa_{2}}{5})^{2}}{4\tilde{C}}\right].$

By choosing
$n>\max\\{N(\theta_{0},\theta_{U}),N(\theta_{0}),N^{\prime}(\theta_{0})\\},$
the probabilities of Type 1 and Type 2 error of test (4.14) is dominated by
$\exp(-\tilde{c}n\epsilon^{2})$ where
$\tilde{c}=\min\\{\frac{81\kappa_{2}^{2}}{256\tilde{C}},\frac{(\frac{9\kappa_{1}}{16}-\frac{\kappa_{2}}{5})^{2}}{4\tilde{C}}\\}$.
Furthermore, this result holds for any open ball $U_{\xi}$ of radius
$\xi\epsilon$ such that
$U_{\xi}\subset\\{\theta\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}<2\epsilon\\}$.

_Step 3: For any $\epsilon>0$ and sufficiently large $M>0$, show the existence
of a test for testing $\operatorname{\mathbb{P}}_{\theta_{0}}$ versus the
complement
$\\{\operatorname{\mathbb{P}}_{\theta}:\theta\in\Theta,\|\theta-\theta_{0}\|_{2}\geq
M\epsilon\\}$ with exponentially decaying error probabilities._

We want to show the existence of a test $\phi^{\epsilon}_{M}$ with
exponentially decaying Type 1 and 2 error probabilities; that is,

$\mathbb{E}_{\theta_{0}}(\phi^{\epsilon}_{M})\lesssim
e^{-c_{1}n\epsilon^{2}M^{2}},\quad\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|_{2}>M/\sqrt{n}}\mathbb{E}_{\theta}(1-\phi^{\epsilon}_{M})\lesssim
e^{-c_{2}n\epsilon^{2}M^{2}}.$

We start by defining a collection of sets that are used to construct our test
later. For any $\epsilon>0$ and $\xi\in(0,1)$, the compactness of the closure
of
$\\{\theta^{\prime}\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2\epsilon\\}$ implies that there exist a finite number of open balls
$\\{U_{l,1}\\}_{l=1}^{N(\xi,\epsilon)}$ with radius $\xi\epsilon$ such that

$\\{\theta^{\prime}\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}<2\epsilon\\}\subseteq\bigcup_{l=1}^{N(\xi,\epsilon)}U_{l,1},$

where
$N(\xi,\epsilon)=N(\xi\epsilon,\\{\theta^{\prime}\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2\epsilon\\},\|\cdot\|_{2})$ is the minimum number of balls with radius
$\xi\epsilon$ under metric $\|\cdot\|_{2}$ needed to cover the set
$\\{\theta^{\prime}\in\Theta,\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2\epsilon\\}$. Extending this argument further, we have that for a large
number $M$ and any integer $j\geq M$, there are $N(\xi
j\epsilon,\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2j\epsilon\\},\|\cdot\|_{2})$ many open balls with radius $\xi j\epsilon$
under metric $\|\cdot\|_{2}$ that cover the set
$\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2j\epsilon\\}$. For ease of notation and following $N(\xi,\epsilon)$ defined
earlier, denote

$N(\xi,j\epsilon)=N(\xi
j\epsilon,\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2j\epsilon\\},\|\cdot\|_{2})$

for any $\epsilon>0$, $0<\xi<1$, and $j\geq M$. The compactness of the closure
of
$\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2j\epsilon\\}$ for every integer $j\geq M$ implies that we have a collection
of open balls $\\{U_{l,j}\\}_{l=1}^{N(\xi,j\epsilon)}$ with radius
$\xi\epsilon$ satisfying

$\displaystyle\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}\leq
2j\epsilon\\}\subseteq\bigcup_{l=1}^{N(\xi,j\epsilon)}U_{l,j},\quad j\geq M.$

We define the test $\phi^{\epsilon}_{M}$ using the test constructed in Step 2
for different choices of the open ball $U$. For each ball in
$\\{U_{l,j}\\}_{l=1}^{N(\xi,j\epsilon)}$, we can find a collection of tests
$\\{\psi_{U_{l,j}}^{n}\\}_{l=1}^{N(\xi,j\epsilon)}$ with Type 1 and 2 error
rates dominated by $\exp(-\tilde{c}nj^{2}\epsilon^{2})$; see Step 2 for the
definition of $\tilde{c}$. Let $\phi_{M}^{\epsilon}$ be the supremum of
countably many tests obtained this way:

$\phi_{M}^{\epsilon}=\sup_{j\geq M}\sup_{1\leq l\leq
N(\xi,j\epsilon)}\psi_{U_{j,l}}^{n}.$

Then, for a sufficiently large $n$,

$\displaystyle\mathbb{E}_{\theta_{0}}\phi_{M}^{\epsilon}$
$\displaystyle\leq\sum_{j=M}^{\infty}\sum_{l=1}^{N(\xi,j\epsilon)}\mathbb{E}_{\theta_{0}}\psi^{n}_{U_{j,l}}\overset{(i)}{\leq}\sum_{j=M}^{\infty}N(\xi,j\epsilon)e^{-\tilde{c}nj^{2}\epsilon^{2}}\leq\sup_{j\geq
1}N(\xi,j\xi)\sum_{j=M}^{\infty}e^{-\tilde{c}nj^{2}\epsilon^{2}}$
$\displaystyle\leq\frac{\sup_{j\geq
1}N(\xi,j\xi)}{1-e^{-\tilde{c}n\epsilon^{2}}}e^{-\tilde{c}nM^{2}\epsilon^{2}},$
(4.19)

where $(i)$ follows from the upper bound on the Type 1 error probability
obtained in Step 2. Similarly, the upper bound on the Type 2 error probability
obtained in Step 2 implies that, for a sufficiently large $n$,

$\displaystyle\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|_{2}>M\epsilon}\mathbb{E}_{\theta}(1-\phi_{M}^{\epsilon})$
$\displaystyle\overset{(i)}{\leq}\sup_{U_{l,j},\;\;j\geq
M}\mathbb{E}_{\theta}(1-\psi_{U_{l,j}}^{n})$ $\displaystyle\leq\sup_{j\geq
M}2e^{-\tilde{c}nj^{2}\epsilon^{2}}\leq 2e^{-\tilde{c}nM^{2}\epsilon^{2}},$
(4.20)

where $(i)$ follows from the construction of $\phi_{M}^{\epsilon}$ because for
each $\theta\in\\{\theta\in\Theta:\|\theta-\theta_{0}\|_{2}>M\epsilon\\}$,
there exists a $j\geq M$ and a test $\psi_{U_{l,j}}^{n}$ with
$\phi\geq\psi_{U_{l,j}}^{n}$ satisfying
$\mathbb{E}_{\theta}(1-\psi_{U_{l,j}}^{n})\leq
e^{-\tilde{c}nj^{2}\epsilon^{2}}.$

Let $\epsilon_{n}>0$, $\epsilon_{n}\rightarrow 0$ and
$(n\epsilon_{n}^{2})^{-1}=O(1)$. Then, Ghosal et al. [14, Example 7.1] implies
that

$\sup_{j\geq 1}N(\xi,j\xi)=\sup_{j\geq 1}N(\xi
j\epsilon,\\{\theta^{\prime}\in\Theta,j\epsilon<\|\theta^{\prime}-\theta_{0}\|_{2}<2j\epsilon\\},\|\cdot\|_{2})\leq\left(\frac{12}{\xi}\right)^{d};$

therefore, let $\xi=\frac{1}{4}$; (4.19) and (4.20) imply that there exists a
constant

$\tilde{c}=\min\\{\frac{81\kappa_{2}^{2}}{256L^{2}C_{H}(1+\delta)^{4}},\frac{(\frac{9\kappa_{1}}{16}-\frac{\kappa_{2}}{5})^{2}}{4L^{2}C_{H}(1+\delta)^{4}}\\}$

and a sequence of test functions $\phi_{n}=\phi_{M}^{\epsilon_{n}}$ for large
enough $M$ such that for every sufficiently large $n$,

$\mathbb{E}_{\theta_{0}}(\phi_{n})\leq\frac{\left(48\right)^{d}}{1-e^{-\tilde{c}}}e^{-\tilde{c}nM^{2}\epsilon_{n}^{2}},\quad\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|>M\epsilon_{n}}\mathbb{E}_{\theta}(1-\phi_{n})\leq
2e^{-\tilde{c}nM^{2}\epsilon_{n}^{2}}.$

The proof is complete by observing that

$\sup_{\theta\in\Theta:M\epsilon_{n}<\|\theta-\theta_{0}\|_{2}\leq
2M\epsilon_{n}}\mathbb{E}_{\theta}(1-\phi_{n})\leq\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|_{2}>M\epsilon_{n}}\mathbb{E}_{\theta}(1-\phi_{n}).$

$\square$

## Proof of Corollary 3.2

The statement (3.7) follows by verifying the conditions in [15, Theorem 3].
$\square$

## Proof of Corollary 3.3

The proof is based on [23, Theorem 10.1]. First we verify the contiguity
property of $\operatorname{\mathbb{P}}_{\theta}$ and the existence of tests as
specified in van der Vaart [23, Theorem 10.1].

_Contiguity property:_ From local asymptotic normality condition (3.8), under
$\operatorname{\mathbb{P}}_{0}$-probability, we have that for any
$h\in\sqrt{n}(\Theta-\theta_{0})$

$\displaystyle
l_{n}(\theta_{0}+\frac{h}{\sqrt{n}})-l_{n}(\theta_{0})=\frac{1}{\sqrt{n}}h^{\top}l^{\prime}_{n}(\theta_{0})-\frac{1}{2}h^{\top}I_{\theta_{0}}h+o_{\operatorname{\mathbb{P}}_{\theta_{0}}}(1),$
(4.21)

where $l_{n}^{\prime}(\theta_{0})$ weakly converges to
$N_{d}(0,I_{\theta_{0}})$ under
$\operatorname{\mathbb{P}}_{\theta_{0}}$-probability. Let
$\operatorname{\mathbb{P}}_{\theta_{n}}$ denote the distribution of
$(Y_{1},\ldots,Y_{n})$ under parameter
$\theta_{n}=\theta_{0}+\frac{h}{\sqrt{n}}$ for some
$h\in\sqrt{n}(\Theta-\theta_{0})$. LeCam’s first lemma [21] implies that
$\operatorname{\mathbb{P}}_{\theta_{n}}$ is mutually contiguous to
$\operatorname{\mathbb{P}}_{\theta_{0}}$.

_Existence of test:_ If the assumptions of Theorem 3.1 hold, then for any
sequence $\epsilon$ satisfying $\epsilon_{n}>0$, $\epsilon_{n}\rightarrow 0$,
and $(n\epsilon_{n}^{2})^{-1}=O(1)$, there exists a constant $K$ and a
sequence of test $\phi_{n}$, such that for every sufficiently large $n$,

$\mathbb{E}_{\theta_{0}}(\phi_{n})\lesssim
e^{-KM^{2}n\epsilon_{n}^{2}},\;\quad\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|>M\epsilon_{n}}\mathbb{E}_{\theta}(1-\phi_{n})\leq
2e^{-KM^{2}n\epsilon_{n}^{2}}.$

By setting $\epsilon_{n}=n^{-1/2}$ and $r_{n}$ be any sequence tending to
infinity, we can find a sequence of tests $\phi_{n}$ satisfying as
$n\rightarrow\infty$,

$\displaystyle\mathbb{E}_{\theta_{0}}(\phi_{n})\lesssim
e^{-Kr_{n}^{2}}\rightarrow
0,\quad\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|>r_{n}/\sqrt{n}}\mathbb{E}_{\theta}(1-\phi_{n})\leq
2e^{-Kr_{n}^{2}}\rightarrow 0.$ (4.22)

Now we prove Corollary 3.3 in two steps. In the first step, we show that the
total variance distance between the posterior distribution of
$h=\sqrt{n}(\theta-\theta_{0})$ relative to prior $\Pi_{n}$ and the restricted
prior $\Pi_{n}^{C_{n}}$, where $C_{n}$ is the ball with radius $r_{n}$, is
asmptotically negligible as $r_{n}\rightarrow\infty.$ By (4.22), for every
$r_{n}\rightarrow\infty$, we can find a constant $K$ and a sequence of tests
$\phi_{n}$, such that as $n\rightarrow\infty$,

$\displaystyle\mathbb{E}_{\theta_{0}}\phi_{n}\rightarrow
0,\quad\sup_{\theta\in\Theta:\|\theta-\theta_{0}\|>r_{n}/\sqrt{n}}\mathbb{E}_{\theta}(1-\phi_{n})\leq
2e^{-Kr_{n}^{2}}.$ (4.23)

Let $\mathcal{F}$ denote the Borel $\sigma$-field on
$\sqrt{n}(\Theta-\theta_{0})$ and $\Pi_{h|Y_{1}^{n}}^{C_{n}}$ denote the
posterior distribution of $h$ conditional on $Y_{1}^{n}$ with respect to the
restricted prior $\Pi_{n}^{C_{n}}$. We have that

$\displaystyle\|\Pi_{h|Y_{1}^{n}}-\Pi_{h|Y_{1}^{n}}^{C_{n}}\|_{\text{TV}}=\sup_{B\in\mathcal{F}}\left|\Pi_{h|Y_{1}^{n}}(B)-\Pi_{h|Y_{1}^{n}}^{C_{n}}(B)\right|$
$\displaystyle=\sup_{B\in\mathcal{F}}\left|\frac{\int
1_{B}(h)\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh}{\int\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}-\frac{\int
1_{C_{n}}(h)1_{B}(h)\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh}{\int
1_{C_{n}}(s)\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}\right|$
$\displaystyle=\sup_{B\in\mathcal{F}}\left|\frac{\int\\{1_{B}(h)-1_{B}(h)1_{C_{n}}(h)\\}\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh}{\int\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}-\int
1_{C_{n}}(h)1_{B}(h)\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh\times\right.$
$\displaystyle\left.\left[\\{\int
1_{C_{n}}(s)\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds\\}^{-1}-\\{\int\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds\\}^{-1}\right]\right|$
$\displaystyle=\sup_{B\in\mathcal{F}}\left|\frac{\int\\{1_{B}(h)-1_{B}(h)1_{C_{n}}(h)\\}\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh}{\int\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}-\right.$
$\displaystyle\left.\frac{\int
1_{C_{n}}(h)1_{B}(h)\pi(h)p_{\theta+h/\sqrt{n}}(Y_{1}^{n})dh}{\int
1_{C_{n}}(s)\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}\times\frac{\int\\{1-1_{C_{n}}(s)\\}\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}{\int\pi(s)p_{\theta+s/\sqrt{n}}(Y_{1}^{n})ds}\right|$
$\displaystyle=\sup_{B\in\mathcal{F}}|P_{h|Y_{1}^{n}}(B\cap
C_{n}^{c})-P_{h|Y_{1}^{n}}^{C_{n}}(B)P_{h|Y_{1}^{n}}(C_{n}^{c})|$
$\displaystyle\leq 2P_{h|Y_{1}^{n}}(C_{n}^{c})$ (4.24)

Let $U$ be a ball with fixed radius. Define
$\operatorname{\mathbb{P}}_{n,U}=\int\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}d\Pi^{U}_{n}(h)$
and the associated expectation as $\operatorname{\mathbb{E}}_{n,U}$. Using
(3.8) and boundedness of $\pi(\cdot)$, $\operatorname{\mathbb{P}}_{n,U}$ is
mutually contiguous with $\operatorname{\mathbb{P}}_{\theta_{0}}$; so using
(4.22), we have that

$\displaystyle\mathbb{E}_{n,U}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})$
$\displaystyle=\mathbb{E}_{n,U}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})+\mathbb{E}_{0}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})\phi_{n}+o(1)$
$\displaystyle=\mathbb{E}_{n,U}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})+o(1),$

where $\mathbb{E}_{n,U}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})$ can be
expressed as

$\displaystyle\mathbb{E}_{n,U}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})$
$\displaystyle=\int\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})d\Pi^{U}_{n}(h)$
$\displaystyle=\frac{1}{\Pi_{n}(U)}\int
1_{U}(h)\pi(h)\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}\Pi_{h|Y_{1}^{n}}(C_{n}^{c})(1-\phi_{n})dh$
$\displaystyle=\frac{\Pi_{n}(C_{n}^{c})}{\Pi_{n}(U)}\int\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}\Pi_{h|Y_{1}^{n}}(U)(1-\phi_{n})d\Pi^{C_{n}^{c}}_{n}(h)$
$\displaystyle\leq\frac{1}{\Pi_{n}(U)}\int_{C_{n}^{c}}\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}(1-\phi_{n})d\Pi_{n}(h)$
$\displaystyle\leq\frac{1}{\Pi_{n}(U)}\int_{\|h\|\geq
r_{n}}e^{-Kr_{n}^{2}}d\Pi_{n}(h),$ (4.25)

where the last inequality follows from (4.22) and $\frac{1}{\Pi_{n}(U)}$ is
bounded above by order $n^{d/2}$ due to the continuity of the density $\pi$ at
$\theta_{0}$; therefore, (Proof of Corollary 3.3) is bounded by
$\int_{\|h\|\geq r_{n}}\exp(-Kr_{n}^{2})dh$, which converges to zero as
$n,r_{n}\rightarrow\infty.$

Let $N^{C}(\mu,\Sigma)$ denote the restriction of normal distribution with
mean $\mu$ and covariance matrix $\Sigma$ on a set $C$. In the second step of
the proof, we want to show that for a ball $C$ with fixed radius $M$,

$\mathbb{E}_{n,C}\|\Pi^{C}_{h|Y_{1}^{n}}-N^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{\text{TV}}\rightarrow
0,$

where
$\Delta_{n,0}=\frac{1}{\sqrt{n}}I_{\theta_{0}}^{-1}l^{\prime}(\theta_{0}).$ We
have that

$\displaystyle\mathbb{E}_{n,C}\|\Pi^{C}_{h|Y_{1}^{n}}-N^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{\text{TV}}$
$\displaystyle=2\int\int\left\\{1-\frac{dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(h)}{1_{C}(h)d\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}(Y_{1}^{n})\pi_{n}(h)/\int_{C}d\operatorname{\mathbb{P}}_{\theta+g/\sqrt{n}}(Y_{1}^{n})\pi_{n}(g)dg}\right\\}^{+}d\Pi_{h|Y_{1}^{n}}^{C}(dh)\operatorname{\mathbb{P}}_{n,C}(dY_{1}^{n})$
$\displaystyle\overset{(i)}{\leq}2\int\left\\{1-\frac{1_{C}(h)d\operatorname{\mathbb{P}}_{\theta+g/\sqrt{n}}(Y_{1}^{n})\pi_{n}(g)dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(h)}{1_{C}(h)d\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}(Y_{1}^{n})\pi_{n}(h)N^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(g)}\right\\}^{+}dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(g)d\Pi_{h|Y_{1}^{n}}^{C}(dh)\operatorname{\mathbb{P}}_{n,C}(dY_{1}^{n}),$
(4.26)

where $(i)$ follows from Jensen’s inequality of $f(x)=(1-x)^{+}$. By
dominated-convergence theorem and
$dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(g)\leq C\lambda^{C}(g)$ for some
constant $C>0$; therefore, it is suffices to show under measure
$\operatorname{\mathbb{P}}_{n,C}(dy_{1}^{n})\Pi_{h|Y_{1}^{n}}^{C}(dh)\lambda^{C}(dg)$,

$\displaystyle\frac{d\operatorname{\mathbb{P}}_{\theta+g/\sqrt{n}}(Y_{1}^{n})\pi_{n}(g)dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(h)}{d\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}(Y_{1}^{n})\pi_{n}(h)dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(g)}\longrightarrow
1\;\text{in probability},$ (4.27)

where $\lambda^{C}$ is the Lebesgue measure on $C$. From the definition of
posterior distribution, we have

$\operatorname{\mathbb{P}}_{n,C}(dy_{1}^{n})\Pi_{h|Y_{1}^{n}}^{C}(dh)\lambda^{C}(dg)=\Pi_{n}^{C}(dh)\operatorname{\mathbb{P}}_{\theta+h/\sqrt{n}}(dy_{1}^{n})\lambda^{C}(dg).$

Using local asymptotic normality condition (3.8) and continuity of $\pi$, we
know that
$\operatorname{\mathbb{P}}_{n,C}(dy_{1}^{n})\Pi_{h|Y_{1}^{n}}^{C}(dh)\lambda^{C}(dg)$
is contiguous to $\lambda^{C}(dh)P_{\theta_{0}}(dy_{1}^{n})\lambda^{C}(dg);$
therefore, it is suffices to show that under measure
$\lambda^{C}(dh)\operatorname{\mathbb{P}}_{\theta_{0}}(dy_{1}^{n})\lambda^{C}(dg)$,

$\frac{d\operatorname{\mathbb{P}}_{n,g}(Y_{1}^{n})\pi_{n}(g)dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(h)}{d\operatorname{\mathbb{P}}_{n,h}(Y_{1}^{n})\pi_{n}(h)dN^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})(g)}\longrightarrow
1\;\text{in probability},$

which is true under the local asymptotic normality condition (3.8). We have
for fixed $C$,

$\mathbb{E}_{\theta_{0}}\|\Pi^{C}_{h|Y_{1}^{n}}-N^{C}(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{TV}\rightarrow
0.$

Combining this with step one, we have for $n,r_{n}\rightarrow\infty,$

$\displaystyle\mathbb{E}_{\theta_{0}}\|\Pi_{h|Y_{1}^{n}}^{C_{n}}-\Pi_{h|Y_{1}^{n}}\|_{\text{TV}}\rightarrow
0,$
$\displaystyle\|N^{C_{n}}(\Delta_{n,0},I_{\theta_{0}}^{-1})-N(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{TV}\rightarrow
0.$ (4.28)

By diagonal argument,

$\mathbb{E}_{\theta_{0}}\|\Pi_{h|Y_{1}^{n}}-N(\Delta_{n,0},I_{\theta_{0}}^{-1})\|_{TV}\rightarrow
0,$

which completes the proof. $\square$

## Acknowledgements

Chunlei Wang and Sanvesh Srivastava are partially supported by grants from the
Office of Naval Research (ONR-BAA N000141812741) and the National Science
Foundation (DMS-1854667/1854662).

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