# Dropout Injection at Test Time for Post Hoc Uncertainty Quantification in
Neural Networks

Emanuele Ledda<EMAIL_ADDRESS>Giorgio Fumera<EMAIL_ADDRESS>Fabio Roli<EMAIL_ADDRESS>
###### Abstract

Among Bayesian methods, Monte-Carlo dropout provides principled tools for
evaluating the _epistemic_ uncertainty of neural networks. Its popularity
recently led to seminal works that proposed activating the dropout layers only
during inference for evaluating uncertainty. This approach, which we call
dropout _injection_ , provides clear benefits over its traditional counterpart
(which we call _embedded_ dropout) since it allows one to obtain a post hoc
uncertainty measure for any existing network previously trained without
dropout, avoiding an additional, time-consuming training process.
Unfortunately, no previous work compared injected and embedded dropout;
therefore, we provide the first thorough investigation, focusing on regression
problems. The main contribution of our work is to provide guidelines on the
effective use of injected dropout so that it can be a practical alternative to
the current use of embedded dropout. In particular, we show that its
effectiveness strongly relies on a suitable scaling of the corresponding
uncertainty measure, and we discuss the trade-off between negative log-
likelihood and calibration error as a function of the scale factor.
Experimental results on UCI data sets and crowd counting benchmarks support
our claim that dropout injection can effectively behave as a competitive post
hoc uncertainty quantification technique.

###### keywords:

Uncertainty Quantification , Epistemic Uncertainty , Monte Carlo Dropout ,
Trustworthy AI , Crowd Counting

††journal: Information Sciences

[inst1]organization=Department of Computer, Control and Management
Engineering, Sapienza University of Rome, addressline=Via Ariosto 25,
city=Rome, postcode=00185, country=Italy [inst2]organization=Department of
Informatics, Bioengineering, Robotics, and Systems Engineering, University of
Genova, addressline=Via Dodecaneso 35, city=Genova, postcode=16146,
country=Italy [inst3]organization=Department of Electric and Electronic
Engineering, University of Cagliari, addressline=Via Marengo 3, city=Cagliari,
postcode=09100, country=Italy

##  1 Introduction

Systems based on Artificial Intelligence are nowadays widespread in industry,
workplaces and everyday life. They often operate in critical contexts where
errors can cause considerable harm to humans (e.g., self-driving cars), where
transparency and accountability are required (e.g., medical and legal
domains), and where end users need to be aware of the uncertainty/reliability
of AI predictions (e.g., intelligent video surveillance). In particular, given
that AI moved out of research laboratories to real-world scenarios, the need
for trustworthiness guarantees led to the growth of an entire research field
devoted to quantifying _uncertainty_ of machine learning models [1, 2].

In this field, it is common to distinguish between two possible sources of
uncertainty: one caused by intrinsic data noise (aleatoric uncertainty, or
data uncertainty) and one caused by the lack of knowledge on the “correct”
prediction model (epistemic uncertainty, or model uncertainty) [3]. In
particular, most of the work on deep neural networks uses their Bayesian
extension for quantifying _uncertainty_ , modeling the network weights with a
probability distribution [4, 5]. The simple and effective Monte Carlo Dropout
technique [6] is one of the leading and most widely used techniques for
evaluating _epistemic_ uncertainty according to the Bayesian extension of
neural networks. It is based on _dropout_ , a stochastic regularization
technique designed for neural networks, consisting of randomly dropping out
some neural units and their associated connections during training, with a
predefined

Figure 1: A schematic diagram of the three main steps that we propose to
effectively implement injected dropout: 1) select a neural network trained for
a regression problem without dropout (network output:
$\hat{y}\in\mathcal{R}$); 2) add dropout layers to the network; 3) compute the
dropout rate $\Phi$ and the scaling factor $C$ of the uncertainty measure by
solving the optimization problem in (9) for obtaining the scaled measure
$\xi^{2}$.

probability [7]. The main idea behind Monte Carlo Dropout is to exploit the
weight randomization induced by the dropout layers by keeping them active
during inference; this allows one to approximate the distribution of the
neural network weights with respect to training data. However, whereas dropout
has been originally designed as a stochastic regularization technique to be
used only during training, when it is used for uncertainty quantification its
purpose is to estimate the weight distribution, and it is no more conceived as
a regularizer. This suggests activating dropout layers during training may not
be necessary if dropout is used only for uncertainty quantification.
Accordingly, a recent work by Loquercio et al. [8] proposed to activate
dropout layers for uncertainty quantification only during inference, instead
of using it also during training. The latter approach, which we call dropout
injection, appears very interesting, as it would allow the _integration_ of a
measure of epistemic uncertainty into _any_ network that was previously
trained with no dropout, without the computational burden of a further
training procedure required by the original Monte Carlo Dropout, which we call
embedded dropout, nor the need to optimize the related hyper-parameters.

Loquercio et al. [8] provided empirical evidence that injected dropout,
combined with assumed density filtering [9] for aleatoric uncertainty
evaluation, can be an effective tool for evaluating the _predictive_
uncertainty of a predictor. However, no previous work, including the one by
Loquercio et al. [8], provided a thorough analysis of injected dropout as an
epistemic uncertainty evaluation technique, nor a comparison with classical
embedded dropout, in order to provide guidelines for its effective use.

Based on the above motivations, in this work we carry out a deep investigation
of injected dropout, focusing on regression problems. First, our analysis
shows that its effectiveness critically relies on a suitable scaling of the
corresponding uncertainty measure, differently from embedded dropout. To this
aim, we extend the original formulation by Loquercio et al. [8] to introduce
and compute a suitable scaling factor [10]. We also address the resulting
issue of achieving a trade-off between different aspects related to the
quality of the corresponding uncertainty measure, i.e., negative log-
likelihood and calibration error, as a function of the scale factor. We then
carry out experiments on eight UCI data sets as well as five benchmarks for
crowd counting, which is a challenging computer vision task, to thoroughly
evaluate the behavior of injected dropout in terms of prediction accuracy and
the quality of the corresponding uncertainty measure, providing experimental
evidence of its effectiveness over embedded dropout.

Our results shed light on the practical operation of injected dropout and show
that it can effectively evaluate epistemic uncertainty at test time without a
time-consuming training process with dropout layers.

In the following, we present background concepts on Bayesian methods for
uncertainty evaluation and Monte Carlo Dropout in Sect 2, a theoretical
analysis of injected dropout and the proposed method for injected dropout in
Sect. 3, and experimental results in Sect. 4. Discussion and conclusions are
given in Sect. 5, where future work on this topic is also discussed.

##  2 Background Concepts

We first summarise the main concepts of uncertainty quantification in neural
networks based on Bayesian methods, then we focus on Monte Carlo dropout for
epistemic uncertainty quantification.

###  2.1 Uncertainty Evaluation through Bayesian Inference

To obtain a network capable of providing predictions with an associated level
of uncertainty, one possible approach is a Bayesian extension. In particular,
for a regression problem, a traditional neural network implements a predictor
$\hat{y}=f(x;w)\in\mathbb{R}$, where $w$ denotes the connection weights.
Bayesian neural networks output a full distribution instead of a point
prediction $\hat{y}$, which is usually modeled with a Gaussian distribution
$\mathcal{N}(\hat{\mu},\hat{\sigma}^{2})$, albeit it is not the only possible
choice (e.g., Gaussian Mixture distributions or Generalized Gaussian
distributions have also been proposed [11, 12]). In this case, the mean
$\hat{\mu}$ is interpreted as the point prediction $\hat{y}$, whereas the
variance $\hat{\sigma}^{2}$ represents the desired uncertainty measure;
ideally, one wants $\hat{\mu}$ and $\hat{\sigma}^{2}$ to approximate well the
true target distribution $\mathcal{N}(\mu,\sigma^{2})$.

Under a Bayesian framework, the full probability distribution of the target
variable $y$ for an input query $x$ is also conditioned on the training set
$\mathcal{D}$:

$p(y|x,\mathcal{D})=\int_{w}p(y|x,w)p(w|\mathcal{D})\mathrm{d}w\ .$ (1)

This is equivalent to taking the expectation under a posterior on the weights,
$p(w|\mathcal{D})$, that provides the desired predictive distribution.
However, since obtaining $p(w|\mathcal{D})$ is intractable for most neural
networks, an approximation $q(w)$ is usually considered. By substituting the
true distribution with its approximation, the posterior can be computed as:

$p(y|x,\mathcal{D})\approx\int_{w}p(y|x,w)q(w)\mathrm{d}w\ .$ (2)

A suitable approximation $q(w)$ is sought among a set of possible candidates
by minimizing its Kullback-Leibler divergence (KL-divergence)
$KL[p(w|\mathcal{D})||q(w)]$ with the real distribution. From variational
inference, it is known that the KL-divergence can be minimised by minimizing
the negative log-likelihood (NLL), that for regression problems is
approximated on a given data set of $N$ samples as:

$\frac{1}{N}\sum_{n=1}^{N}\frac{1}{2}\frac{(y_{n}-\hat{y}_{n})^{2}}{\hat{\sigma}_{n}^{2}}+\frac{1}{2}\log(\hat{\sigma}_{n}^{2})\
.$ (3)

###  2.2 Monte Carlo Dropout as a Bayesian Approximation

To model an approximated distribution $q(w)$, one may consider incorporating a
stochastic component inside a network. One possible way to introduce
randomness on the network weights to estimate a full probability distribution
over them is to use Stochastic Regularization Techniques [2], originally
proposed for network regularisation. In particular, it has been shown both
theoretically [6] and practically [6, 13, 3] that dropout can be used to this
aim.

The standard dropout envisages the deactivation of each neuron belonging to a
specific network layer with a probability, named dropout rate, that follows
the Bernoulli distribution $\mathcal{B}(\Phi)$, where the parameter $\Phi$
corresponds to the dropout rate. The dropout layers are kept active during
training and are deactivated at inference time, which produces a
regularisation effect.

Unlike standard dropout, Monte Carlo dropout keeps the dropout layers active
at inference time; this allows one to derive a full probability distribution
$q(w|\Phi)$ approximating $p(w|\mathcal{D})$. The optimal dropout rate $\Phi$
is the one that minimizes the divergence between the real and the approximated
distributions; in practice, a suitable $\Phi$ is found by minimizing an
approximation of NLL computed, e.g., on a validation set of $N$ samples:

$\Phi=\arg\min_{\varphi}\frac{1}{N}\sum_{n=1}^{N}\frac{1}{2}\frac{(y_{n}-\hat{y}_{n}(\varphi))^{2}}{\hat{\sigma}_{n}^{2}(\varphi)}+\frac{1}{2}\log(\hat{\sigma}_{n}^{2}(\varphi))\
.$ (4)

Once such a value of $\Phi$ and the trained network are obtained, one can use
the approximate distribution $q(w|\Phi)$ for computing a prediction and
associated predictive uncertainty, modeled as the mean and the variance of the
posterior distribution:

$p(y|x,\mathcal{D})\approx\frac{1}{T}\sum_{t=1}^{T}p(y|x,w_{t})\sim\mathcal{N}(\hat{\mu};\hat{y},\hat{\sigma}^{2})\
,$ (5)

where $T$ denotes the number of Monte Carlo samples obtained by querying the
network multiple times with the same sample $x$.

Two widely used techniques for solving the optimization problem (4) are grid
search [6] and concrete dropout [13]. In embedded dropout, after solving (4)
using a training and a validation set, one can use the obtained dropout rate
in the inference stage. To improve performance, a different dropout rate can
be used for each network layer [13]. On the contrary, in the injected dropout
approach [8] dropout layers are added to an already-trained network, and the
dropout rate $\Phi$ is sought by minimizing the NLL (4) of the trained
network, e.g., on a validation set.

##  3 The proposed method for Monte Carlo Dropout Injection

A notable difference between embedded and injected dropout is that the former
also performs a regularization, whereas the latter does not. This can lead to
differences in the prediction error of the corresponding networks, as well as
in their uncertainty measures, which we shall investigate in this section. In
particular, we analyze, theoretically and empirically, how prediction error
and quality of the uncertainty measure behave as a function of the dropout
rate $\Phi$, for both embedded and injected dropout (Sect. 3.1). The results
of this analysis will point out an issue of injected dropout, that we propose
to mitigate by suitably _scaling_ the uncertainty measure; to this aim, we
propose an extension of the original formulation in Eq. (4), using the
$\sigma$-scaling technique [10] taking into account how the scaling factor
affects the calibration of the uncertainty measure(Sect. 3.2).

For simplicity, we shall consider a fixed dropout rate across all network
layers, but all our results apply to the most general case when different
dropout rates are allowed.

###  3.1 Analysis of Monte Carlo Dropout Injection

In Sect. 2.2 we mentioned that the dropout rate $\Phi$ is usually chosen by
minimizing the NLL, both in embedded and injected dropout. Now, let us
consider the weights vector $w$ of a neural network: one can obtain $w$ by
using a standard optimization process that may or may not involve the
activation of dropout layers during _training_ for regularizing the network
weights (i.e., embedded or injected dropout, respectively). Dropout layers are
then activated at _inference_ time, both when performing embedded and injected
dropout, with dropout rate $\varphi$, in order to obtain a so-called Monte-
Carlo iteration, i.e. a prediction taken using the network with stochastic
weights deactivation. This is equivalent to applying a binary mask
$m(\varphi)=diag(z_{1}(\varphi)\dots z_{L}(\varphi))$ – where
$z_{\ell}(\varphi)\sim\mathcal{B}(\varphi)$ and $L$ denotes the size of $w$ –
to the network weights. Applying $T$ stochastic iterations is equivalent to
applying a series of $T$ masks, each one denoted by $m_{t}(\varphi)$, to
obtain a series of instances of the network weights, $w_{1}(\varphi)\dots
w_{T}(\varphi)$ with $w_{t}(\varphi)=w\cdot m_{t}(\varphi)$, where $\cdot$
denotes the matrix product. It is easy to see that $w_{t}(\varphi)$ always
depends upon $\varphi$ through the binary mask $m_{t}(\varphi)$, both for
embedded and injected dropout. We can now rewrite Eq. (5) for modelling the
prediction $\hat{y}$ and the corresponding uncertainty $\hat{\sigma}^{2}$ of a
given instance $x$ when using a neural network with dropout rate $\varphi$:

$\frac{1}{T}\sum_{t=1}^{T}p(y|x,w_{t}(\varphi))=\frac{1}{T}\sum_{t=1}^{T}f(x;w_{t}(\varphi))\sim\mathcal{N}(\hat{\mu};\hat{y},\hat{\sigma}^{2})\
,$ (6)

where $f(x;w_{t}(\varphi))$ denotes the prediction of the network parametrized
with $w_{t}(\varphi)$ on the instance $x$.

With abuse of notation, let us denote with $x$, $y$, $\hat{y}$ and
$\hat{\sigma}^{2}$ the _vectors_ of the instances, their ground truths, and
the corresponding predictions and uncertainty measures, respectively, of a
data set $\mathcal{D}$ of size $N$. It is easy to see that, as the dropout
rate changes, the prediction vector $\hat{y}=[\hat{y}_{1}\dots\hat{y}_{N}]$
can change, and as a consequence also the quadratic error vector
$\epsilon^{2}=(y-\hat{y})^{2}$ and the uncertainty vector
$\hat{\sigma}^{2}=[\hat{\sigma}_{1}^{2}\dots\hat{\sigma}_{N}^{2}]$ will
change. In particular, choosing the dropout rate $\Phi$ by solving the
optimization problem (4) amounts to pursuing two goals: (i) minimizing the
quadratic error $\epsilon_{n}^{2}=(y_{n}-\hat{y}_{n})^{2}$; (ii) minimizing
the absolute difference between the prediction’s quadratic error and the
variance, $|\epsilon_{n}^{2}-\hat{\sigma}_{n}^{2}|$. It is easy to show that
the “best” trade-off between them according to Eq. (4), i.e., the minimum NLL,
is achieved when the following condition holds (a simple proof is reported in
A):

$\hat{\sigma}^{2}_{n}=(y_{n}-\hat{y}_{n})^{2}\quad\forall n\in\mathcal{D}\ .$
(7)

A significant difference can now be noticed in the behavior of $\epsilon^{2}$
and $\hat{\sigma}^{2}$, as a function of the dropout rate $\varphi$, between
embedded and injected dropout. As we already pointed out, when injecting
dropout the choice of $\varphi$ does not affect the network weights $w$; in
embedded dropout, instead, changing $\varphi$ results in explicitly modifying
the vector $w$. Under the assumption of not having an under-parametrization,
the influence of $\varphi$ will lead to a regularization effect on $w$ when
using embedded dropout, but not when using injected dropout: since a network
with dropout regularization is usually more robust to dropping network
weights, it can be expected that a lower quadratic error is attained when
using embedded dropout. Moreover, when using injected dropout we argue that,
as $\varphi$ increases, prediction quality quickly deteriorates, i.e., the
quadratic error increases. This behavior penalizes injected dropout, limiting
to a tiny range $[0,\varphi_{\rm max}]$, for some $\varphi_{\rm max}$, the
values of dropout rate $\varphi$ that lead to acceptably small values of the
quadratic error $\epsilon^{2}$. However, for very small values of dropout
rate, the variability induced on the network predictions, and thus
$\hat{\sigma}^{2}$, are so small that the resulting NLL is very high, which in
turn is an indicator of poor quality of the uncertainty measure. Therefore,
the range of “useful” values of $\varphi$ further reduces to $[\varphi_{\rm
min},\varphi_{\rm max}]$, for some value $\varphi_{\rm min}$. We shall provide
experimental evidence of this behavior in Sect. 4.2. In the next section, we
propose to address the above issue of injected dropout through a suitable
rescaling of its uncertainty measure.

###  3.2 The Proposed Injection Method

The solution we propose to mitigate the above limitation of injected dropout
pointed out in Sect. 3.1 is summarized in Fig. 1 and consists in _rescaling_
the uncertainty measure $\hat{\sigma}^{2}$.

Rescaling the uncertainty measure – Let us consider a rescaled uncertainty
measure $\xi^{2}(\Phi)=C\cdot\hat{\sigma}^{2}(\Phi)$, where $C\in(0,+\infty)$.
A suitable value of $C$ can be sought by optimizing the NLL as a function of
$C$, besides $\Phi$, through the following modification of the optimization
problem (4):

$\Phi,C=\arg\min_{\varphi,c}\frac{1}{N}\sum_{n=1}^{N}\frac{1}{2}\frac{(y_{n}-\hat{y}_{n}(\varphi))^{2}}{c\cdot\hat{\sigma}_{n}^{2}(\varphi)}+\frac{1}{2}\log(c\cdot\hat{\sigma}_{n}^{2}(\varphi))\
.$ (8)

Figure 2: These plots (taken from experiments in Sect. 4.2) point out the
difference between the scale-agnostic optimization process of Eq. (4) (left)
and the scale-aware one of Eq. (8) (right). The latter can potentially lead to
different dropout rates with a lower NLL than the former, thanks to a
rescaling of the uncertainty measure.

We argue that such a scaled measure can lead to a much better NLL value than
the original formulation (4). To solve the optimization problem (8) it is
possible to exploit the results by Laves et al. [10], which show that, given a
trained regressor providing a prediction $\hat{y}$ and a corresponding
uncertainty measure $\hat{\sigma}^{2}$, one can further reduce the NLL with
respect to the solution of (3) by rescaling $\hat{\sigma}^{2}$ as
$C\cdot\hat{\sigma}^{2}$, and that the optimal value of the scale factor $C$,
for any _given_ quadratic error and uncertainty measure vectors
$\epsilon_{n}^{2}$ and $\hat{\sigma}^{2}_{n}$, is analytically given by:

$C=\frac{1}{N}\sum_{n=1}^{N}\frac{\epsilon_{n}^{2}}{\hat{\sigma}^{2}_{n}}.$
(9)

Accordingly, the optimal scale factor can be interpreted as an intrinsic
indicator of how much the magnitude of the uncertainty vector is close to the
magnitude of the quadratic error: if $C>1$, the uncertainty measure
$\hat{\sigma}_{n}$ is smaller, on average (over the $N$ samples), than the
quadratic error $\epsilon_{n}$; on the contrary, if $C<1$, then
$\hat{\sigma}_{n}$ is on average larger than $\epsilon_{n}$. Finally, since
the scale factor is analytically known for a given uncertainty measure, Eq.
(9) can be exploited to rewrite $C$ as a function of $\Phi$ in the
optimization problem (8), which can therefore be simplified as:

$\Phi=\arg\min_{\varphi}\frac{1}{N}\sum_{n=1}^{N}\frac{1}{2}\frac{(y_{n}-\hat{y}_{n}(\varphi))^{2}}{C(\varphi)\cdot\hat{\sigma}_{n}^{2}(\varphi)}+\frac{1}{2}\log(C(\varphi)\cdot\hat{\sigma}_{n}^{2}(\varphi))\
.$ (10)

Accordingly, both the dropout rate and the scale factor can be found by
optimizing the NLL only with respect to the dropout rate, similarly to the
original formulation (4).

When introducing a scale factor, the values of the unscaled uncertainty
measure $\hat{\sigma}_{n}^{2}$ do not need anymore to be close to the
corresponding quadratic error $\epsilon^{2}_{n}$ to keep the NLL small, and
are therefore not penalized as in the original formulation (4). We point out
that, although rescaling can also be applied to embedded dropout, it is much
more advantageous for injected dropout to mitigate the issue described in
Sect. 3.1, which instead does not affect embedded dropout to a significant
extent. In particular, such mitigation is expected to be more beneficial for
large data sets, complex tasks, and networks with a large number of weights:
we shall provide experimental evidence of this fact in Sect. 4.2. As shown in
Fig. 2, the proposed formulation (10) potentially allows one to explore
regions of the NLL surface (considered as a function of both the dropout rate
and the scale factor), that would otherwise be inaccessible by the original
formulation (4), leading to a better NLL.

Calibration of the uncertainty measure as a function of the scale factor – So
far, we have focused on NLL minimization, since this allows one to minimize
the distance between the actual prior distribution of the weights, $p(w)$, and
the approximated one, $q(w)$. However, the quality of an uncertainty measure
also depends on its _calibration_ , and minimizing NLL does not guarantee to
obtain a well-calibrated measure. Given the relevance of the scaling factor to
injected dropout, it is interesting to investigate how it affects the trade-
off between minimizing NLL and obtaining a well-calibrated measure. To this
aim, we first summarise the notion of calibration of an uncertainty measure
for regression problems. As mentioned in Sect. 2, the distribution
$p(y|x,\mathcal{D})$ is assumed to be Normal, $\mathcal{N}(\mu,\sigma^{2})$,
as well as its approximation in Eq. (5),
$\mathcal{N}(\hat{\mu},\hat{\sigma}^{2})$; to make the latter as close as
possible to the former, one minimizes the NLL on a validation set. Considering
$\mu$ and $\sigma^{2}$ as the point prediction and the corresponding
uncertainty for a given instance $(x,y)$, one can construct a confidence
interval with probability $\alpha$ for a generic sample $(x,y)$:

$CI_{\alpha}=[\mu-z_{\alpha}\cdot\sigma^{2},\mu+z_{\alpha}\cdot\sigma^{2}]\ ,$
(11)

which is characterized by the following:

$p(y\in CI_{\alpha})=\alpha,\quad\forall\alpha\in[0,1]\ .$ (12)

One would like that Eq. (12) holds for any instance $(x,y)$ also with respect
to a confidence interval $\hat{CI}_{\alpha}$ computed from the estimated
distribution:

$\hat{CI}_{\alpha}=[\hat{\mu}-z_{\alpha}\cdot\hat{\sigma}^{2},\hat{\mu}+z_{\alpha}\cdot\hat{\sigma}^{2}]\
.$ (13)

If so, the uncertainty measure $\hat{\sigma}^{2}$ is said to be perfectly
calibrated. This is, however, not guaranteed in practice.

To empirically evaluate the calibration of an uncertainty measure, the
probability $p(y\in CI_{\alpha})$ can be estimated on a data set (referred to
as validation [14] or calibration set [15]), as the fraction of instances
belonging to the confidence interval $\hat{CI}_{\alpha}$, for several values
of $\alpha$. We call this fraction the _observed frequency_ , and denote it by
$\pi(\alpha)$; ideally, $\pi(\alpha)$ should be equal to $\alpha$, which we
call the _expected frequency_. Calibration quality can therefore be evaluated
through a calibration curve, i.e., a plot of the observed frequency as a
function of the expected one, as in the example shown in Fig. 3. Note that a
perfect calibration corresponds to a calibration curve equal to the bisector
of the X-Y axes. When the observed frequency $\pi(\alpha)$ is below the
bisector, the uncertainty measure is _overconfident_ for the corresponding
values of $\alpha$ (because the expected frequency is greater than the
observed one); on the contrary, where $\pi(\alpha)$ is above the bisector, the
uncertainty measure is _underconfident_.111Note that the calibration curve can
cross the bisector for zero, one or more points, besides the ones
corresponding to $\alpha=0$ and $\alpha=1$.

Calibration quality can now be quantitatively evaluated as a scalar measure,
in terms of “how far” the calibration curve is from the ideal one. In this
work, we employ the area subtended by the calibration curve with respect to
the bisector, known in the literature by the name of Miscalibration Area (MA)
[16], which corresponds to the shaded regions in Fig. 3:

${\rm MA}=\int_{0}^{1}|\pi(\alpha)-\alpha|{\rm d}\alpha\ .$ (14)

In practice, the MA can be empirically estimated from a validation set as the
average difference in absolute value between the expected and the observed
frequency, over a set of $M$ values of $\alpha$:

${\rm MA}=\frac{1}{M}\sum_{m=1}^{M}|\pi(\alpha_{m})-\alpha_{m}|\ .$ (15)

As mentioned above, the scale factor $C$ obtained by solving problem (10) does
not guarantee that the corresponding MA is null, i.e., perfect calibration. To
analyze the trade-off between MA and NLL that can be obtained as a function of
the scale factor, we devised an iterative procedure that, starting from the
obtained scale factor $C$, modifies it to reduce MA as much as possible, by
increasing or reducing the confidence of each prediction of the calibration
set, depending on whether the uncertainty measure is over- or under-confident
on that prediction, respectively.

Our procedure (reported as Algorithm 1) aims at reducing to zero the average
difference between the expected and the observed frequency (which we refer to
as $Balance$) as a proxy for minimizing the MA. In such a setting, the value
of $Balance$ depends upon the chosen scale factor $c$; we denote as
$Balance_{c}$ such dependence, which is also reflected on the observed
frequency $\pi_{c}(\alpha)$. Since $Balance_{c}$ is a monotonically non-
decreasing function of $c$, one can empirically construct a suitable interval
$[C_{\rm low},C_{\rm high}]$ around the scaling factor $C$ obtained by solving
problem (10), such that $Balance_{C_{\rm low}}$ is negative, and
$Balance_{C_{\rm high}}$ is positive; then, our algorithm uses the bisection
method to find a “relaxed” scaling factor $C_{\rm r}$ such that
$Balance_{C_{\rm r}}=0$ (in practice, the objective is defined as
$Balance_{C_{\rm r}}<\tau$, where $\tau$ is a given tolerance threshold).

By analyzing the “path” of the scale factor between $C$ and $C_{\rm r}$, i.e.,
the corresponding values of NLL and ME, it is possible to explore the trade-
off between these two measures. Although acting on the scale factor does not
guarantee to reduce MA to arbitrarily small values (i.e., to approach perfect
calibration), empirical results reported in Sect. 4 show that small enough MA
values can nevertheless be obtained. Interestingly, our results also show that
the corresponding increase of NLL can be very limited: therefore, although
standard post-hoc techniques can be applied (e.g., Kuleshov calibration [15])
to enhance the calibration quality, the above result suggests that, when using
injected dropout, a good trade-off between minimizing NLL and obtaining a
well-calibrated uncertainty measure can also be achieved by first computing
the dropout rate and the corresponding scale factor through problem (10), and
then suitably adjusting the latter.

Algorithm 1 Modifying the scale factor to improve calibration

vectors of uncertainty measures $\hat{\sigma}^{2}$, network predictions
$\hat{y}$ and ground truth $y$ for a data set of $N$ instances; a suitable
interval $[C_{\rm low},C_{\rm high}]$ around the initial scaling factor $C$;
$M$ values $\alpha_{1},\ldots,\alpha_{M}$; tolerance threshold $\tau>0$

the relaxed scaling factor $C_{\rm r}$

do

$C_{\rm r}\leftarrow(C_{\rm low}+C_{\rm high})/2$

$Balance_{C_{\rm r}}\leftarrow\frac{1}{M}\sum_{m=1}^{M}(\pi_{C_{\rm
r}}(\alpha_{m})-\alpha_{m})$

if $Balance_{C_{\rm r}}<0$ then

$C_{\rm low}\leftarrow C_{\rm r}$

else if $Balance_{C_{\rm r}}>0$ then

$C_{\rm high}\leftarrow C_{\rm r}$

end if

while $Balance_{C_{\rm r}}\geq\tau$

Figure 3: Example of a calibration curve, which plots the observed frequency
of instances belonging to the confidence interval $CI_{\alpha}$ with
probability $\alpha$ as a function of the respective expected frequency. In
this example, an under-confidence and lower-confidence regions are present
(see the text for their meaning).

##  4 Experiments

We carried out two groups of experiments. We first compared the behavior of
embedded and injected dropout on eight data sets related to regression
problems taken from the UCI
repository222https://archive.ics.uci.edu/ml/datasets.php, using a multi-layer
feed-forward network as the regression model. In the second group of
experiments, we evaluated injected dropout on five benchmark data sets for
crowd counting [17], which is a challenging computer vision application using
a state-of-the-art CNN architecture for this task. In both experiments, we
implemented injected dropout by setting the dropout rate and scaling factor
according to Eq. (10), using a validation set; in the experiments on crowd
counting we also considered the relaxed scaling factor obtained as described
in Sect. 3.2 on the same validation set.

We evaluated both embedded and injected dropout, as a function of the dropout
rate, in terms of the three metrics considered in Sect. 3: root mean squared
error (RMSE), NLL, and MA. In particular, we evaluated NLL and MA using both
the unscaled values $\hat{\sigma}^{2}(\varphi)$ and the scaled ones
$\xi^{2}(\varphi)$; for reference, we also computed the “ideal” uncertainty
measure of Eq. (7), defined as the error vector $\epsilon^{2}$ computed on the
_validation set_

###  4.1 Experiments on UCI Data Sets

We selected eight data sets related to several multivariate regression
problems that have been used in the literature to evaluate uncertainty
measures: Protein-Tertiary-Structure, Boston-Housing, Concrete, Energy,
Kin8nm, Power-Plant, Wine-Quality-Red, and Yacht. We adopted the same set-up
proposed by Hernandez-Lobato et al. [18], described in the following. We also
open-sourced the code for the UCI experiment reproducibility at the following
link https://github.com/EmanueleLedda97/Dropout_Injection.

Experimental set-up – We first distinguished between the challenging Protein-
Tertiary-Structure data set and the remaining simpler ones. We used a one-
hidden-layer network with 50 hidden units for the latter and 100 for the
former. We randomly split each data set into a training set made up of 90% of
the original instances and a testing set containing the remaining 10%. This
procedure is repeated five times for Protein-Tertiary-Structure and 20 times
for the other data sets. In each split, we used 20% of the training partition
as a validation set, which is crucial not only for proper hyper-parameter
tuning but also – and primarily – for scaling the uncertainty measure.

Differently from the original set-up [18], we used a mini-batch stochastic
gradient descent; we empirically tuned the batch size, the learning rate, and
the number of epochs on the validation set. We considered $15$ possible
dropout rates within the interval $[0.001,0.5]$. We chose these rather extreme
values to properly investigate the issue mentioned in Sect. 3.2, where we
hypothesized a significant deterioration of the NLL of the unscaled measure
for such values of dropout rate.

Experimental results – Results are reported in Fig. 4.

Figure 4: RMSE and NLL as a function of dropout rate on UCI data sets, for
injected (blue) and embedded (red) dropout. Two NLL plots are reported for
each data set, related to different ranges of dropout rate: $[0.001,0.5]$ (top
plot, same range as the RMSE plot) and $[0.05,0.5]$ (bottom plot), to better
visualize the Y-axis scale.

We can first notice that the RMSE of embedded dropout tends to deteriorate as
the dropout rate increases. This behavior suggests that the network’s capacity
is very low and that the regularization effect of dropout leads to under-
fitting. This over-penalization makes injected dropout more accurate on
average, in terms of average RMSE and standard deviation, although a different
behavior can be observed on different data sets. Another phenomenon we can
observe is that, for the smallest dropout rates, the RMSE is low, but the
unscaled NLL is almost always severely affected. This trend reveals that – as
we hypothesized in Sect. 3.2 – unscaled uncertainty measures suffer from the
correlation between the dropout rate and the corresponding network
perturbation’s magnitude. When the dropout rate is extremely low, the final
perturbation (and therefore the uncertainty measure) is so small that it
completely disagrees with the quadratic prediction error, worsening the NLL.

Figure 5: NLL as a function of dropout rate on UCI data sets, for the scaled
(dotted line) and ideal (solid line) uncertainty measures, both for injected
(blue) and embedded (red) dropout.

Fig. 5 shows that the scaled measures exhibit significantly different behavior
instead. In this case, it is evident that even for extremely low dropout
rates, the NLL does not deteriorate as much as the unscaled one, in line with
our hypothesis in Sect. 3.2. Although the observed trends are always in favor
of the injected dropout, it may be possible that this behavior was due to the
use of a multi-layer network with a hidden layer of 50 or 100 units, and – as
we said before – there is clear evidence that using the dropout at training
time in this set-up leads to under-fitting the considered data sets. This
deterioration is also visible by looking at the ideal NLL: the solid red line
in Fig. 5 tends to be higher than the solid blue line, which means that the
ideal uncertainty measure (which is equal to the prediction error, for each
sample) is already penalized. Therefore, the observed phenomena are entirely
in line with our theoretical considerations in Sect. 3.2.

Looking at the calibration error in Fig. 6, we can see that dropout injection
can provide well-calibrated uncertainty measurements. Analyzing the trends of
the calibration error in detail, we can notice the following behaviors: first
of all, the unscaled measures are clearly more sensitive to changes in dropout
rate compared to their scaled counterparts. However, although the scaled
measures seem much more stable, there are cases where the unscaled measure of
embedded dropout performs comparably to or even outperforms the scaled one;
this phenomenon occurs more sporadically – and much less clearly – for
injected dropout. This is a crucial issue if we consider that, for the
corresponding dropout rates, a trade-off can be observed between RMSE, NLL,
and MA; this suggests that the embedded dropout inherently achieves a trade-
off between these metrics. A further investigation of this issue is an
interesting direction for future work.

Figure 6: Calibration error as a function of dropout rate on UCI data sets,
using the unscaled (dotted line) and scaled (solid line) uncertainty measures,
both for injected (blue) and embedded (red) dropout.

###  4.2 Experiments on Crowd Counting

The experiments done on UCI data sets provide useful insights about the
behavior of injected vs. embedded dropout and evidence that also the former
can be an effective approach to epistemic uncertainty quantification. In the
next experiments, we focused on a much more challenging computer vision
problem, i.e., crowd counting. State-of-the-art CNN models solve it by first
estimating the crowd density map, which is an image-to-image regression
problem.

Experimental set-up – We used three benchmark crowd counting data sets, UCSD
[19], Mall [20], and PETS [21]. For the latter, we used three different
partitions corresponding to different crowd scenes, named PETSview1,
PETSview2, and PETSview3 [22]. We used the Multi-Column Neural Network
architecture [23], a well-known state-of-the-art CNN for crowd counting. We
implemented dropout injection right before each convolutional layer, except
for the first one, following the suggestion of previous works on Monte Carlo
dropout [13]. Unlike the previous experiments, we found an exponential
increase in the RMSE using the same range of dropout rates. Therefore, we
followed the suggestion by Loquercio et al. [8] to use very small dropout
rates, which leads to a quasi-deterministic network. In particular, we
considered ten dropout rates in the interval $[0.001,0.01]$.

Experimental results – Fig. 7 reports the RMSE, NLL (scaled and unscaled), and
MA as a function of dropout rate. Since there is a significant difference in
the magnitude of the NLL for the unscaled measures and the scaled ones, we
reported them into two different plots.

Figure 7: RMSE, NLL, and calibration error of the injected dropout as a
function of dropout rate for the crowd counting data sets. Two plots are
reported for NLL, corresponding to the unscaled (top plot) and scaled (bottom
plot) uncertainty measure. The plots of scaled NLL and calibration error
report also a comparison with the ideal scale (the former) and with the
unscaled measure (the latter)

We can observe an increase in RMSE in three of five data sets, while for two
of them (PETSview1 and PETSview3), the RMSE decreased. This apparently
different behavior is due to the very small dropout rates used in these
experiments; we observed indeed that for higher dropout rates, the RMSE
started to increase also for PETSview1 and PETSview3.

Regarding the NLL, a large difference between the scaled and the unscaled
measures can be observed, even in this experiment: similarly to the UCI data
sets, the smallest dropout rates lead to a poor unscaled measure, whereas the
scaled measure attained considerably better results. It is also possible to
see how the best dropout rate values differ between scaled and unscaled
measures: for example, in UCSD, the minimum NLL is around $\Phi=0.04$ for the
scaled measure whereas, for the unscaled measure, it is around $\Phi=0.01$.
Similar considerations can be made for Mall. We can also notice that the
scaled measure is much more stable than the unscaled one, even with the
smallest dropout rates: this can be advantageous when using injected dropout,
as it allows one to make minimal changes to the network to obtain suitable
measures of uncertainty. Although re-scaling performs well in terms of the
NLL, the same does not hold for the calibration error: in this case, sub-
optimal performances can be observed, with – in most cases – a miscalibration
area greater than $0.3$, which is an indicator of poor calibration. Therefore,
in this case, a suitable trade-off between NLL and calibration error should be
sought, e.g., by tuning the scaling factor obtained from Eq. (10) through the
relaxation procedure presented in Sect. 3.2.

Figure 8: Trade-off between RMSE, NLL, and MA of the injected dropout, as a
function of dropout rate, on the crowd counting data sets, for the optimal
scaling (warm color gradient) and the relaxed one (cold color gradient).

Using a relaxed version of the scaling factor allows one to attain a trade-off
between RMSE, NLL, and MA, instead of minimizing only the NLL, by manually
tuning the scaling factor; Fig. 8 shows the value of the three analyzed
metrics as a function of dropout when using both $C$ (cold-colored gradient)
and $C_{\rm r}$ (warm-colored gradient). It is easy to notice that when using
$C_{r}$, the NLL is not far from the values obtained using $C$: this suggests
that the relaxed measure behaves as a reasonable compromise between NLL and
MA.

Finally, the calibration curves in Fig. 9 clearly show that the proposed
relaxation procedure effectively reduces the calibration error. In all cases,
the relaxed curve is much closer to the axes’ bisector, supporting our
hypothesis in Sect. 3.2.

Figure 9: Calibration curves of injected dropout on the crowd counting data
sets, attained using the unscaled measure (blue), the optimal scaling factor
$C$ (red), the relaxation $C_{r}$ (orange). For reference, the perfect
calibration line is also shown (dotted line).

Practical utility of uncertainty quantification by dropout injection in video
surveillance applications – We first show how the operator of a video
surveillance system can benefit from updating its crowd density estimator
software by integrating an uncertainty estimate through dropout injection
(i.e., without requiring further training), and then how uncertainty
evaluation could be automatically exploited to improve the accuracy of the
estimated density maps and of the corresponding people count.

Crowd counting by density estimation uses density maps as ground truth: the
standard method for obtaining such maps starts by annotating each image
location where one person’s head is present, then the density map is obtained
by adding a kernel (e.g., Gaussian) with a unit area centered on each head
location. Therefore, the people count corresponds to the sum of each pixel of
the density map. In this context, a crowd counting predictor will output an
estimated density map reconstructing the original one. The image-to-image
nature of this task, similar to what happens in Monocular Depth Estimation and
Semantic Segmentation, implies that also the uncertainty estimate consists of
a 2D array, associating a level of uncertainty to each pixel-wise predicted
density (i.e., its variance). In turn, the pixel-wise uncertainty on the
predicted density allows the construction of a confidence interval on the
people count, as seen in the examples in Fig. 10, similarly to [17], but in
our case, in a post hoc fashion. Having a notion of the confidence level
associated with system predictions makes video surveillance operators aware of
their reliability, allowing them to make more informed decisions and
increasing their trust in the prediction system compared to a point estimator.

An in-depth analysis of our experimental results on crowd counting benchmarks
reveals that uncertainty evaluation could also be helpful to detect some
inaccuracies in the estimated density maps that correspond to “false positive”
pedestrian localization, i.e., image regions where the predicted density is
not zero (thus contributing to the pedestrian count) despite the absence of
pedestrians. As seen from the examples in Fig. 10, we observed that false
positive localizations are often characterized by isolated regions with
relatively low estimated density and high uncertainty. This suggests that they
could be detected and automatically corrected (by setting to zero the
corresponding values in the density map and disregarding their contribution to
the people count), thus improving the accuracy of both the density map and the
people count. As an example, a simple policy for automatic rejection of such
regions may consist in rejecting isolated regions with high pixel-level
uncertainty/prediction ratio, but other solutions are conceivable. Regardless
of the policy, it is intriguing to see that such automatic rejection
mechanisms are theoretically possible and practically applicable by
integrating dropout injection in the toolbox of video surveillance software. A
further investigation of this issue is an interesting topic for future work.

Figure 10: Two examples of “false positive” detections enabled by uncertainty
estimation on crowd-counting images (top: UCSD, bottom: PETSview2). From left
to right: the original frame, ground truth, predicted density map, and
uncertainty map. Each density map is normalized: darker pixels on the gradient
palette ( for ground truth and predicted maps,  for the uncertainty maps)
correspond to lower values. “False positive” regions, corresponding to low
predicted density and high uncertainty are highlighted with red boxes.

##  5 Conclusions

This work provides the first investigation of injected dropout for epistemic
uncertainty evaluation in order to assess if and how this method can be a
practical alternative to the current use of embedded dropout which needs time-
consuming uncertainty-aware training (i.e., should be designed at training
time). Our analysis focused on regression problems and encompassed two
different dimensions related to the behavior of injected dropout, namely
prediction accuracy, measured in terms of RMSE, and quality of the uncertainty
measure, evaluated in terms of NLL and MA (i.e., calibration).

We analyzed, theoretically and empirically, how prediction error and quality
of the uncertainty measure behave as a function of the dropout rate and
proposed a novel injection method grounded on the results of our analysis.

Experimental results on several benchmark data sets, including a challenging
computer vision task, provided clear evidence that dropout injection can be an
effective alternative to embedded dropout, also considering that it does not
require a computationally costly re-training procedure of already-trained
networks. Our results also confirm that the effectiveness of injected dropout
critically depends on a suitable re-scaling of its uncertainty measure.

We point out two limitations of our analysis, that can be addressed in future
work: (i) We considered only regression problems; an obvious extension is to
analyze dropout injection for classification problems. (ii) We considered the
same dropout rate across different network layers; however, our analysis can
be easily extended to the case of different dropout rates across the layers,
which is known to be potentially beneficial to Monte Carlo Dropout [13].

Finally, another interesting issue for future work is to investigate the
effectiveness of injected dropout for detecting out-of-distribution samples,
which is potentially useful under domain shifts.

## Acknowledgments

This work was partially supported by the projects: “Law Enforcement agencies
human factor methods and Toolkit for the Security and protection of CROWDs in
mass gatherings” (LETSCROWD), EU Horizon 2020 programme, grant agreement No.
740466; “IMaging MAnagement Guidelines and Informatics Network for law
enforcement Agencies” (IMMAGINA), European Space Agency, ARTES Integrated
Applications Promotion Programme, contract No. 4000133110/20/NL/AF; “Science
and engineering Of Security of Artificial Intelligence” (S.O.S. AI) included
in the Spoke 3 - Attacks and Defences of the Research and Innovation Program
PE00000014, “SEcurity and RIghts in the CyberSpace (SERICS)”, under the
National Recovery and Resilience Plan, Mission 4 ”Education and Research” -
Component 2 ”From Research to Enterprise” - Investment 1.3, funded by the
European Union - NextGenerationEU.

Emanuele Ledda is affiliated with the Italian National PhD in Artificial
Intelligence, Sapienza University of Rome. He also acknowledges the
cooperation with and the support from the Pattern Recognition and Applications
Lab of the University of Cagliari.

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##  Appendix A Proof of Eq. (7)

In the following we prove that, given a data set
$\mathcal{D}=\\{(x_{n},y_{n})\\}_{n=1}^{N}$ for a regression problem, and
denoting the corresponding predictions of a given regressor by $\hat{y}_{n}$,
the ideal uncertainty measure $\sigma^{*2}$, i.e., the one that minimizes the
NLL, is obtained when the following condition holds:

$\sigma^{*2}_{n}=(y_{n}-\hat{y}_{n})^{2},\quad n=1,\ldots,n\ .$ (16)

Consider any instance $x_{n}\in\mathcal{D}$. The derivative of the NLL with
respect to the corresponding $\sigma^{2}_{n}$ is:

$\displaystyle\frac{\partial\left(\frac{1}{2}\cdot\frac{(y_{n}-\hat{y}_{n})^{2}}{\sigma^{2}_{n}}+\frac{1}{2}\cdot\log(\sigma^{2}_{n})\right)}{\partial\sigma^{2}_{n}}=$
(17)
$\displaystyle\frac{1}{2}\cdot\left((y_{n}-\hat{y_{n}})^{2}\cdot\frac{\partial(\frac{1}{\sigma^{2}_{n}})}{\partial\sigma^{2}_{n}}+\frac{\partial(\log(\sigma^{2}_{n}))}{\partial\sigma^{2}_{n}}\right)=$
(18)
$\displaystyle\frac{1}{2}\cdot\left(\frac{-(y_{n}-\hat{y}_{n})^{2}}{(\sigma^{2}_{n})^{2}}+\frac{1}{\sigma^{2}_{n}}\right)=$
(19)
$\displaystyle\frac{\sigma^{2}_{n}-(y_{n}-\hat{y}_{n})^{2}}{2(\sigma^{2}_{n})^{2}}$
(20)

Setting the above derivative to $0$ and solving for $\sigma^{2}_{n}$ we
obtain:

$\frac{\sigma^{2}_{n}-(y_{n}-\hat{y}_{n})^{2}}{2(\sigma^{2}_{n})^{2}}=0\ ,$
(21)

$\sigma^{2}_{n}=(y_{n}-\hat{y}_{n})^{2},\quad\sigma^{2}_{n}\neq 0$ (22)