# Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods
For Uncertainty Estimation

Dennis Ulmer<EMAIL_ADDRESS>
Christian Hardmeier<EMAIL_ADDRESS>
Jes Frellsen,*<EMAIL_ADDRESS>
IT University of Copenhagen, Technical University of Denmark, *Pioneer Centre
for Artificial Intelligence

###### Abstract

Popular approaches for quantifying predictive uncertainty in deep neural
networks often involve distributions over weights or multiple models, for
instance via Markov Chain sampling, ensembling, or Monte Carlo dropout. These
techniques usually incur overhead by having to train multiple model instances
or do not produce very diverse predictions. This comprehensive and extensive
survey aims to familiarize the reader with an alternative class of models
based on the concept of _Evidential Deep Learning_ : For unfamiliar data, they
aim to admit “what they don’t know”, and fall back onto a prior belief.
Furthermore, they allow uncertainty estimation in a single model and forward
pass by parameterizing _distributions over distributions_. This survey
recapitulates existing works, focusing on the implementation in a
classification setting, before surveying the application of the same paradigm
to regression. We also reflect on the strengths and weaknesses compared to
other existing methods and provide the most fundamental derivations using a
unified notation to aid future research.

##  1 Introduction

Figure 1: Taxonomy of surveyed approaches, divided into tractable
parameterizations of the prior or posterior on one axis (see Tables 1 and 2
for an overview) and into approaches for classification and regression on the
other. Regression methods are outlined in Table 3.

Many existing methods for uncertainty estimation leverage the concept of
Bayesian model averaging: These include ensembling (Lakshminarayanan et al.,
2017; Wilson & Izmailov, 2020), Markov chain Monte Carlo sampling (de Freitas,
2003; Andrieu et al., 2000) as well as variational inference approaches
(Mackay, 1992; MacKay, 1995; Hinton & Van Camp, 1993; Neal, 2012), including
approaches such as Monte Carlo (MC) dropout (Gal & Ghahramani, 2016) and
Bayes-by-backprop (Blundell et al., 2015). Bayesian model averaging for neural
networks usually involves the approximation of an otherwise infeasible
integral using MC samples. This causes the following problems: Firstly, the
quality of the MC approximation depends on the veracity and diversity of
samples from the weight posterior. Secondly, the approach often involves
increasing the number of parameters in a model or training more model
instances altogether. Recently, a new class of models has been proposed to
side-step this conundrum by using a different factorization of the posterior
predictive distribution. This allows computing uncertainty in a single forward
pass and with a single set of weights. These models are grounded in a concept
coined _Evidential Deep Learning_ : For out-of-distribution (OOD) inputs, they
are encouraged to fall back onto a prior. This is often described as _knowing
what they don’t know_.

In this paper, we summarize the existing literature and provide an overview of
Evidential Deep Learning approaches. We give an overview over all discussed
work in Figure 1, where we distinguish surveyed works for classification
between models parameterizing a Dirichlet prior (Section 3.4.1) or posterior
(Section 3.4.2). We further discuss similar methods for regression problems
(Section 4). As we will see, obtaining well-behaving uncertainty estimates can
be challenging in the Evidential Deep Learning framework; proposed solutions
that are also reflected in Figure 1 are the usage of OOD examples during
training (Malinin & Gales, 2018; 2019; Nandy et al., 2020; Shen et al., 2020;
Chen et al., 2018; Zhao et al., 2019; Hu et al., 2021; Sensoy et al., 2020),
knowledge distillation (Malinin et al., 2020b; a) or the incorporation of
density estimation (Charpentier et al., 2020; 2022; Stadler et al., 2021),
which we discuss in more detail in Section 6. This survey aims to both serve
as an accessible introduction to this model family to the unfamiliar reader as
well as an informative overview, in order to promote a wider application
outside the uncertainty quantification literature. We also provide a
collection of the most important derivations for the Dirichlet distribution
for Machine Learning, which plays a central role in many of the discussed
approaches.

##  2 Background

We first introduce the central concepts for this survey, including Bayesian
inference in Section 2.1, Bayesian model averaging in Section 2.2 and
Evidential Deep Learning in Section 2.3.111Note that in the following we will
use the suggested notation of the TMLR journal, e.g. by using $P$ for
probability mass and $p$ for probability density functions.

###  2.1 Bayesian Inference

The foundation of the following sections is Bayesian inference: Given some
prior belief $p(\bm{\theta})$ about parameters of interest $\bm{\theta}$, we
use available observations $\mathbb{D}=\\{(x_{i},y_{i})\\}_{i=1}^{N}$ and
their likelihood $p(\mathbb{D}|\bm{\theta})$ to obtain an updated belief in
form of the posterior $p(\bm{\theta}|\mathbb{D})\propto
p(\mathbb{D}|\bm{\theta})p(\bm{\theta})$. This update rule is derived from
Bayes’ rule, namely

$p(\bm{\theta}|\mathbb{D})=\frac{p(\mathbb{D}|\bm{\theta})p(\bm{\theta})}{p(\mathbb{D})}=\frac{p(\mathbb{D}|\bm{\theta})p(\bm{\theta})}{\int
p(\mathbb{D}|\bm{\theta})p(\bm{\theta})d\bm{\theta}},$ (1)

where we often try to avoid computing the term in the denominator since
marginalization over a large (continuous) parameter space of $\bm{\theta}$ is
usually intractable. In order to perform a prediction $y$ for a new data point
$\mathbf{x}$, we can now utilize the _posterior predictive distribution_
defined as

$P(y|\mathbf{x},\mathbb{D})=\int
P(y|\mathbf{x},\bm{\theta})p(\bm{\theta}|\mathbb{D})d\bm{\theta}.$ (2)

Since we integrate over the entire parameter space of $\bm{\theta}$, weighting
each prediction by the posterior probability of its parameters to obtain the
final result, this process is referred to as _Bayesian model averaging_ (BMA).
Here, predictions $P(y|\mathbf{x},\bm{\theta})$ stemming from parameters that
are plausible given the observed data will receive a higher weight
$p(\bm{\theta}|\mathbb{D})$ in the final prediction
$P(y|\mathbf{x},\mathbb{D})$. As we will see in the following section, this
factorization of the predictive predictive distribution also has beneficial
properties for analyzing the uncertainty of a model.

###  2.2 Predictive Uncertainty in Neural Networks

In probabilistic modelling, uncertainty is commonly divided into aleatoric and
epistemic uncertainty (Der Kiureghian & Ditlevsen, 2009; Kendall & Gal, 2017;
Hüllermeier & Waegeman, 2021). Aleatoric uncertainty refers to the uncertainty
that is induced by the data-generating process, for instance noise or inherent
overlap between observed instances of classes. Epistemic uncertainty is the
type of uncertainty about the optimal model parameters (or even hypothesis
class). It is reducible with an increasing amount of data, as fewer and fewer
possible models become a plausible fit. These two notions resurface when
formulating the posterior predictive distribution for a new data point
$\mathbf{x}$:222Note that the predictive distribution in Equation 2
generalizes the common case for a single network prediction where
$P(y|\mathbf{x},\mathbb{D})\approx P(y|\mathbf{x},\hat{\bm{\theta}})$.
Mathematically, this is expressed by replacing the posterior
$p(\bm{\theta}|\mathbb{D})$ by a Dirac delta distribution as in Equation 5,
where all probability density rests on a single parameter configuration.

$P(y|\mathbf{x},\mathbb{D})=\int\underbrace{P(y|\mathbf{x},\bm{\theta})}_{\text{Aleatoric}}\underbrace{p(\bm{\theta}|\mathbb{D})}_{\text{Epistemic}}d\bm{\theta}.$
(3)

Here, the first factor captures the aleatoric uncertainty about the correct
prediction, while the second one expresses uncertainty about the correct model
parameters—the more data we observe, the more density of
$p(\bm{\theta}|\mathbb{D})$ should lie on reasonable parameter values for
$\bm{\theta}$. For high-dimensional real-valued parameters $\bm{\theta}$ like
in neural networks, this integral becomes intractable, and is usually
approximated using Monte Carlo samples:333For easier distributions, the
integral can often be evaluated analytically exploiting conjugacy. Another
approach for more complex distributions can be the method of moments (see e.g.
Duan, 2021).

$P(y|\mathbf{x},\mathbb{D})\approx\frac{1}{K}\sum_{k=1}^{K}P(y|\mathbf{x},\bm{\theta}^{(k)});\quad\bm{\theta}^{(k)}\sim
p(\bm{\theta}|\mathbb{D})$ (4)

based on $K$ different sets of parameters $\bm{\theta}^{(k)}$. Since this
requires obtaining multiple versions of model parameters through some
additional procedure, this however comes with the aforementioned problems of
computational overhead and approximation errors, motivating the approaches
discussed in this survey.

###  2.3 Evidential Deep Learning

Since the traditional approach to predictive uncertainty estimation requires
multiple parameter sets and can only approximate the predictive posterior, we
can factorize Equation 2 further to obtain a tractable form:

$\displaystyle p(y|\mathbf{x},\mathbb{D})$
$\displaystyle=\iint\underbrace{P(y|\bm{\pi})}_{\vphantom{\big{[}}\text{Aleatoric}}\underbrace{p(\bm{\pi}|\mathbf{x},\bm{\theta})}_{\vphantom{\big{[}}\text{\
Distributional\
}}\underbrace{p(\bm{\theta}|\mathbb{D})}_{\vphantom{\big{[}}\text{Epistemic}}d\bm{\pi}d\bm{\theta}\approx\int
P(y|\bm{\pi})\underbrace{\vphantom{\big{[}}p(\bm{\pi}|\mathbf{x},\hat{\bm{\theta}})}_{p(\bm{\theta}|\mathbb{D})\approx\delta(\bm{\theta}-\hat{\bm{\theta}})}d\bm{\pi}.$
(5)

This factorization contains another type of uncertainty, which Malinin & Gales
(2018) call the _distributional_ uncertainty, uncertainty caused by the
mismatch of training and test data distributions. In the last step, Malinin &
Gales (2018) replace $p(\bm{\theta}|\mathbb{D})$ by a point estimate
$\hat{\bm{\theta}}$ using the Dirac delta function, i.e. a single trained
neural network, to get rid of the intractable integral. Although another
integral remains, retrieving the uncertainty from this predictive distribution
actually has a closed-form analytical solution for the Dirichlet (see Section
3.3). The advantage of this approach is further that it allows us to
distinguish uncertainty about a data point because it is ambiguous from points
coming from an entirely different data distribution. As an example, consider a
binary classification problem, in which the data manifold consists of two
overlapping clusters. As we are classifying a new data point, we obtain a
distribution $P(y|\mathbf{x},\bm{\theta})$ which is uniform over the two
classes. What does this mean? The model might either be confident that the
point lies in the region of overlap and is inherently ambiguous, or that the
model is uncertain about the correct class. Without further context, we cannot
distinguish between these two cases (Bengs et al., 2022; Hüllermeier, 2022).
Compare that to instead predicting $p(\bm{\pi}|\mathbf{x},\bm{\theta})$: If
the data point is ambiguous, the resulting distribution will be centered on
$0.5$, if the model is generally uncertain, the distribution will be uniform,
allowing this distinction. We will illustrate this principle further in the
upcoming Sections 2.4 and 3.3.

In the neural network context in Equation 5, it should be noted that
restricting oneself to a point estimate of the parameters prevent the
estimation of epistemic uncertainty like in earlier works through the weight
posterior $p(\bm{\theta}|\mathbb{D})$, as discussed in the next section.
However, there are works like Haussmann et al. (2019); Zhao et al. (2020) that
combine both approaches.

The term _Evidential Deep Learning_ (EDL) originates from the work of Sensoy
et al. (2018) and is based on the _Theory of Evidence_ (Dempster, 1968; Audun,
2018): Within the theory, belief mass is assigned to set of possible states,
e.g. class labels, and can also express a lack of evidence, i.e. an “I don’t
know”. We can for instance generalize the predicted output of a neural
classifier using the Dirichlet distribution, allowing us to express a lack of
evidence through a uniform Dirichlet. This is different from a uniform
Categorical distribution, which does not distinguish an equal probability for
all classes from the lack of evidence. For the purpose of this survey, we
define Evidential Deep Learning as a family of approaches in which a neural
network can fall back onto a uniform prior for unknown inputs. While neural
networks usually parameterize likelihood functions, approaches in this survey
parameterize prior or posterior distributions instead. The advantages of this
methodology are now demonstrated using the example in the following section.

###  2.4 An Illustrating Example: The Iris Dataset

(a) _Iris setosa_

(b) _Iris versicolor_

(c) _Iris virginica_

Figure 2: Illustration of different approaches to uncertainty quantifying on
the Iris dataset, with examples for the classes given on the left (Figures
2(a), 2(b) and 2(c)). On the right, the data is plotted alongside some
predictions of a prior network (lighter colors indicate higher density) and an
ensemble and MC Dropout model on the probability simplex, with $50$
predictions each. Iris images were taken from Wikimedia Commons, 2022a; b; c.

To illustrate the advantages of EDL, we choose a classification problem based
on the Iris dataset (Fisher, 1936). It contains measurements of three
different species of iris flowers (shown in Figures 2(a), 2(b) and 2(c)). We
use the dataset as made available through scikit-learn (Pedregosa et al.,
2011) and plot the relationship between the width and lengths measurements of
the flowers’ petals in Figure 2.

We train an deep neural network ensemble (Lakshminarayanan et al., 2017) with
$50$ model instances, a model with MC Dropout (Gal & Ghahramani, 2016) with
$50$ predictions and a prior network (Sensoy et al., 2018), an example of EDL,
on all available data points, and plot their predictions on three test points
on the 3-probability simplex in Figure 2.444For information about training and
model details, see Section A.1. On these simplices, each point signifies a
Categorical distribution, with the proximity to one of the corners indicating
a higher probability for the corresponding class. EDL methods for
classification do not predict a single output distribution, but an entire
_density over output distributions_.

Test point 3 lies in a region of overlap between instances of _Iris
versicolor_ and _Iris virginica_ , thus inducing high aleatoric uncertainty.
In this case, we can see that the prior network places all of its density on
between these two classes, similar to most of the predictions of the ensemble
and MC Dropout (bottom right). However, some of the latter predictions still
land in the center of the simplex. The point 1 is located in an area without
training examples between instances of _Iris versicolor_ and _setosa_ , as
well as close to a single _virginica_ outlier. As shown in the top left,
ensemble and MC Dropout predictions agree that the point belongs to either the
_setosa_ or _versicolor_ class, with a slight preference for the former. The
prior network concentrates its prediction on _versicolor_ , but admits some
uncertainty towards the two other choices. The last test point 2 is placed in
an area of the feature space devoid of any data, roughly equidistant from the
three clusters of flowers. Similar to the previous example, the ensemble and
MC dropout predictions on the top right show a preference for _Iris setosa_
and _versicolor_ , albeit with higher uncertainy. The prior network however
shows an almost uniform density, admitting distributional uncertainty about
this particular input.

This simple example provides some insights into the potential advantages of
EDL: First of all, the prior network was able to provide reasonable
uncertainty estimates in comparison with BMA methods. Secondly, the prior
network is able to admit its lack of knowledge for the OOD data point by
predicting an almost uniform prior, something that the other models are not
able to. As laid out in Section 3.3, EDL actually allows the user to
disentangle model uncertainty due to a simple lack of data and due to the
input being out-of-distribution. Lastly, training the prior network only
required a single model, which is a noticeable speed-up compared to MC Dropout
and especially the training of ensembles.

##  3 Evidential Deep Learning for Classification

In order to introduce EDL methods for classification, we first give a brief
introduction to the Dirichlet distribution and its role as a conjugate prior
in Bayesian inference in Section 3.1. We then show in Section 3.2 how neural
networks can parameterize Dirichlet distributions, while Section 3.3 reveals
how such a parameterization can be exploited for efficient uncertainty
estimation. The remaining sections enumerate different examples from the
literature parameterizing either a prior (Section 3.4.1) or posterior
Dirichlet distribution (Section 3.4.2).

###  3.1 The Dirichlet distribution

Figure 3: A prior Dirichlet distribution is updated with a vector of class
observations. The posterior Dirichlet then shifts density towards the classes
$k$ with more observed instances.

Modelling for instance a binary classification problem is commonly done using
the Bernoulli likelihood. The Bernoulli likelihood has a single parameter
$\pi$, indicating the probability of success (or of the positive class), and
is given by

$\text{Bernoulli}(y|\pi)=\pi^{y}(1-\pi)^{(1-y)}.$ (6)

Within Bayesian inference as introduced in Section 2, the Beta distribution is
a commonly used prior for a Bernoulli likelihood. It defines a probability
distribution over the parameter $\pi$, itself possessing two shape parameters
$\alpha_{1}$ and $\alpha_{2}$:

$\text{Beta}(\pi;\alpha_{1},\alpha_{2})=\frac{1}{B(\alpha_{1},\alpha_{2})}\pi^{\alpha_{1}-1}(1-\pi)^{\alpha_{2}-1};\quad
B(\alpha_{1},\alpha_{2})=\frac{\Gamma(\alpha_{1})\Gamma(\alpha_{2})}{\Gamma(\alpha_{1}+\alpha_{2})};$
(7)

where $\Gamma(\cdot)$ stands for the gamma function, a generalization of the
factorial to the real numbers, and $B(\cdot)$ is called the Beta function (not
to be confused with the distribution). When extending the classification
problem from two to an arbitrary number of classes, we use a Categorical
likelihood:

$\text{Categorical}(y|\bm{\pi})=\prod_{k=1}^{K}\pi_{k}^{\mathbf{1}_{y=k}},$
(8)

in which $K$ denotes the number of categories or classes, and the class
probabilities are expressed using a vector $\bm{\pi}\in[0,1]^{K}$with
$\sum_{k}\pi_{k}=1$, and $\mathbf{1}_{(\cdot)}$ is the indicator function.
This distribution appears for instance in classification problems when using
neural networks, since most neural networks for classification use a softmax
function after their last layer to produce a Categorical distribution of
classes s.t. $\pi_{k}\equiv P(y=k|x)$. In this setting, the Dirichlet
distribution arises as a suitable prior and multivariate generalization of the
Beta distribution (and is thus also called the _multivariate Beta
distribution_):

$\text{Dir}(\bm{\pi};\bm{\alpha})=\frac{1}{B(\bm{\alpha})}\prod_{k=1}^{K}\pi_{k}^{\alpha_{k}-1};\quad
B(\bm{\alpha})=\frac{\prod_{k=1}^{K}\Gamma(\alpha_{k})}{\Gamma(\alpha_{0})};\quad\alpha_{0}=\sum_{k=1}^{K}\alpha_{k};\quad\alpha_{k}\in\mathbb{R}^{+};$
(9)

where the Beta function $B(\cdot)$ is now defined for $K$ shape parameters
compared to Equation 7. For notational convenience, we also define
$\mathbb{K}=\\{1,\ldots,K\\}$ as the set of all classes. The distribution is
characterized by its _concentration parameters_ $\bm{\alpha}$, the sum of
which, often denoted as $\alpha_{0}$, is called the _precision_.555The
precision is analogous to the precision of a Gaussian, where a larger
$\alpha_{0}$ signifies a sharper distribution. The Dirichlet is a _conjugate
prior_ for such a Categorical likelihood, meaning that according to Bayes’
rule in Equation 1, they produce a Dirichlet posterior with parameters
$\bm{\beta}$, given a data set $\mathbb{D}=\\{(x_{i},y_{i})\\}_{i=1}^{N}$ of
$N$ observations with corresponding labels:

$\displaystyle p(\bm{\pi}|\mathbb{D},\bm{\alpha})$ $\displaystyle\propto
p\big{(}\\{y_{i}\\}_{i=1}^{N}|\bm{\pi},\\{x_{i}\\}_{i=1}^{N}\big{)}p(\bm{\pi}|\bm{\alpha})=\prod_{i=1}^{N}\prod_{k=1}^{K}\pi_{k}^{\mathbf{1}_{y_{i}=k}}\frac{1}{B(\bm{\alpha})}\prod_{k=1}^{K}\pi_{k}^{\alpha_{k}-1}$
(10)
$\displaystyle=\prod_{k=1}^{K}\pi_{k}^{\big{(}\sum_{i=1}^{N}\mathbf{1}_{y_{i}=k}\big{)}}\frac{1}{B(\bm{\alpha})}\prod_{k=1}^{K}\pi_{k}^{\alpha_{k}-1}=\frac{1}{B(\bm{\alpha})}\prod_{k=1}^{K}\pi_{k}^{N_{k}+\alpha_{k}-1}\propto\text{Dir}(\bm{\pi};\bm{\beta}),$

where $\bm{\beta}$ is a vector with $\beta_{k}=\alpha_{k}+N_{k}$, with $N_{k}$
denoting the number of observations for class $k$. Intuitively, this implies
that the prior belief encoded by the initial Dirichlet is updated using the
actual data, sharpening the distribution for classes for which many instances
have been observed. Similar to the Beta distribution in Equation 7, the
Dirichlet is a _distribution over Categorical distributions_ on the $K-1$
probability simplex; we show an example with its concentration parameters and
the Bayesian update in Figure 3.

###  3.2 Parameterization

For a classification problem with $K$ classes, a neural classifier is usually
realized as a function
$f_{\bm{\theta}}:\mathbb{R}^{D}\rightarrow\mathbb{R}^{K}$, mapping an input
$\mathbf{x}\in\mathbb{R}^{D}$ to _logits_ for each class. Followed by a
softmax function, this then defines a Categorical distribution over classes
with a vector $\bm{\pi}$ with $\pi_{k}\equiv p(y=k|\mathbf{x},\bm{\theta})$.
The same underlying architecture can be used without any major modification to
instead parameterize a _Dirichlet_ distribution, predicting a distribution
_over Categorical distributions_ $p(\bm{\pi}|\mathbf{x},\hat{\bm{\theta}})$ as
in Equation 9.666The only thing to note here is that the every $\alpha_{k}$
has to be strictly positive, which can for instance be enforced by using an
additional softplus, exponential or ReLU function (Sensoy et al., 2018;
Malinin & Gales, 2018; Sensoy et al., 2020). In order to classify a data point
$\mathbf{x}$, a Categorical distribution is created from the predicted
concentration parameters of the Dirichlet as follows (this corresponds to the
mean of the Dirichlet, see Section C.1):

$\bm{\alpha}=\exp\big{(}f_{\bm{\theta}}(\mathbf{x})\big{)};\quad\pi_{k}=\frac{\alpha_{k}}{\alpha_{0}};\quad\hat{y}=\operatorname*{arg\,max}_{k\in\mathbb{K}}\
\pi_{1},\ldots,\pi_{K}.$ (11)

Parameterizing a Dirichlet posterior distribution follows a similar logic, as
we will discuss in Section 3.4.2.

###  3.3 Uncertainty Estimation with Dirichlet Networks

(a) Categorical distributions predicted by a neural ensemble on the
probability simplex.

(b) Probability simplex for a confident prediction, for with the density
concentrated in a single corner.

(c) Dirichlet distribution for a case of data uncertainty, with the density
concentrated in the center.

(d) Dirichlet distribution for a case of model uncertainty, with the density
spread out more.

(e) Dirichlet for a case of distributional uncertainty, with the density
spread across the whole simplex.

(f) Alternative approach to distributional uncertainty called representation
gap, with density concentrated along the edges.

Figure 4: Examples of the probability simplex for a $K=3$ classification
problem, where every corner corresponds to a class and every point to a
Categorical distribution. Brighter colors correspond to higher density. (a)
Predicted Categorical distributions by an ensemble of discriminators. (b) –
(e) (Desired) Behavior of Dirichlet in different scenarios by Malinin & Gales
(2018): (b) For a confident prediction, the density is concentrated in the
corner of the simplex corresponding to the assumed class. (c) In the case of
aleatoric uncertainty, the density is concentrated in the center, and thus
uniform Categorical distributions are most likely. (d) In the case of model
uncertainty, the density may still be concentrated in a corner, but more
spread out, expressing the uncertainty about the right prediction. (e) In the
case of an OOD input, a uniform Dirichlet expresses that any Categorical
distribution is equally likely, since there is no evidence for any known
class. (f) Representation gap by Nandy et al. (2020), proposed as an
alternative behavior for OOD data. Here, the density is instead concentrated
solely on the edges of the simplex.

Let us now turn our attention to how to estimate the aleatoric, epistemic and
distributional uncertainty as laid out in Section 2.2 within the Dirichlet
framework. In Figure 4, we show different shapes of a Dirichlet distribution
parameterized by a neural network, corresponding to different cases of
uncertainty, where each point on the simplex represents a Categorical
distribution, with proximity to a corner indicating a high probability for the
corresponding class. Figure 4(a) displays the predictions of an ensemble of
classifiers as a point cloud on the simplex. Using a Dirichlet, this finite
set of distributions can be extended to a continuous density over the whole
simplex. As we will see in the following sections, parameterizing a Dirichlet
distribution with a neural network enables us to distinguish different
scenarios using the shape of its density, as shown in Figures 4(b), 4(c),
4(d), 4(e) and 4(f), which we will discuss in more detail along the way.

However, since we do not want to inspect Dirichlets visually, we instead use
closed form expression to quantify uncertainty, which we will discuss now.
Although stated for the prior parameters $\bm{\alpha}$, the following methods
can also be applied to the posterior parameters $\bm{\beta}$ without loss of
generality.

##### Data (aleatoric) uncertainty

To obtain a measure of data uncertainty, we can evaluate the expected entropy
of the data distribution $p(y|\bm{\pi})$ (similar to previous works like e.g.
Gal & Ghahramani, 2016). As the entropy captures the “peakiness” of the output
distribution, a lower entropy indicates that the model is concentrating most
probability mass on a single class, while high entropy characterizes a more
uniform distribution—the model is undecided about the right prediction. For
Dirichlet networks, this quantity has a closed-form solution (for the full
derivation, refer to Section D.1):

$\mathbb{E}_{#1}\bigg{[}H\Big{[}P(y|\bm{\pi})\Big{]}\bigg{]}=-\sum_{k=1}^{K}\frac{\alpha_{k}}{\alpha_{0}}\bigg{(}\psi(\alpha_{k}+1)-\psi(\alpha_{0}+1)\bigg{)}$
(12)

where $\psi$ denotes the digamma function, defined as
$\psi(x)=\frac{d}{dx}\log\Gamma(x)$, and $H$ the Shannon entropy.

##### Model (epistemic) uncertainty

As we saw in Section 2.2, most approaches in the Dirichlet framework avoid the
intractable integral over network parameters $\bm{\theta}$ by using a point
estimate $\hat{\bm{\theta}}$.777With exceptions such as Haussmann et al.
(2019); Zhao et al. (2020). When the distribution over parameters in Equation
5 is retained, alternate expressions of the aleatoric and epistemic
uncertainty are derived by Woo (2022). This means that computing the model
uncertainty via the weight posterior $p(\bm{\theta}|\mathbb{D})$ like in
Blundell et al. (2015); Gal & Ghahramani (2016); Smith & Gal (2018) is not
possible. Nevertheless, a key property of Dirichlet networks is that epistemic
uncertainty is expressed in the spread of the Dirichlet distribution (for
instance in Figure 4 (d) and (e)). Therefore, the epistemic uncertainty can be
quantified considering the concentration parameters $\bm{\alpha}$ that shape
this distribution: Charpentier et al. (2020) simply consider the maximum
$\alpha_{k}$ as a score akin to the maximum probability score by Hendrycks &
Gimpel (2017), while Sensoy et al. (2018) compute it by
$K/\sum_{k=1}^{K}(\alpha_{k}+1)$ or simply $\alpha_{0}$ (Charpentier et al.,
2020). In both cases, the underlying intuition is that larger $\alpha_{k}$
produce a sharper density, and thus indicate increased confidence in a
prediction.

##### Distributional uncertainty

Another appealing property of this model family is being able to distinguish
uncertainty due to model underspecification (Figure 4(d)) from uncertainty due
to unknown inputs (Figure 4(e)). In the Dirichlet framework, the
distributional uncertainty can be quantified by computing the difference
between the total amount of uncertainty and the data uncertainty, which can be
expressed through the mutual information between the label $y$ and its
Categorical distribution $\bm{\pi}$:

$I\Big{[}y,\bm{\pi}\Big{|}\mathbf{x},\mathbb{D}\Big{]}=\underbrace{H\bigg{[}\mathbb{E}_{#1}\Big{[}P(y|\bm{\pi})\Big{]}\bigg{]}}_{\text{Total
Uncertainty}}-\underbrace{\mathbb{E}_{#1}\bigg{[}H\Big{[}P(y|\bm{\pi})\Big{]}\bigg{]}}_{\text{Data
Uncertainty}}$ (13)

This quantity expresses how much information we would receive about $\bm{\pi}$
if we were given the label $y$, conditioned on the new input $\mathbf{x}$ and
the training data $\mathbb{D}$. In regions in which the model is well-defined,
receiving $y$ should not provide much new information about $\bm{\pi}$—and
thus the mutual information would be low. Yet, such knowledge should be very
informative in regions in which few data have been observed, and there this
mutual information would indicate higher distributional uncertainty. Given
that $\mathbb{E}_{#1}[\pi_{k}]=\frac{\alpha_{k}}{\alpha_{0}}$ (Section C.1)
and assuming the point estimate $p(\bm{\pi}|\mathbf{x},\mathbb{D})\approx
p(\bm{\pi}|\mathbf{x},\hat{\bm{\theta}})$ to be sufficient (Malinin & Gales,
2018), we obtain an expression very similar to Equation 12:

$I\Big{[}y,\bm{\pi}\Big{|}\mathbf{x},\mathbb{D}\Big{]}=-\sum_{k=1}^{K}\frac{\alpha_{k}}{\alpha_{0}}\bigg{(}\log\frac{\alpha_{k}}{\alpha_{0}}-\psi(\alpha_{k}+1)+\psi(\alpha_{0}+1)\bigg{)}$
(14)

##### Note on epistemic uncertainty estimation

The introduction of distributional uncertainty, a notion that is non-existent
in the Bayesian Model Averaging framework, warrants a note on the estimation
of epistemic uncertainty in general. Firstly, since we often use the point
estimate
$p(\bm{\theta}|\mathbb{D})\approx\delta(\bm{\theta}-\hat{\bm{\theta}})$ from
Equation 5 in Evidential Deep Learning, model uncertainty usually is no longer
estimated via the uncertainty in the weight posterior, but instead through the
parameters of the prior or posterior distribution. Furthermore, even though
they appear similar, distributional uncertainty is different from epistemic
uncertainty, since it is the uncertainty in the distribution
$p(\bm{\pi}|\mathbf{x},\bm{\theta})$. Distinguishing epistemic from
distributional uncertainty also allows us to differentiate uncertainty due to
underspecification from uncertainty due to a lack of evidence. In BMA, these
notions are indistinguishable: In theory, model uncertainty on OOD data should
be high since the model is underspecified on them, however theoretical and
empirical work has shown this is not always the case (Ulmer et al., 2020;
Ulmer & Cinà, 2021; Van Landeghem et al., 2022). Even then, the additive
decomposition of the mutual information has been critized since the model will
also have a great deal of _uncertainty about its aleatoric uncertainty_ in the
beginning of the training process (Hüllermeier, 2022), and thus this
decomposition might not be accurate. Furthermore, even when we obtain the best
possible model within its hypothesis class, using the discussed methods it is
impossible to estimate uncertainty induced by a misspecified hypothesis class.
This can motivate approaches in which a second, auxiliary model directly
predicts model uncertainty of a target model (Lahlou et al., 2022; Zerva et
al., 2022).

###  3.4 Existing Approaches for Dirichlet Networks

Being able to quantify aleatoric, epistemic and distributional uncertainty in
a single forward pass and in closed form are desirable traits, as they
simplify the process of obtaining different uncertainty scores. However, it is
important to note that the behavior of the Dirichlet distributions in Figure 4
is idealized. In the usual way of training neural networks through empirical
risk minimization, Dirichlet networks are not incentivized to behave in the
depicted way. Thus, when comparing existing approaches for parameterizing
Dirichlet priors in Section 3.4.1 and posteriors in Section 3.4.2,888Even
though the term _prior_ and _posterior network_ were coined by Malinin & Gales
(2018) and Charpentier et al. (2020) for their respective approaches, we use
them in the following as an umbrella term for all methods targeting a prior or
posterior distribution. we mainly focus on the different ways in which authors
try to tackle this problem by means of loss functions and training procedures.
We give an overview over the discussed works in Tables 1 and 2 in these
respective sections. For additional details, we refer the reader to Appendix C
for general derivations concerning the Dirichlet distribution. We dedicate
Appendix D to derivations of the different loss functions and regularizers and
give a detailed overview over their mathematical forms in Appendix E.
Available code repositories for all works surveyed are listed in Section A.2.

####  3.4.1 Prior Networks

Table 1: Overview over prior networks for classification. $(*)$ OOD samples
were created inspired by the approach of Liang et al. (2018). ID: Using in-
distribution data samples.

Method | Loss function | Architecture |  OOD-free training?
---|---|---|---
Prior network (Malinin & Gales, 2018) |  ID KL w.r.t smoothed label & OOD KL w.r.t. uniform prior | MLP / CNN | ✗
Prior networks (Malinin & Gales, 2019) |  Reverse KL of Malinin & Gales (2018) | CNN | ✗
Information Robust Dirichlet Networks (Tsiligkaridis, 2019) |  $l_{p}$ norm w.r.t one-hot label & Approx. Rényi divergence w.r.t. uniform prior | CNN | ✓
Dirichlet via Function Decomposition (Biloš et al., 2019) |  Uncertainty Cross-entropy & mean & variance regularizer | RNN | ✓
Prior network with PAC Regularization (Haussmann et al., 2019) |  Negative log-likelihood loss + PAC regularizer | BNN | ✓
Ensemble Distribution Distillation (Malinin et al., 2020b) | Knowledge distillation objective | MLP / CNN | ✓
Self-Distribution Distillation (Fathullah & Gales, 2022) | Knowledge distillation objective | CNN | ✓
Prior networks with representation gap (Nandy et al., 2020) |  ID & OOD Cross-entropy + precision regularizer | MLP / CNN | ✗
Prior RNN (Shen et al., 2020) | Cross-entropy + entropy regularizer | RNN | (✗)∗
Graph-based Kernel Dirichlet distribution estimation (GKDE) (Zhao et al., 2020) |  $l_{2}$ norm w.r.t. one-hot label & KL reg. with node-level distance prior & Knowledge distillation objective | GNN | ✓

The key challenge in training Dirichlet networks is to ensure both high
classification performance and the intended behavior under OOD inputs. For
this reason, most discussed works follow a loss function design using two
parts: One optimizing for task accuracy to achieve the former goal, the other
optimizing for a flat Dirichlet distribution, as flatness suggests a lack of
evidence. To enforce flatness, the predicted Dirichlet is compared to a
uniform distribution using some probabilistic divergence measure. We divide
prior networks into two groups: Approaches using additional OOD data for this
purpose (_OOD-dependent approaches_), and those which do not required OOD data
(_OOD-free approaches_), as listed in Table 1.

##### OOD-free approaches

Apart from a standard negative log-likelihood loss (NLL) as used by Haussmann
et al. (2019), one simple approach to optimizing the model is to impose a
$l_{p}$-loss between the one-hot encoding $\mathbf{y}$ of the original label
$y$ and the Categorical distribution $\bm{\pi}$. Tsiligkaridis (2019) show
that since the values of $\bm{\pi}$ depend directly on the predicted
concentration parameters $\bm{\alpha}$, a generalized loss can be derived to
be upper-bounded by the following expression (see the full derivation given in
Section D.3):

$\mathbb{E}_{#1}\big{[}||\mathbf{y}-\bm{\pi}||_{p}\big{]}\leq\bigg{(}\frac{\Gamma(\alpha_{0})}{\Gamma(\alpha_{0}+p)}\bigg{)}^{\frac{1}{p}}\bBigg@{4}(\frac{\Gamma\Big{(}\sum_{k\neq
y}\alpha_{k}+p\Big{)}}{\Gamma\Big{(}\sum_{k\neq
y}\alpha_{k}\Big{)}}+\sum_{k\neq
y}\frac{\Gamma(\alpha_{k}+p)}{\Gamma(\alpha_{k})}\bBigg@{4})^{\frac{1}{p}}$
(15)

Since the sum over concentration parameters excludes the one corresponding to
the true label, this loss can be seen as reducing the density on the areas of
the probability simplex that do not correspond to the target class. Sensoy et
al. (2018) specifically utilize the $l_{2}$ loss, which has the following form
(see Section D.4):

$\mathbb{E}_{#1}\Big{[}||\mathbf{y}-\bm{\pi}||_{2}^{2}\Big{]}=\sum_{k=1}^{K}\Big{(}\mathbf{1}_{y=k}-\frac{\alpha_{k}}{\alpha_{0}}\Big{)}^{2}+\frac{\alpha_{k}(\alpha_{0}-\alpha_{k})}{\alpha_{0}^{2}(\alpha_{0}+1)}$
(16)

where $\mathbf{1}_{(\cdot)}$ denotes the indicator function. Since
$\alpha_{k}/\alpha_{0}\leq 1$, we can see that the term with the indicator
functions penalizes the network when the concentration parameter $\alpha_{k}$
corresponding to the correct label does not exceed the others. The remaining
aspect lies in the regularization: To achieve reliable predictive uncertainty,
the density associated with incorrect classes should be reduced. One such
option is to decrease the Kullback-Leibler divergence from a uniform Dirichlet
(see Section C.3):

$\text{KL}\Big{[}p(\bm{\pi}|\bm{\alpha})\Big{|}\Big{|}p(\bm{\pi}|\bm{1})\Big{]}=\log\frac{\Gamma(K)}{B(\bm{\alpha})}+\sum_{k=1}^{K}(\alpha_{k}-1)\big{(}\psi(\alpha_{k})-\psi(\alpha_{0})\big{)}$
(17)

In the case of Zhao et al. (2020), who apply their model to graph structures,
they do not decrease the divergence from a uniform Dirichlet, but incorporate
information about the local graph neighborhood into the reference distribution
by considering the distance from and label of close nodes.999They also add
another knowledge distillation term (Hinton et al., 2015) to their loss, for
which the model tries to imitate the predictions of a vanilla Graph Neural
Network that functions as the teacher network. Nevertheless, the KL-divergence
w.r.t. a uniform Dirichlet is used by many of the following works. Other
divergence measures are also possible: Tsiligkaridis (2019) instead use a
local approximation of the Rényi divergence.101010The Kullback-Leibler
divergence can be seen as a special case of the Rényi divergence (van Erven &
Harremoës, 2014), where the latter has a stronger information-theoretic
underpinning. First, the concentration parameter for the correct class
$\alpha_{y}$ is removed from the Dirichlet by creating
$\tilde{\bm{\alpha}}=(1-\mathbf{y})\cdot\bm{\alpha}+\mathbf{y}$. Then, the
remaining concentration parameters are pushed towards uniformity by the
divergence measure, which can be derived to be

$\text{R\'{e}nyi}\Big{[}p(\bm{\pi}|\tilde{\bm{\alpha}})\Big{|}\Big{|}p(\bm{\pi}|\bm{1})\Big{]}\approx\frac{1}{2}\Big{[}\sum_{k\neq
y}\big{(}\alpha_{k}-1\big{)}^{2}\big{(}\psi^{(1)}(\alpha_{j})-\psi^{(1)}(\tilde{\alpha}_{0})\big{)}-\psi^{(1)}(\tilde{\alpha}_{0})\sum_{\begin{subarray}{c}k\neq
k^{\prime}\\\ k\neq y,\ k^{\prime}\neq
y\end{subarray}}\big{(}\alpha_{k}-1\big{)}\big{(}\alpha_{k^{\prime}}-1\big{)}\Big{]}$
(18)

where $\psi^{(1)}$ denotes the first-order polygamma function, defined as
$\psi^{(1)}(x)=\frac{d}{dx}\psi(x)$. Since the sums ignore the concentration
parameter of the correct class, only the ones of the incorrect classes are
penalized. Haussmann et al. (2019) derive an entirely different regularizer
using Probably Approximately Correct (PAC) bounds from learning theory, that
together with the negative log-likelihood gives a proven bound to the expected
true risk of the classifier. Setting a scalar $\delta$ allows one to set the
desired risk, i.e. the model’s expected risk is guaranteed to be the same or
less than the derived PAC bound with a probability of $1-\delta$. For a
problem with $N$ available training data points, the following _upper bound_
is presented:

$\sqrt{\frac{\text{KL}\big{[}p(\bm{\pi}|\bm{\alpha})\big{|}\big{|}p(\bm{\pi}|\mathbf{1})\big{]}-\log\delta}{N}-1}.$
(19)

This upper bound is then used as the actual regularizer term in practice. We
see that even from the learning-theoretic perspective, this method follows the
intuition of the original KL regularizer in a shifted and scaled form.
Haussmann et al. (2019) also admit that in this form, the regularizer does not
allow for a direct PAC interpretation anymore, since its approximates only
admits a loose bound on the risk. Yet, they demonstrate its usefulness in
their experiments. Summarizing all of the presented approaches thus far, we
can see that they try to force the model to concentrate the Dirichlet’s
density solely on the parameter corresponding to the right label—expecting a
more flat density for difficult or unknown inputs.

##### Knowledge distillation

A way to avoid the use of OOD examples while still using external information
for regularization is to use _knowledge distillation_ (Hinton et al., 2015).
Here, the core idea lies in a student model learning to imitate the
predictions of a more complex teacher model. Malinin et al. (2020b) exploit
this idea and show that prior networks can also be distilled using an ensemble
of classifiers and their predicted Categorical distributions (akin to learning
Figure 4(e) from Figure 4(a)), which does not require regularization at all,
but comes at the cost of having to train an entire ensemble a priori. Trying
to solve this shortcoming, Fathullah & Gales (2022) propose to use a shared
feature extractor between the student and the teacher network. Instead of
training an ensemble, diverse predictions are obtained from the teacher
network through the use of Gaussian dropout, which are distilled into a
Dirichlet distribution as in Malinin et al. (2020b).

##### OOD-dependent approaches

A uniform Dirichlet in the face of unknown inputs can also be achieved
explicitly by training with OOD inputs and learning to be uncertain on them.
We discuss a series of works utilizing this direction next. Malinin & Gales
(2018) simply minimize the KL divergence to a uniform Dirichlet on OOD data
points. This way, the model is encouraged to be agnostic about its prediction
in the face of unknown inputs. Further, instead of an $l_{p}$ norm, they
utilize another KL term to train the model on predicting the correct label,
minimizing the distance between the predicted concentration parameters and the
true label. However, since only a gold _label_ and not a gold _distribution_
is available, they create one by re-distributing some of the density from the
correct class onto the rest of the simplex (see Appendix E for full form). In
their follow-up work, Malinin & Gales (2019) argue that the asymmetry of the
KL divergence as the main objective creates undesirable properties in
producing the correct behavior of the predicted Dirichlet, since it creates a
multi- instead of unimodal target distribution. They therefore propose to use
the reverse KL instead (see Section D.5 for the derivation), which enforces
the desired unimodal target. Nandy et al. (2020) refine this idea further,
stating that even with reverse KL training high epistemic and high
distributional uncertainty (Figures 4(d) and 4(e)) might be confused, and
instead propose novel loss functions producing a _representation gap_ (Figure
4(f)), which aims to be more easily distinguishable. In this case, spread out
densities signify epistemic uncertainty, whereas densities concentrated
entirely on the edges of the simplex indicate distributional uncertainty. The
way they achieve this goal is two-fold: In addition to minimizing the negative
log-likelihood loss on in-domain and maximizing the entropy on OOD examples,
they also penalize the precision of the Dirichlet (see Appendix E for full
form). Maximizing the entropy on OOD examples hereby serves the same function
as minimizing the KL w.r.t. to a uniform distribution, and can be implemented
using the closed-form solution in Section C.2:

$H\big{[}p(\bm{\pi}|\bm{\alpha})\big{]}=\log
B(\bm{\alpha})+(\alpha_{0}-K)\psi(\alpha_{0})-\sum_{k=1}^{K}(\alpha_{k}-1)\psi(\alpha_{k})$
(20)

##### Sequential models

We also have identified two sequential applications of prior networks in the
literature: For Natural Language Processing, Shen et al. (2020) train a
recurrent neural network for spoken language understanding using a simple
cross-entropy loss. Instead of using OOD examples for training, they maximize
the entropy of the model on data inputs given a learned, noisy version of the
predicted concentration parameters. In comparison, Biloš et al. (2019) apply
their model to asynchronous event classification and note that the standard
cross-entropy loss only involves a point estimate of a Categorical
distribution, discarding all the information contained in the predicted
Dirichlet. For this reason, they propose an _uncertainty-aware_ cross-entropy
(UCE) loss instead, which has a closed-form solution in the Dirichlet case
(see Section D.6)

$\mathcal{L}_{\text{UCE}}=\psi(\alpha_{y})-\psi(\alpha_{0}),$ (21)

with $\psi$ referring to the digamma function. By mimizing the difference
between the digamma values of $\alpha_{y}$ and $\alpha_{0}$, the model learns
to concentrate density on the correct class. Since their final concentration
parameters are created using additional information from a class-specific
Gaussian process, they further regularize the mean and variance for OOD data
points using an extra loss term, incentivizing a loss mean and a variance
corresponding to a pre-defined hyperparameter.

####  3.4.2 Posterior Networks

Table 2: Overview over posterior networks for classification. OOD data is
created using $(\dagger)$ the fast-sign gradient method (Kurakin et al.,
2017), a $(\ddagger)$ Variational Auto-Encoder (VAE; Kingma & Welling, 2014)
or $(\mathsection)$ a Wasserstein GAN (WGAN; Arjovsky et al., 2017). NLL:
Negative log-likelihood. CE: Cross-entropy.

Method | Loss function | Architecture |  OOD-free training?
---|---|---|---
Evidential Deep Learning (Sensoy et al., 2018) |  $l_{2}$ norm w.r.t. one-hot label + KL w.r.t. uniform prior | CNN | ✓
Regularized ENN (Zhao et al., 2019) |  $l_{2}$ norm w.r.t. one-hot label + Uncertainty regularizer on OOD/ difficult samples | MLP / CNN | ✗
WGAN–ENN (Hu et al., 2021) |  $l_{2}$ norm w.r.t. one-hot label + Uncertainty regularizer on synth. OOD |  MLP / CNN + WGAN | (✗)§
Variational Dirichlet (Chen et al., 2018) |  ELBO + Contrastive Adversarial Loss | CNN | (✗)†
Dirichlet Meta-Model (Shen et al., 2022) |  ELBO + KL w.r.t. uniform prior | CNN | ✓
Belief Matching (Joo et al., 2020) |  ELBO | CNN | ✓
Posterior Networks (Charpentier et al., 2020) |  Uncertainty CE (Biloš et al., 2019) \+ Entropy regularizer |  MLP / CNN + Norm. Flow | ✓
Graph Posterior Networks (Stadler et al., 2021) | Same as Charpentier et al. (2020) | GNN | ✓
Generative Evidential Neural Networks (Sensoy et al., 2020) |  Contrastive NLL + KL between uniform & Dirichlet of wrong classes | CNN | (✗)‡

As elaborated on in Section 3.1, choosing a Dirichlet prior, due to its
conjugacy to the Categorical distribution, induces a Dirichlet posterior
distribution. Like the prior before, surveyed works listed in Table 2
parameterize the posterior with a neural network. The challenges hereby are
two-fold: Accounting for the number of class observations $N_{k}$ that make up
part of the posterior density parameters $\bm{\beta}$ (Equation 10), and,
similarly to prior networks, ensuring the wanted behavior on the probability
simplex for in- and out-of-distribution inputs. Sensoy et al. (2018) base
their approach on the Dempster-Shafer theory of evidence (Yager & Liu, 2008;
lending its name to the term “Evidential Deep Learning”) and its formalization
via subjective logic (Audun, 2018), where subjective beliefs about
probabilities are expressed through Dirichlet distributions. In doing so, an
agnostic belief in form of a uniform Dirichlet prior $\forall k:\alpha_{k}=1$
is updated using pseudo-counts $N_{k}$, which are predicted by a neural
network. This is different from prior networks, where the prior concentration
parameters $\bm{\alpha}$ are predicted instead. In both cases, this does not
require any modification to a model’s architecture except for replacing the
softmax output function by a ReLU (or similar). Sensoy et al. (2018) for
instance train their model using the same techniques presented in the previous
section: The main objective is the $l_{2}$ loss, penalizing the difference
between the predicted Dirichlet and the one-hot encoded class label (Section
D.4), and the KL divergence from a uniform Dirichlet is used for
regularization.

##### Generating OOD samples using generative models

Since OOD examples are not always readily available, several works try to
create artificial samples using deep generative models. Hu et al. (2021) train
a Wasserstein GAN (Arjovsky et al., 2017) to generate OOD samples, on which
the network’s uncertainty is maximized. The uncertainty is given through
_vacuity_ , defined as $K/\sum_{k}\beta_{k}$. The vacuity compares a uniform
prior belief against the amassed evidence $\sum_{k}\beta_{k}$, and thus is $1$
when there is no additonal evidence available. In a follow-up work, Sensoy et
al. (2020) similarly train a model using a contrastive loss with artificial
OOD samples from a Variational Autoencoder (Kingma & Welling, 2014), and a KL-
based regularizer similar to that of Tsiligkaridis (2019), where the density
for posterior concentration parameters $\beta_{k}$ that do not correspond to
the true label are pushed to the uniform distribution.

(a) Posterior Network (Charpentier et al., 2020).

(b) Natural Posterior Network (Charpentier et al., 2022).

Figure 5: Schematic of the Posterior Network and Natural Posterior Network,
taken from Charpentier et al. (2020; 2022), respectively. In both cases, an
encoder $f_{\bm{\theta}}$ maps inputs to a latent representation $\mathbf{z}$.
NFs then model the latent densities, which are used together with the prior
concentration to produce the posterior parameters. In (a), the latent
representation of $\mathbf{x}^{(1)}$ lies right in the modelled density of the
first class, and thus receives a confident prediction. The latent
$\mathbf{z}^{(2)}$ lies between densities, creating aleatoric uncertainty.
$\mathbf{x}^{(3)}$ is an OOD input, is mapped to a low-density area of the
latent space and thus produces an uncertain prediction. The differences in the
two approaches is that the Posterior Network in (a) uses one NF per class,
while only one NF is used in (b). Furthermore, (b) constitutes a
generalization to different exponential family distributions, and is not
restricted to classification problems (see main text for more detail).

##### Posterior networks via Normalizing Flows

Charpentier et al. (2020) also set $\bm{\alpha}$ to a uniform prior, but
obtain the pseudo-observations $N_{k}$ in a different way: Instead of a model
predicting them directly, $N_{k}$ is determined by the number of examples of a
certain class in the training set. This quantity is further modified in the
following way: An encoder model $f_{\bm{\theta}}$ produces a latent
representation $\mathbf{z}$ of some input. A (class-specific) normalizing
flow111111A NF is a generative model, estimating a density in the feature
space by mapping it to a Gaussian in a latent space by a series of invertible,
bijective transformations. The probability of an input can then be estimated
by calculating the probability of its latent encoding under that Gaussian and
applying the change-of-variable formula, traversing the flow in reverse.
Instead of mapping from the feature space into latent space, the flows in
Charpentier et al. (2020) map from the encoder latent space into a separate,
second latent space. (NF; Rezende & Mohamed, 2015) with parameters $\bm{\phi}$
then assigns a probability to this latent representation, which is used to
weight $N_{k}$:

$\beta_{k}=\alpha_{k}+N_{k}\cdot
p(\mathbf{z}|y=k,\bm{\phi});\quad\mathbf{z}=f_{\bm{\theta}}(\mathbf{x}).$ (22)

This has the advantage of producing low probabilities for strange inputs like
the noise as depicted in Figure 5(a), which in turn translate to low
concentration parameters of the posterior Dirichlet, as it falls back onto the
uniform prior. The model is optimized using the same uncertainty-aware cross-
entropy loss as in Biloš et al. (2019) with an additional entropy regularizer,
encouraging density only around the correct class. This scheme is also applied
to Graph Neural Networks by Stadler et al. (2021): In order to take the
neighborhood structure of the graph into account, the authors also use a
Personalized Page Rank scheme to diffuse node-specific posterior parameters
$\bm{\beta}$ between neighboring nodes. The Page Rank scores, reflecting the
importance of a neighboring node to the current node, can be approximated
using power iteration (Klicpera et al., 2019) and used to aggregate the
originally predicted concentration parameters $\bm{\beta}$ on a per-node
basis.

A generalization of the posterior network method to exponential family
distributions is given by Charpentier et al. (2022). Akin to the update for
the posterior Dirichlet parameters, the authors formulate a general Bayesian
update rule as

$\bm{\chi}_{i}^{\text{post}}=\frac{n^{\text{prior}}\bm{\chi}^{\text{prior}}+n_{i}\bm{\chi}_{i}}{n^{\text{prior}}+n_{i}};\quad\mathbf{z}_{i}=f_{\bm{\theta}}(\mathbf{x}_{i});\quad
n_{i}=N\cdot
p(\mathbf{z}|\bm{\phi});\quad\bm{\chi}_{i}=g_{\bm{\psi}}(\mathbf{x}_{i}).$
(23)

$\bm{\chi}$ here denotes the parameters of the exponential family distribution
and $n$ the evidence. Thus, posterior parameters for a sample $\mathbf{x}_{i}$
are obtained by updating the prior parameter and some prior evidence by some
input-dependent pseudo-evidence $n_{i}$ and parameters $\bm{\chi}_{i}$: Again,
given a latent representation by an encoder $\mathbf{z}$, a (this time single)
normalizing flow predicts $n_{i}=N_{H}\cdot p(\mathbf{z}|\bm{\phi})$ based on
some pre-defined certainty budget $N_{H}$.121212The certainty budget can
simply be set to the number of available datapoints, however Charpentier et
al. (2022) suggest to set it to $\log
N_{H}=\frac{1}{2}\big{(}H\log(2\pi)+\log(H+1)\big{)}$ to better scale with the
dimensionality $H$ of the latent space. The update parameters $\bm{\chi}_{i}$
are predicted by an additional network
$\bm{\chi}_{i}=g_{\bm{\psi}}(\mathbf{z})$, see Figure 5(b). For
classification, $n^{\text{prior}}=1$ and $\bm{\chi}^{\text{prior}}$
corresponds to the uniform Dirichlet, while $\bm{\chi}_{i}$ are concentration
parameters predicted by an output layer based on the latent encoding. For
unfamiliar inputs, this method will again result in a small pseudo-evidence
term $n_{i}$, reflecting high model uncertainty. Since the generalization to
the exponential family implies the application of this scheme to normal
distributions, we will discuss the same method applied to regression in the
next section.

##### Posterior networks via variational inference

Another route lies in directly parameterizing the posterior parameters
$\bm{\beta}$. Given a target distribution defined by a uniform Dirichlet prior
plus the number of times an input is associated with a specific label, Chen et
al. (2018) optimize a distribution matching objective, i.e. the KL-divergence
between the posterior parameters predicted by a neural network and the target
distribution. Since this objective is intractable to optimize directly, this
leaves us to instead model an _approximate posterior_ using variational
inference methods. As the KL divergence between the true and approximate
posterior is infeasible to estimate, variational methods usually optimize the
_evidence lower bound_ (ELBO):

$\mathcal{L}_{\text{ELBO}}=\underbrace{\psi(\beta_{y})-\psi(\beta_{0})}_{\text{UCE
loss}}-\underbrace{\log\frac{B(\bm{\beta})}{B(\bm{\gamma})}+\sum_{k=1}^{K}(\beta_{k}-\gamma_{k})\big{(}\psi(\beta_{k})-\psi(\beta_{0})\big{)}}_{\text{KL-
divergence}}$ (24)

in which we can identify to consist of the uncertainty-aware cross-entropy
(UCE) loss used by Biloš et al. (2019); Charpentier et al. (2020; 2022) and
the KL-divergence between two Dirichlets (Section C.3). This approach is also
employed by Joo et al. (2020), Chen et al. (2018) and Shen et al. (2022),
while the latter predict posterior parameters based on the activations of
different layers of a pre-trained feature extractor.

##  4 Evidential Deep Learning for Regression

Table 3: Overview over Evidential Deep Learning methods for regression.

Method |  Parameterized distribution | Loss function | Model
---|---|---|---
Deep Evidential Regression (Amini et al., 2020) |  Normal-Inverse Gamma Prior |  Negative log-likelihood loss + KL w.r.t. uniform prior | MLP / CNN
Deep Evidential Regression with Multi-task Learning (Oh & Shin, 2022) |  Normal-Inverse Gamma Prior |  Like Amini et al. (2020), with additional Lipschitz-modified MSE loss | MLP / CNN
Multivariate Deep Evidential Regression (Meinert & Lavin, 2021) |  Normal-Inverse Wishart Prior |  Like Amini et al. (2020), but tying two predicted params. instead of using a regularizer | MLP
Regression Prior Network (Malinin et al., 2020a) | Normal-Wishart Prior |  Reverse KL (Malinin & Gales, 2019) | MLP / CNN
Natural Posterior Network (Charpentier et al., 2022) |  Inverse-$\chi^{2}$ Posterior |  Uncertainty Cross-entropy (Biloš et al., 2019) \+ Entropy regularizer |  MLP / CNN + Norm. Flow

Because the EDL framework provides convenient uncertainty estimation, the
question naturally arises of whether it can be extended to regression problems
as well. The answer is affirmative, although the Dirichlet distribution is not
an appropriate choice in this case. It is very common to model a regression
problem using a normal likelihood (Bishop, 2006; Chapter 3.3). As such, there
are multiple potential choices for a prior distribution. The methods listed in
Table 3 either choose the Normal-Inverse Gamma distribution (Amini et al.,
2020; Charpentier et al., 2022), inducing a scaled inverse-$\chi^{2}$
posterior (Gelman et al., 1995),131313The form of the Normal-Inverse Gamma
posterior and the Normal Inverse-$\chi^{2}$ posterior are interchangable using
some parameter substitutions (Murphy, 2007). or a Normal-Wishart prior
(Malinin et al., 2020a). We will discuss these approaches in turn.

Figure 6: Example of an application of Evidential Deep Learning for
regression, taken from Amini et al. (2020). The neural network predicts an
Normal Inverse-Gamma prior, whose corresponding normal likelihoods display
decreasing variance (and thus uncertainty) in the face of stronger evidence.

##### Univariate regression

Amini et al. (2020) model the regression problem as a normal distribution with
unknown mean and variance $\mathcal{N}(y;\pi,\sigma^{2})$, and use a normal
prior for the mean with $\pi\sim\mathcal{N}(\gamma,\sigma^{2}\nu^{-1})$ and an
inverse Gamma prior for the variance with
$\sigma^{2}\sim\Gamma^{-1}(\alpha,\beta)$, resulting in a combined Inverse-
Gamma prior with parameters $\gamma,\nu,\alpha,\beta$, shown in Figure 6.
These are predicted by different “heads” of a neural network. For predictions,
the expectation of the mean corresponds to $\mathbb{E}_{#1}[\pi]=\gamma$, and
aleatoric and epistemic uncertainty can then be estimated using the expected
value of the variance as well as the variance of the mean, respectively, which
have closed form solutions under this parameterization:

$\mathbb{E}_{#1}[\sigma^{2}]=\frac{\beta}{\alpha-1};\quad\text{Var}[\pi]=\frac{\beta}{\nu(\alpha-1)}$
(25)

By choosing to optimize using a negative log-likelihood objective, we can
actually evaluate the loss function analytically, since the likelihood
function corresponds to a Student’s t-distribution with $\gamma$ degrees of
freedom, mean $\beta(1+\nu)/(\nu\alpha)$ and $2\alpha$ variance:

$\mathcal{L}_{\text{NLL}}=\frac{1}{2}\log\Big{(}\frac{\pi}{\nu}\Big{)}-\alpha\log\Omega+\Big{(}\alpha+\frac{1}{2}\Big{)}\log\Big{(}(y_{i}-\gamma)^{2}\nu+\Omega\Big{)}+\log\bigg{(}\frac{\Gamma(\alpha)}{\Gamma(\alpha+\frac{1}{2})}\bigg{)}$
(26)

using $\Omega=2\beta(1+\nu)$. Akin to the entropy regularizer for Dirichlet
networks, Amini et al. (2020) propose a regularization term that only allows
for concentrating density on the correct prediction:

$\mathcal{L}_{\text{reg}}=|y_{i}-\gamma|\cdot(2\nu+\alpha)$ (27)

Since $\mathbb{E}_{#1}[\pi]=\gamma$ is the prediction of the network, the
second term in the product will be scaled by the degree to which the current
prediction deviates from the target value. Since $\nu$ and $\alpha$ control
the variance of the mean and the variance of the normal likelihood, this term
encourages the network to decrease the evidence for mispredicted data samples.
As Amini et al. (2020) point out, large amounts of evidence are not punished
in cases where the prediction is close to the target. However, Oh & Shin
(2022) argue that this combination of objectives might create adverse
incentives for the model during training: Since the difference between the
prediction and target in Equation 26 is scaled by $\nu$, the model could learn
to increase the predictive uncertainty by decreasing $\nu$ instead of
improving its prediction. They propose to ameliorate this issue by using a
third loss term of the form

$\mathcal{L}_{\text{MSE}}=\begin{cases}(y_{i}-\gamma)^{2}&\quad\text{if
}(y_{i}-\gamma)^{2}<U_{\nu,\alpha}\\\
2\sqrt{U_{\nu,\alpha}}|y_{i}-\gamma|-U_{\nu,\alpha}&\quad\text{if
}(y_{i}-\gamma)^{2}\geq U_{\nu,\alpha}\end{cases}$ (28)

where $U_{\nu,\alpha}=\min(U_{\nu},U_{\alpha})$ denotes the minimum value for
the uncertainty thresholds for $\nu,\alpha$ given over a mini-batch, which are
themselves defined as

$U_{\nu}=\frac{\beta(\nu+1)}{\alpha\nu};\quad
U_{\alpha}=\frac{2\beta(\nu+1)}{\nu}\Big{[}\exp\Big{(}\psi\Big{(}\alpha+\frac{1}{2}\Big{)}-\psi(\alpha))-1\Big{)}\Big{]}.\\\
$ (29)

These expression are obtained by taking the derivatives
$\partial\mathcal{L}_{\text{NLL}}/\partial\nu$,
$\partial\mathcal{L}_{\text{NLL}}/\partial\alpha$ and solving for the
parameters, thus giving us the values for $\nu$ and $\alpha$ for which the
loss gradients are maximal. In combination with Equation 28, Equation 29
ensures that, should the model error exceed $U_{\nu,\alpha}$, the error is
rescaled. Thus, this rescaling bounds the Lipschitz constant of the loss
function, motivating the model to ensure the correctness of its prediction,
since its ability to increase uncertainty to decrease its loss is now limited.

##### Posterior networks for regression

Another approach for regression is the Natural Posterior Network by
Charpentier et al. (2022), which was already discussed for classification in
Section 3.4.2. But since the proposed approach is a generalization for
exponential family distributions, it can be applied to regression as well,
using a Normal likelihood and Normal Inverse-Gamma prior. The Bayesian update
rule in Equation 23 is adapted as follows: $n$ is set to $n=\lambda=2\alpha$,
and $\bm{\chi}=\big{[}\pi_{0}\ |\ \pi_{0}^{2}+2\beta/n\big{]}^{T}$. Feeding an
input into the natural posterior network again first produces a latent
encoding $\mathbf{z}$, from which a NF predicts $n_{i}=N_{H}\cdot
p(\mathbf{z}|\bm{\phi})$, and an additional network produces
$\bm{\chi}_{i}=g_{\bm{\psi}}(\mathbf{z})$, which are used in Equation 23 to
produce $\bm{\chi}^{\text{post}}$ and $n^{\text{post}}$, from which the
parameters of the posterior Normal Inverse-Gamma can be derived. The authors
also produce a general exponential family form of the UCE loss by Biloš et al.
(2019), consisting of expected log-likelihood and an entropy regularizer,
which they derive for the regression parameterization. Again, this approach
relies on the density estimation capabilities of the NF to produce an agnostic
belief about the right prediction for OOD examples (see Figure 5(b)).

##### Multivariate evidential regression

There are also some works offering solutions for multivariate regression
problems: Malinin et al. (2020a) can be seen as a multivariate generalization
of the work of Amini et al. (2020), where a combined Normal-Wishart prior is
formed to fit the now Multivariate Normal likelihood. Again, the prior
parameters are the output of a neural network, and uncertainty can be
quantified in a similar way. For training purposes, they apply two different
training objectives using the equivalent of the reverse KL objective of
Malinin & Gales (2019) as well as of the knowledge distillation objective of
Malinin et al. (2020b), which does not require OOD data for regularization
purposes. Meinert & Lavin (2021) also provide a solution using a Normal
Inverse-Wishart prior. In a similar vein to Oh & Shin (2022), they argue that
the original objective proposed by Amini et al. (2020) can be minimized by
increasing the network’s uncertainty instead of decreasing the mismatch of its
prediction. As a solution, they simply propose to tie $\beta$ and $\nu$ via a
hyperparameter.

##  5 Related Work

##### Other Approaches to Uncertainty Quantification

The need for the quantification of uncertainty in order to earn the trust of
end-users and stakeholders has been a key driver for research (Bhatt et al.,
2021; Jacovi et al., 2021; Liao & Sundar, 2022). Existing methods can broadly
be divided into frequentist and Bayesian methods, where the former judge the
confidence of a model based on its predicted probabilities. Unfortunately,
standard neural discriminator architectures have been proven to possess
unwanted theoretical properties w.r.t. OOD inputs (Hein et al., 2019; Ulmer &
Cinà, 2021) and might therefore be unable to detect potentially risky
inputs.141414Pearce et al. (2021) argue that some insights might partially be
misled by low-dimensional intuitions, and that empirically OOD data in higher
dimensions tend to be mapped into regions of higher uncertainty. Further, a
large line of research works has questioned the calibration of models (Guo et
al., 2017; Nixon et al., 2019; Desai & Durrett, 2020; Minderer et al., 2021;
Wang et al., 2021b), i.e. to what extend the probability score of a class—also
referred to as its confidence—corresponds to the chance of a correct
prediction. Instead of relying on the confidence score alone, another way lies
in constructing prediction sets consisting of the classes accumulating a
certain share of the total predictive mass (Kompa et al., 2021; Ulmer et al.,
2022). By scoring a held-out population of data points to calibrate these
prediction sets, we can also obtain frequentist guarantees in a procedure
referred to a _conformal prediction_ (Papadopoulos et al., 2002; Vovk et al.,
2005; Lei & Wasserman, 2014; Angelopoulos & Bates, 2021). This however still
does not let us distinguish different notions of uncertainty. A popular
_Bayesian_ way to overcome this blemish by aggregating multiple predictions by
networks in the Bayesian model averaging framework (Mackay, 1992; MacKay,
1995; Hinton & Van Camp, 1993; Neal, 2012; Jeffreys, 1998; Wilson & Izmailov,
2020; Kristiadi et al., 2020; Daxberger et al., 2021; Gal & Ghahramani, 2016;
Blundell et al., 2015; Lakshminarayanan et al., 2017). Nevertheless, many of
these methods have been shown not to produce diverse predictions (Wilson &
Izmailov, 2020; Fort et al., 2019) and to deliver subpar performance and
potentially misleading uncertainty estimates under distributional shift
(Ovadia et al., 2019; Masegosa, 2020; Wenzel et al., 2020; Izmailov et al.,
2021a; b), raising doubts about their efficacy. The most robust method in this
context is often given by an ensemble of neural predictors (Lakshminarayanan
et al., 2017), with multiple works exploring ways to make their training more
efficient (Huang et al., 2017; Wilson & Izmailov, 2020; Wen et al., 2020;
Turkoglu et al., 2022) or to provide theoretical guarantees (Pearce et al.,
2020; Ciosek et al., 2020; He et al., 2020; D’Angelo & Fortuin, 2021).

##### Related Approaches to EDL

Kull et al. (2019) found an appealing use of the Dirichlet distribution as a
post-training calibration map. Hobbhahn et al. (2022) use the Laplace bridge,
a modified inverse based on an idea by MacKay (1998), to map from the model’s
logit space to a Dirichlet distribution. The proposed Posterior Network
(Charpentier et al., 2020; 2022) can furthermore be seen as related to
another, competing approach, namely the combination of neural discriminators
with density estimation methods, for instance in the form of energy-based
models (Grathwohl et al., ; Elflein et al., 2021) or other hybrid
architectures (Lee et al., 2018; Mukhoti et al., 2021). Furthermore, there is
a line of other single-pass uncertainty quantification approaches which do not
originate from the evidential framework, for instance by taking inspiration
from RBF networks (van Amersfoort et al., 2020b) or via Gaussian Process
output layers (Liu et al., 2020; Fortuin et al., 2021; van Amersfoort et al.,
2021).

##### Applications of EDL

Some of the discussed models have already found a variety of applications,
such as in autonomous driving (Capellier et al., 2019; Liu et al., 2021; Petek
et al., 2022; Wang et al., 2021a), remote sensing (Gawlikowski et al., 2022),
medical screening (Ghesu et al., 2019; Gu et al., 2021; Li et al., 2022),
molecular analysis (Soleimany et al., 2021), open set recognition (Bao et al.,
2021), active learning (Hemmer et al., 2022) and model selection (Radev et
al., 2021).

##  6 Discussion

##### What is state-of-the-art?

As apparent from Table 5, evaluation methods and datasets can vary
tremendously between different research works (for an overview, refer to
Appendix B,). This can make it hard to accurately compare different approaches
in a fair manner. Nevertheless, we try to draw some conclusion about the
state-of-art in this research direction to the best extent possible: For image
classification, the posterior (Charpentier et al., 2020) and natural posterior
network (Charpentier et al., 2022) provide the best results on the tested
benchmarks, both in terms of task performance and uncertainty quality. When
the training an extra normalizing flow creates too much computational
overhead, prior networks (Malinin & Gales, 2018) with the PAC-based
regularizer (Haussmann et al., 2019; see Table 6 for final form) or a simple
entropy regularizer (Section C.2) can be used. In the case of regression
problems, the natural posterior network (Stadler et al., 2021) performs better
or on par with the evidential regression by Amini et al. (2020) or an ensemble
Lakshminarayanan et al. (2017) or MC Dropout (Gal & Ghahramani, 2016). For
graph neural networks, the graph posterior network (Stadler et al., 2021) and
a ensemble provide similar performance, but with the former displaying better
uncertainty results. Again, this model requires training a NF, so a simpler
fallback is provided by evidential regression (Amini et al., 2020) with the
improvement by Oh & Shin (2022). For NLP and count prediction, the works of
Shen et al. (2020) and Charpentier et al. (2022) are the only available
instances from this model family, respectively. In the latter case, ensembles
and the evidential regression framework (Amini et al., 2020) produce a lower
root mean-squared error, but worse uncertainty estimates on OOD.

##### Computational Cost

When it comes to the computational requirements, most of the proposed methods
in this survey incur the same cost as a single deterministic network using a
softmax output, since most of the architecture remains unchanged. Additional
cost is mostly only produced when using knowledge distillation (Malinin et
al., 2020b; Fathullah & Gales, 2022), adding normalizing flow components like
for posterior networks (Charpentier et al., 2020; 2022; Stadler et al., 2021)
or using generative models to produce synthetic OOD data (Chen et al., 2018;
Sensoy et al., 2020; Hu et al., 2021).

##### Comparison to Other Approaches to Uncertainty Quantification

As discussed in Section 5, several existing approaches to uncertainty
quantification equally suffer from shortcomings with respect to their
reliability. One possible explanation for this behavior might lie in the
insight that neural networks trained in the empirical risk minimization
framework tend to learn spurious but highly predictive features (Ilyas et al.,
2019; Nagarajan et al., 2021). This way, inputs stemming from the training
distribution can be mapped to similar parts of the latent space as data points
outside the distribution even though they display (from a human perspective)
blatant semantic differences, simply because these semantic features were not
useful to optimize for the training objective. This can result in ID and OOD
points having assigned similar feature representations by a network, a
phenomenon has been coined “feature collapse” (Nalisnick et al., 2019; van
Amersfoort et al., 2021; Havtorn et al., 2021). One strategy to mitigate (but
not solve) this issue has been to enforce a constraint on the smoothness of
the neural network function (Wei et al., 2018; van Amersfoort et al., 2020a;
2021; Liu et al., 2020), thereby maintaining both a sensitivity to semantic
changes in the input and robustness against adversarial inputs (Yu et al.,
2019). Another approach lies in the usage of OOD data as well, sometimes
dubbed “outlier exposure” (Fort et al., 2021), but displaying the same
shortcomings as in the EDL case. A generally promising strategy seems to seek
functional diversity through ensembling: Juneja et al. (2022) show how model
instances ending up in different low-loss modes correspond to distinct
generalization strategies, indicating that combining diverse strategies may
lead to better generalization and thus potentially more reliable uncertainty.
Attaining different solutions still creates computational overhead, despite
new methods to reduce it (Garipov et al., 2018; Dusenberry et al., 2020;
Benton et al., 2021).

##### Bayesian model averaging

One of the most fundamental differences between EDL and existing approaches is
the sacrifice of Bayesian model averaging (Equations 2 and 5): In principle,
combining multiple parameter estimates is supposed to result in a lower
predictive risk (Fragoso et al., 2018). The Machine Learning community has
ascribed further desiderata to this approach, such as better generalization
and robustness to distributional shifts. Recent studies with exact Bayesian
Neural Networks however have cast doubts on these assumptions (Izmailov et
al., 2021a; b). Nevertheless, ensembles, that approximate Equation 2 via Monte
Carlo estimates, remain state-of-the-art on many uncertainty benchmarks. EDL
abandons modelling epistemic uncertainty through the learnable parameters, and
instead expresses it through the uncertainty in prior / posterior parameters.
This loses functional diversity which could aid generalization, while
sidestepping computational costs. Future research could therefore explore the
combination of both paradigms, as proposed by Haussmann et al. (2019); Zhao et
al. (2020); Charpentier et al. (2022).

##### Challenges

Despite their advantages, the last chapters have pointed out key weaknesses of
Dirichlet networks as well: In order to achieve the right behavior of the
distribution and thus guarantee sensible uncertainty estimates (since ground
truth estimates are not available), the surveyed literature proposes a variety
of loss functions. Bengs et al. (2022) show formally that many of the loss
functions used so far are _not_ appropriate and violate basic asymptotic
assumptions about epistemic uncertainty: With increasing amount of data,
epistemic uncertainty should vanish, but this is not guaranteed using the
commonly used loss functions. Furthermore, some approaches (Malinin & Gales,
2018; 2019; Nandy et al., 2020; Malinin et al., 2020a) require out-of-
distribution data points during training. This comes with two problems: Such
data is often not available or in the first place, or cannot guarantee
robustness against _other_ kinds of unseen OOD data, of which infinite types
exist in a real-valued feature space.151515The same applies to the artificial
OOD data in Chen et al. (2018); Shen et al. (2020); Sensoy et al. (2020).
Indeed, Kopetzki et al. (2021) found OOD detection to deteriorate across a
family of EDL models under adversarial perturbation and OOD data. Stadler et
al. (2021) point out that much of the ability of posterior networks stems from
the addition of a NF, which have been shown to also sometimes behave
unreliably on OOD data (Nalisnick et al., 2019). Although the NFs in posterior
networks operate on the latent and not the feature space, they are also
restricted to operate on features that the underlying network has learned to
recognize. Recent work by Dietterich & Guyer (2022) has hinted at the fact
that networks might identify OOD by the absence of known features, and not by
the presence of new ones, providing a case in which posterior networks are
likely to fail. Such evidence on OOD data and adversarial examples has indeed
been identified by a study by Kopetzki et al. (2021).

##### Future Research Directions

Overall, the following directions for future research on EDL crystallize from
our previous reflections: _(1) Explicit epistemic uncertainty estimation:_
Since we often employ the point estimate in Equation 5 to avoid the posterior
$p(\bm{\theta}|\mathbb{D})$, explicit estimation of the epistemic uncertainty
is not possible, and some summary statistic of the concentration parameters is
used for classification problems instead (Section 3.3). Estimating model
uncertainty through modelling the (approximate) posterior
$p(\bm{\theta}|\mathbb{D})$ in Bayesian model averaging is a popular technique
(Houlsby et al., 2011; Gal et al., 2016; Smith & Gal, 2018; Ulmer et al.,
2020), but comes with the disadvantage of additional computational overhead.
However, Sharma et al. (2022) recently showed that a Bayesian treatment of all
model parameters may not be necessary, potentially allowing for a compromise.
_(2) Robustness to diverse OOD data:_ The emprical evidence compiled by
Kopetzki et al. (2021) indicates that EDL classification models are not
completely able to robustly classify and detect OOD and adversarial inputs.
These findings hold both for prior networks trained with OOD data, or for
posterior networks using density estimators. We speculate that through the
information bottleneck principle (Tishby & Zaslavsky, 2015), EDL models might
not learn input features that are useful to indicate uncertainty in their
prediction, or at best identify the absence of known features, but not the
presence of new ones (Dietterich & Guyer, 2022). Finding a way to have models
identify unusual features could this help to mitigate this problem. _(3)
Theoretical guarantees:_ Even though some guarantees have been derived for EDL
classifiers w.r.t. OOD data points (Charpentier et al., 2020; Stadler et al.,
2021), Bengs et al. (2022) point out the flaws of current training regimes for
epistemic uncertainty in the limit of infinite limit. Furthermore, Hüllermeier
& Waegeman (2021) argue that even uncertainty estimates are affected by
uncertainty themselves, impacting their usefulness.

##  7 Conclusion

This survey has given an overview over contemporary approaches for uncertainty
estimation using neural networks to parameterize conjugate priors or the
corresponding posteriors instead of likelihoods, called Evidential Deep
Learning. We highlighted their appealing theoretical properties allowing for
uncertainty estimation with minimal computational overhead, rendering them as
a viable alternative to existing strategies. We also emphasized practical
problems: In order to nudge models towards the desired behavior in the face of
unseen or out-of-distribution samples, the design of the model architecture
and loss function have to be carefully considered. Based on a summary and
discussion of experimental findings in Section 6, the entropy regularizer
seems to be a sensible choice in prior networks when OOD data is not
available. Combining discriminators with generative models like normalizing
flows as in Charpentier et al. (2020; 2022), embedded in a sturdy Bayesian
framework, also appears as an exciting direction for practical applications.
In summary, we believe that recent advances show promising results for
Evidential Deep Learning, making it a viable option in uncertainty estimation
to improve safety and trustworthiness in Machine Learning systems.

## Acknowledgements

We would like to thank Giovanni Cinà, Max Müller-Eberstein, Daniel Varab and
Mike Zhang for reading early versions of this draft and providing tremendously
useful feedback. Further, we would like to explicitly thank Mike Zhang for
helping to improve Figure 1. We also would like to thank Alexander Amini for
providing a long list of references that helped to further improve the
coverage of this work and the anonymous reviewers for their suggestions.
Lastly, we owe our gratitude to the anonymous reviewers that helped us such
much to improve the different versions of this paper.

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