Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimation
Authors: Dennis Thomas Ulmer
Christian Hardmeier
Jes Frellsen
Venue: Accepted by TMLR ()

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 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.

Published in Transactions on Machine Learning Research (04/2023)
Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimation
Dennis Ulmer O
dennis, ulmer ulmer@mailozz org chrha@itu. dk jefv@dla. dk
Christian Hardmeier Jes Frellsen c x
IT University of Copenhagen, L
Technical University of Denmark, X Pioneer Centre for Artificial Intelligence
Reviewed on OpenReview: https: // openreview.net forum? id- TXuZaK
Abstract
Popular approaches for quantifying predictive uncertainty in deep neural networks often in- volve 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 comprehen- sive 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 prior belief. Furthermore, they allow uncer- tainty estimation in single model and forward pass by paramcterizing distributions over distributions. This survey recapitulates existing works, focusing on the implementation in 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
Prior
Posterior
Many existing methods for uncertainty estimation leverage the concept of Bayesian model averag- ing: These include ensembling (Lakshminarayanan et al., 2017; Wilson Ixmailov, 2020), Markov chain Monte Carlo sampling (de Freitas, 2003; Andrieu et al., 2000) as well variational inference ap- proaches (Mackay, 1992; MacKay, 1995; Hinton & Van Camp, 1993; Neal, 2012), including approaches such as Monte Carlo (MC) dropout (Gal& & Ghahra- mani, 2016) and Bayes-by-backproy (Blundell et al., 2015). Bayesian model averaging for neural net- works usually involves the approximation of an oth- erwise infeasible integral using MC samples. This causes the following problems: Firstly, the quality of the MC approximation depends 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, new class of models has been proposed to side-step this co- nundrum by using different factorization of the posterior predictive distríbution. This allows computing
Tsilingkaridis (2019) Bilos et al. (2019)
Sensoy (2018)
Variational
(2020)
Joo al. (2018)
Haussmann al. (2019)
Shen et al. (2022)
Classiication
Requires
Chen (2018)
Malinin & Gales (2018)
Zhao et al. (2019)
Malinin Gales (2019)
Hu (2021)
Nandy et al. (2020)
Sensoy et al. (2020)
Shen (2020)
Knowledge Distillaionn
Flow based
Malinin (2020h)
Charpentier (2020)
Fathullah & Gales (2022)
Stadler et al. (2021)
Regression
Malinin et al. (2020a)
Charpentier al. (2021)
Amini al. (2020)
Oh & Shin (2021)
Lavin (2021)
Figure 1: Taxonomy of surveyed approaches, divided into tractable parameterizations of the prior or poste- rior one axis (see Tables and 2 for overview) and into approaches for classification and regression the other. Regression methods are outlined in Table 3.

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uncertainty in a single forward pass and with a single set of weights. These models are grounded in 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 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 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 promole wider application outside the uncertainly quantification literature. We also provide 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.'
2.1 Bayesian Inference
The foundation of the following sections is Bayesian inference: Given some prior belief p(0) about parameters of interest 0, we use available observations D = {xx; })) and their likelihood p(D)() to obtain an updated belief in form of the posterior p() x p(D)O)p(O). This update rule is derived from Bayes' rule, namely
p pD))))p(O) p())))(()
P(D)
where often try to avoid computing the term in the denominator since marginalization over a large (continuous) parameter space of 0 is usually intractable. In order to perform prediction for a new data point x, we can now utilize the posterior predictive distribution defined as
Plylx,D) - / P(u|x, O))00
Since we integrate over the entire parameter space of 0, weighting each prediction by the posterior probability of its parameters to obtain the final result, this process is referred to Bayesian model averaging (BMA). Here, predictions P(y|x, 0) stemming from parameters that are plausible given the observed dala will receive a higher weight p( P( in the final prediction Ply x, 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 alcatoric and epistemic uncertainty (Der Ki- ureghian & Ditlevsen, 2009; Kendall & Gal, 2017: Hüllermeier & Waegernan, 2021). Aleatoric uncertainly refers to the uncertainty that is induced by the gattrrtnn 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
Note that the following we will use suggested notation of the TMLR journal, e.g. by using P probability mass and p for probability density functions.
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fewer possible models become plausible fit. These two notions resurface when formulating the posterior predictive distribution for a new data point x:
/ Ply (yxx00) ((D d0
P(y|x, D)
Aleatoric Epistomic
Here, the first factor captures the alcatoric uncortainty 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(O( should lie reasonable parameter values for 0. For high-dimensional real-valued parameters O like in neural networks, this integral becomes intractable, and is usually approximated using Monte Carlo samples:
K P(yxx 0(h)); 0 ~ p())
1 K
P =
(4)
based on K different sets of parameters O(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:
/f (ly p 0
/ P(ulT)) p( 0)
ply x1 D)
O DD
drdd
dr.
(5)
Aleatoric Distributional Epistemic
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, Ma- linin & Gales (2018) replace p(0[D) by point estimate O using the Dirac delta function, i.e. 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 Son 3.3). The advantage of this approach further that it allows us lo distinguish uncertainly about a data point because it 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, obtain distribution P(y|x, 0) 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, 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(T|x, x, 0): Tf the data point is ambiguous, the resulting distribution will be cen- tered on 0.5, if the model is generally uncertain, the distribution will 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 point estimate of the parameters prevent the estimation of epistemic uncertainty like in earlier works through the weight posterior D) as discussed in the next section. However, there are works like Haussmann et al. (2019); Zhao et al. (2020) that combine both approaches.
Note that the predictive distribution in Equation (2) generalizes the common case for a single network prediction where P(y|x, 2 P(y|x, 0). Mathematically, this expressed by replacing the posterior p(0ID) by Dirac delta distribution in Equation (5), where all probability deusity rests single parameter configuration.
For easier distributions, the integral can often evaluated analytically exploiting conjugacy. Another approach more complex distributions be the method of moments (see egg. Duan, 2021).
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Bropout
NC Dropout
4.5
Setosa
(a) Iris selosa
Versicolor
4.0
Vírginica
Test point
Sepal width
3.5
3
(b) Iris versicolor
3.0
2.5
Ensemble MC Dropout:
2.0
(c) Iris virginica
4.5
5.0
5.5
6.5 Sepal length
6.0
7.0
7.5
8.0
Figure 2: Illustration of different approaches to uncertainty quantifying on the Tris dataset, with examples for the classes given on the left (Figuros 2a to 2c). On the right, the data plotted alongsidc some predictions of 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;bre.
The term Evidential Deep Learning (EDL) originates from the work of Sensoy et al. (2018) and based on the Theory of Evidence (Dempster, 1968; Audun, 2018): Within the theory, belief mass is assigned set of possible states, e.g. class labels, and can also express lack of evidence, i.e. an don't know". We can for instance generalize the predicted output of a neural classifier using the Dirichlet distribution, allowing to express lack of evidence through a uniform Dirichlet. This is different from 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 family of approaches in which a neural network can fall back onto a uniform prior for unknown inputs. While neural networks usually paramcterize 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
To illustrate the advantages of EDL, we choose a classification problem based on the Iris dataset (Fisher, 1936). contains measurements of three different species of iris flowers (shown in Figures 2a to 2c). We use the dataset as made available through scikit-learn (Pedregosa al., 2011) and plot the relationship between the width and lengths measurements of the flowers' petals in Figure 2.
We train deep neural network ensemble (Lakshminarayanan et al., 2017) with 50 model instances, model with MC Dropout (Gal & Ghahramani, 2016) with 50 predictions and prior network (Sensay 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 2.4 On these simplices, each point signifies a Categorical distribution, with
4Fr information about training and model details, Appendix A.1.
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Observations N
11
+
4
8
a [1.5, 1.5, 1.5]
B [13.5, 5.5, 9.5]T
Figure 3: A prior Dirichlet distribution is updated with a vector of class observations. The posterior Dirichlet then shifts density towards the classes with more observed instances.
the proximity to one the corners indicating a higher probability for the correspouding class. EDL methods for classification do not predict single output distribution, but an entire density over output distributions.
Test point 3 lies in region of overlap between instances of Tris versicolor and Tris virginica, thus inducing high aleatoric uncertainty. In this case, 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 the simplex. The point ① is located in an area without training examples between instances of Irís versicolor and setosa, as well as close to a single viryiníca 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 prodiction 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.
versicolor, but admits some uncertainty towards the two other choices. The
This simple example provides some insights into the potential advantages of EDL: First of all, the prior network was able 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 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 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.
almost
3 Evidential Deep Learning for Classification
In order to introduce EDL methods for classification, we first give a brief introduction the Dirichlet dis- tribution and its role a conjugate prior 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 parame- terization 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
Modelling for instance a binary classification problem is commonly done using the Bernoulli likelihood. The Bernoulli likelihood has single parameter æ, indicating the probability of success (or of the positive class), and is given by
Bernoull(())m) n (1-1)))
(6)
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Within Bayesian inference introduced in Section 2, the Beta distribution is commonly used prior for Bernoulli likelihood. It defines a probability distribution over the parameter n, itself possessing two shape parameters ar and a2:
(eta( a1, a2) = a2) π 1 (1 1) 0=2=-1 B(a1, 02) = [(o1)[((2))
(7)
r(a1 + +22)
where T(-) stands for the gamma function, a generalization of the factorial to the real numbers, and B(-) is called the Beta function (not to be confused with the distribution). When extending the classification problem from two to arbitrary number of classes, we use Categorical likelihood:
C)
II 1y=k k
(8)
in which K' denoles the number of categories or classes, and the class probabilities are expressed using a vector π E [0, 1] with Σx T = 1, and (()) 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 produce a Categorical distribution of classes s.t. T P(y k|x). In this setting, the Dirichlet distribution arises suitable prior and multivariate generalization of the Beta distribution (and is thus also called the multivariate Beta distribution):
I T(ax), I(ao) ag Σ
Dir(T; B(a) II x T B(a)
ag ag
R
(9)
where the Beta function B(-) is now defined for K shape parameters compared to Equation (7). For notational convenience, we also define K = f1..... ......K K} as the set of all classes. The distribution is characterized by its concentration parumelers a, the sum of which, often denoted as on, is called the precision." 5 The Dirichlet is a conjugate prior for such a Categorical likelihood, meaning that according to Bayes' rule Equation (1), they produce a Dirichlet posterior with parameters 3, given a data set ID {(x,, y:) of N obscrvations with corresponding labels:
N
K
K
(2:1
1
T
((a))
k=1
k=1
(10)
K
(IN 1y
K
av-1
Nutae-1 x Di:(T:;))
T
B(a)
T
B(a
where B is vector with 8e an + N₂, with N₂ 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 a distribution over Categorical distributions on the K probability simplex; we show an example with its concentration parameters and the Bayesian update in Figure
3.2 Parameterization
For a classification problem with K classes, neural classifier is usually realized as function fo R - R mapping an input x E R to logils for each class. Followed by a softmax function, this then defines a Categorical distribution over classes with a vector π with F& = p(y = x|x, 0). The same underlying architecture can be used without any major modification to instead parameterize a Dirichlet distribution,
5Te precision analogous the precision of Gaussian, where larger ag signifies sharper distribution.

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(a) Categorical distributions pre- dicted by neural ensemble on the probability simplex.
(b) Probability simplex for fident prediction, for with the den- sity concentrated single corner.
(c) Dirichlet distribution for of data uncertainty, with the den- sity concentrated in the center.
(d) Dirichlet distribution for of model uncertainty, with the den- sity spread out
(e) Dirichlet for of distribu- tional uncertainty, with the density spread across the whole simplex.
(f) Alternative approach distri- butional uncertainly called repre- sentation gap, with density trated along the edges.
Figure 4: Examples of the probability simplex for a K = 3 classification problem, where every corner corresponds to class and every point to a Categorical distribution. Brighter colors correspond to higher density. (a) Predicted Categorical distributions by an ensemble of discriminators.
(b) (c) (Desired) Behavior of Dirichlet in different scenarios by Malinin & Gales (2018): (b) For 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 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 cqually 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.
predicting a distribution over Categorical distributions p(Tr|x, O) as in Equation (9). In order to classify a data point a Categorical distribution is created from the predicted concentration parameters of the Dinichlet as follows (this corresponds to the mean of the Dirichlet, see Appendix C.1):
a = exp (fo(x));
Okk Thk
9 = arg max T1,-..,
(11)
Opp
REK
Parameterizing a Dirichlet. posterior distribution follows similar logic, will discuss in Section 3.4.2.

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3.3 Uncertainty Estimation with Dirichlet Networks
Let now our attention to how to estimate the aleatoric, epistemic and distributional uncertainty laid out in Section 2.2 within the Dirichlet framework. In Figure 4, show different shapes of Dirichlet distribution parameterized by neural network, corresponding to different cases of uncertainty, where cach point on the simplex represents a Categorical distribution, with proximity to a corner indicaling a high probability for the corresponding class. Figure 4a 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 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, shown in Figures 4b to 4f, 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 a, the following methods can also be applicd to the posterior paramcters / without loss of gencrality.
Data (aleatoric) uncertainty To obtain a measure of data uncertainty, we evaluate the expected entropy of the data distribution pTy(T) (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 undecided about the right prediction. For Dirichlet networks, this quantity has a closed-form solution (for the full derivation, refer Appendix D.1):
K
E p( x.0)
H ((()))) =
Σ
ag
vlan + + 18(00 + +1)
(12)
00
where vb denotes the digamma function, defined as v(xx) = d log [ and H the Shannon entropy.
de
Model (epistemic) uncertainty As we saw in Section 2.2, most approaches in the Dirichlet framework avoid the intractable integral over network parameters by using a point estimate 0.7 This means that computing the model uncertainty via the weight posterior p(0 like in Bhundell al. (2015); Gal & Ghahramani (2016); Smith & Gal (2018) not possible. Nevertheless, 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 (c)). Therefore, the epistemic uncertainty can be quantified considering the concentration parameters a that shape this distribution: Charpentier el al. (2020) simply consider the maximum 0g as a score akin to the maximum probability score by Hendrycks & Gimpel (2017), while Sensoy et al. (2018) compute by K/ K (ax + 1) or simply on (Charpentier et al., 2020). In both cases, the underlying intuition is that larger produce a sharper density, and thus indicate increased confidence in prediction.
Distributional uncertainty Another appealing property of this model family is being able to distinguish uncertainty due to model underspecification (Figure 4d) from uncertainty due unknown inputs (Figure 4e). 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 w:
[y.. π D H Bp(=|x.,) p(UIT)]]
EE (p(r) H [P(Wm)]]
(13)
Total Uncertainty
Data Uncertainty
TThe only thing here that the every og has be strictly positive, which can for instance be enforced by using additional softplus, exponential ReLU function (Sensoy al., 2018; Malinin Gales, 2018; Sensoy ct al., 2020). With exceptions such as Haussmann (2019); Zhao al. (2020). When the distribution over parameters in Equation (5)
retained, alternate expressions of the aleatoric and epistemic uncertainty derived by Woo (2022).
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This quantity expresses how much information we would receive about л if we were given the label y: conditioned on the new input x and the training data D. In regions in which the model is well-defined, receiving should not provide much new information about π 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 uncertainly. Given that E[Tx] = a (Appendix C.1) and assuming the point estimate p(TT/x, D) p(Tr/x,) O) to be sufficient (Malinin & Gales, 2018), we obtain an expression very similar to Equation (12):
K
v. xx =
ak
ak log 0x
~baa + 1)+ taa + +1)
(14)
a
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 ID) = 8(0 0) from Equa- tion (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 uncertainly is different from epistemic uncertainly, since it is the uncertainty in the distribution p(T||x, 0). 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 & Cinå, 2021; Van Landeghem al., 2022). Even then, the additive dccompasition of the mutual information has been critized since the model will also have great deal of uncerlainly aboul ils alealoric uncertainly 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 hesot 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 (Tahlou 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, we mainly focus on the different ways in which authors try to tackle this problem by mcans of loss functions and training procedurcs. We give an overview over the discussed works in Tables 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 repositorics for all works surveyed are listed in Appendix A.2.
3.4.1 Prior Networks
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 flat Dirichlet distribution, as flatness suggests lack of evidence. To enforce flatness, the predicted Dirichlet is compared to a uniform distribution using some probabilistic divergence mcasure. We divide prior networks
&ven though the term prior and posterior network were coined by Malinin & Gales (2018) and Charpenticr 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.
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Overview over prior networks for classification. (*) OOD samples were created inspired by the approach of Liang al. (2018). ID: Using in-distribution data samples.
Table 1:
Method
Loss function
Architecture
OOD-free training?
Prior network
TD w.r.t smoothed label & OOD KL. w.r.t. uniform prior
MLP / CNN X
(Malinin & Gales, 2018)
Prior networks
Reverse KL of Malinin & Gales (2018)
CNN
X
(Malinin & Gales, 2019)
Information Robust Dirichlet Networks (Tsiligkaridis, 2019)
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-entrow & mean variance regularizer
RNN
Prior network with PAC Regularization (Haussmann et al., 2019)
Negative log-likelihood + PAC regularizer
BNN
Ensemble Distribution Distillation
Knowledge distillation objective
MLP / CNN
(Malinin et al., 2020b)
Self- Distribution Distillation (Fathullah & Gales, 2022)
Knowledge distillation objective
CNN
Prior networks with representation gap (Nandy et al., 2020)
ID & OOD Cross-entropy +
MLP / CNN
X
precision regularizer
Prior RNN (Shen al., 2020)
Cross-entropy + entropy regularizer
RNN
(x)
based Kernel Dirichlet distribution estimation (GKDE) (Zhao al., 2020)
norm w.r.t. one-hot label KL reg. with node-level distance prior & Knowledge distillation objective
GNN
two groups: Approaches using additional OOD data, for this purpose (OOD-dependent approuches), and those which do not required OOD data (OOD-free approaches), as listed in Table 1.
OOD-free approaches Apart from standard negative log-likeldhood loss (NLL) used by Haussmann et al. (2019), one simple approach to optimizing the model is to impose a lo-loss between the one-hot encoding of the original label and the Categorical distribution π. Tsiligkaridis (2019) show that since the values of æ depend directly on the predicted concentration parameters a, a generalized loss can be derived to be upper-bounded by the following expression (see the full derivation given in Appendix D.3):
Ir( Entt as +p) r
I(ao) r(aa + p)
I + T(ou)
Ep( [1y -Tb/]
(15)
Noyty ax
kgy
Since the concentration parameters excludes the one corresponding to the true label, this loss be seen reducing the density the areas of the probability simplex that do not correspond to the target class. Sensoy et al. (2018) specifically utilize the l2 loss, which has the following form (sce Appendix D.4):
( [|y ा
Σ
(1
O ax) + ax - o8k
(16)
an
az
where 1() denotes the indicator function. Since au/au S 1, we can see that the term with the indicator functions penalizes the network when the concentration parameter og corresponding to the correct label
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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-Leibie divergence from a uniform Dirichlet (see Appendix C.3):
T(K) KL p((1)) = llog B(a)
K
Σ (ax - ()b((a)) - v((00)
(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, 2 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." 10 First, the concentration parameter for the correct class ay removed from the Dirichlet by creating = (l-y) a-yy.. Then, the remaining concentration parameters are pushed towards uniformity by the divergence measure, which can be derived to be
Rényi ((TT|))||((T] ~ 2 kzy
(au (a)) (+))))))((((a)) - (āo)
Σ kgk kgy, k'
(ax- ((xx (18)
where v)(1) denotes the first-order polygammma function, defined as (z)(())) = d ((x). Since the sums de
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 scalar 8 allows one to set the desired risk, i.e. the model's expected risk is guaranteed to be the same less than the derived PAC bound with a probability of 1 11-66 6. For problem with N available training data points, the following upper bound is presented:
KL KL [p(() p(T[1)] log
1.
(19)
N
This upper bound is then used as the actual regularizer term in practice. We see that even from the learning- thcoretic 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 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 the parameter corresponding to the right label-expecting 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 a student model lcarning to imitate the prodictions of a more complex teacher model. Malinin al. (2020b) exploit this idea and show that prior networks can also be disuilled using an ensemble of classifiers and their predicted Categorical distributions (akin to learning Figure 4e from Figure 4a), 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).
"They also add another knowledge distillation the prodictions of vanilla Craph Neural Network that functions as the teacher network.
(Hinton al., 2015) to their loss, for which the model tries imitate
10 Kullback-Leiilee divergence can be seen special case of the Rényl divergence (van Erven & Harremoës, 2014). where the latter has stronger Information-theoree theoretic underpinning.
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OOD-dependent approaches A uniform Dirichlet in the face of unknown imputs 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 lace of unknown inpuls. Further, instead of an lp norm, they utilize another KI. term to train the model on predicting the correct label, minimizing the distance between the predicted concentration paramcters and the true label. However, since only gold label and not a gold distribulion 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 KT. divergence 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 Appendix 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 4d and 4e) might be confused, and instead propose novel loss functions producing representation gap (Figure 4f), 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-likelihodd 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-fort form solution in Appendix C.2:
K H [p(T]a)]] = log B(a) + (ao - K)b(an) Σ (ax
(aa - 1)b(as)
(20)
Sequential models We also have identified two sequential applications of prior networks in the literature: For Natural Language Processing, Shen al. (2020) train a recurrent neural network for spoken language understanding using a simple cross-cntropy loss. Instead of using OOD examples for training, they maximize the entropy of the model on data inputs given 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- 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-auan aware cross-entropy (UCE) loss instead, which has closed-form solution in the Dirichlet case (see Appendix D.6)
LUCE = xbay)) - ((a))
(21)
with ei referring to the digamma function. By mimizing the difference between the digamma values of ay and oo, the model learns to concentrate density on the correct class. Since their final concentration parameters are created using additional information from class-specific Gaussian process, they further regularize the mcan and variance for OOD data points using an extra loss term, incentivizing a loss mean and variance corresponding to a pre-defined hyperparameter.
3.4.2 Posterior Networks
As elaborated on in Section 3.1, choosing a Dirichlet prior, due to its conjugacy to the Categorical distri- bution, induces Dirichlet posterior distribution. Like the prior before, surveyed works listed in Table 2 parameterize the posterior with neural network. The challenges hereby are two-fold: Accounting for the number of class observations N₂ that make up part of the posterior density parameters / (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 probabílities are expressed through Dirichlet distributions. In doing so, an agnostic belief in form of uniform Dirichlet prior Wk og 1 is updated
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Table 2: Overview over posterior networks for classification. OOD data is created using (t) the fast-sign gradient method (Kurakin et al., 2017), (t) Variational Auto-Encoder (VAE; Kingma & Welling, 2014) or (3) 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)
la norm w.r.t. one-hot label +
CNN
KL w.rtt. uniform prior
Regularized ENN (Zhao al., 2019)
le norm w.r.t. one-hot label +
MLP CNN
x
Uncertainty regularizer on OOD/ difficult samples
WGAN-ENN (Hu et al., 2021)
la norm w.r.t. one-hot label + Uncertainty regularizer synth. OOD
MLP / CNN +
(X)S
WGAN
Variational Dirichlet (Chen et al., 2018)
ELBO +
CNN
(0)
Contrastive Adversarial Loss
Dirichlet Meta-Model (Shen al., 2022)
ELBO
CNN
KI. w.r.t. uniform prior
Belief Matching (Joo al., 2020)
ELBO
CNN
Posterior Networks
Uncertainty CE (Biloš et al., 2019) + Entropy regularizer
MLP CNN
(Charpentier et al., 2020)
Norm. Flow
Graph Posterior Networks
Same as Charpentier et al. (2020)
GNN
(Stadler ct. al., 2021)
Generative Evidential Noural Networks (Seusoy et al., 2020)
Contrastive NLL + KL between
CNN
(x)F
uniform & Dirichlet of wrong classes
using pseudo-counts Ns. which are predicted by neural network. This is different from prior networks, where the prior concentration parameters a 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 simi- lar). Sensoy et al. (2018) for instance train their model using the same techniques presented in the previous section: The main objective is the l2 loss, penalizing the difference between the predicted Dirichlet and the one-hot encoded class label (Appendix 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 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/ Ex 8x. The vacuity compares a uniform prior belief against the amassed evidence Σ. 3x, and thus is when there is no additonal evidence available. In follow-up work, Sensoy et al. (2020) similarly train a model using 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 Bx. that do not correspond to the true label are pushed to the uniform distribution.
Posterior networks via Normalizing Flows Charpentier et al. (2020) also set to a uniform prior, but obtain the pseudo-observations N₂ in different way: Instead of model predicting them directly, Ng is determined by the number of examples of a certain class the training set. This quantity is further modified in the following way: An encoder model fe produces latent representation z of some input. A (class-specific) normalizing flow (NF; Rezende & Mohamed, 2015) with parameters d then assigns a probability to this latent representation, which is used to weight N₁:
A NF is generative model, estimating density in the feature space by mapping series of invertible, bijective transformations, The probability of an input then be estimated by calculating the probability of its latent encoding under that Caussian and applying the chango-of-variabll formula, traversing the flow in reverse. Instcad of mapping from the feature space into latent space, the flows Charpentier al. (2020) map from the encoder latent space into a separate, second latent space.
Gaussian in latent space by
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3)
Normalzzing Flow p(a():
Deja (ighmen)
xif i apta
fulxlaiy
G
JP (ooelll
O
4
x
HP NuPiuPhia) x syta"""
Toput
Latent space
Prodiction
Input Spoce
Laten: Space
Exponential Fomily Porameter Space
(a) Posterior Network (Charpentier et al., 2020).
(b) Natural Posterior Network (Charpentier 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 Se maps inputs to a latent representation 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 x(1) lies right in the modelled density of the first class, and thus receives a confident prediction. The latent 2(2) lies between densities, creating aleatoric uncertainty. (3) is an OOD input, is mapped to a low-density area of the latent space and thus produces an uncertain prediction. The differences class, while only one NF used in (b). Furthermore, (b) constitutes generalization to different exponential family distributions, and is not restricted to classification problems (see main text for more detail).
the two approaches that the Posterior Network in (a) uses one NF per
Bg akk + No p(zzy = k Φ); 2 = fo())
(22)
This has the advantage of producing low probabilities for strange inputs like the noise as depicted in Figure 5a, 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-awware 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 take the neighborhood structure of the graph into account, the authors also use Personalized Page Rank scheme to diffuse node- specific posterior parameters 8 between neighboring nodes. The Page Rank scores, reflecting the importance of a neighboring node to the current node, can be approximated using power itcration (Klicpera et al., 2019) and used to aggregate the originally predicted concentration parameters / 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 postcrior Dirichlet parameters, the authors formulate a general Bayesian update rule as
ne prior prior + niXi
x post
prior +n; 2; = fo(;); n4 = N. p(z())) X: = (x))..
(23)
X here denotes the parameters of the exponential family distribution and the evidence. Thus, posterior parameters for sample x; are obtained by updating the prior parameter and some prior evidence by some input-dependent pseudo-evidence n; and parameters X Again, given latent represenlation by encoder z, a (this time single) normalizing flow predicts n, = Nи p(z(Ø) based on some pre-defined certainty budget Ny. The update parameters X; are predicted by an additional network X; = I (z), see Figure 5b. For classification, prioo and prior corresponds to the uniform Dirichlet, while X; are concentration parameters predicted by an output layer based on the latent encoding. For unfamiliar inputs, this method will
The certainty budget can simply be set to the number of available datapoints, however Charpentier et al. (2022) suggest set log Ny = (H 12g( + log(H++ (H++ 1)) better scale with the dimensionality H of the latent space.
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Table 3: Overview over Evidential Deep Learning methods for regression.
Method
Parameteriise distribution
Loss function
Model
Deep Evidential Regression (Amini et al., 2020)
Normal-Inverse Gamma Prior
Negative log- regliiklihood loss + KL w.r.t. uniform prior
MLP CNN
Deep Rvidential Regression with Multi-task Learning (Oh & Shin, 2022)
Normal-Invert
Like Amini et al. (2020), with additional Lipschitz-nodiffed MSE loss
MLP CNN
Gamma Prior
Multivariate Deep Evidential Regression (Meinert & Lavin, 2021)
Normal-Inverse Wishart Prior
Like Amini al. (2020), but tying two predicted params. instead using regularizer
MLP
Regression Prior Network
Normal-Wishan Prior
Reverse KL
MLP CNN
(Malinin et al., 2020a)
(Maliuin & Gales, 2019)
Natural Posterior Network (Charpentier et ul., 2022)
Inverse-x x Posterior
Unoertainby Cross-entropy (Bilos et al., 2019) + Entropy regularizer
MLP CNN + Norm. Flow
again result in small pseudo-evidence term n,, reflecting high model uncertainty, Since the generalization to the exponential family implics 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 lics in directly parameterizing the poste- rior parameters B. Given a target distribution defined by uniform Dirichlet prior plus the number of times an imput associated with a specific label, Chen al. (2018) optimize a distribution matching objective, i.c. the KL-divergence between the posterior parameters predicted by a neural network and the target distribu- tion. Since this objective is intractable to optimize directly, this leaves us to instead model an approzimate 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):
B(B) B(Y) + E (3x (b((3))) xb6)) k=
CELBO - -386)) log
(24)
UCE loss
KI-divergence
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 (Appendix 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 the activations of different layers of a pre-trained feature extractor.
4 Evidential Deep Learning for Regression
Bocause the EDL framework provides convenient uncertainty estimation, the question naturally ariscs of whether it can be extended to regression problems well. The answer is affirmative, although the Dirichlet distribution is not appropriate choice in this case. It is very common to model regression problem using normal likelihood (Bishop, 2006; Chapter 3.3). As such, there are multiple potential choices for prior distribution. The methods listed in Table 3 either choose the Normal-Inverss Gamma distribution (Amini et al., 2020; Charpentier et al., 2022), inducing a scaled inverse-x² posterior (Gelman et al., 1995), 13 or a Normal- Soinal-Wishare Wishart prior (Malinin et al., 2020a). We will discuss these approaches in turn.
Univariate regression Amini et al. (2020) model the regression problem as a normal distribution with unknown mean and variance N(g; ,, o²), and use normal prior for the mean with o(1, and an inverse Gamma prior for the variance with o² [-) B), resulting in a combined Inverse- Gamma
13eTh form of the Novesl-Inverse Gamma posterior and Normal Inverse- posterior are interchangable using some parameter substitutions (Murphy, 2007).
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Higher Order (Evidential):
Lower Order (Likelihood): p(u 14, σ²) p(u ())
p(u, o² hy; A, a) m)
A
B
(, 9) (0. 0.3. 0.3)
Increasing Evidence
Decreasing Variance
72
y/
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- prior, whose corresponding normal likeli- hoods display decreasing variance (and thus uncertainty) the face stronger evidence.
prior with parameters 7, M, at 3, shown in Figure 6. These are predicted by different "heads" of a neural network. For predictions, the expectation of the mean corresponds to R[xr] = y, and aleatoric and epistemic uncertainty can then be estimated using the expected value of the variance well the variance of the mean, respectively, which have closed form solutions under this parameterization:
B
E
Var[m] = v(a 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 Student's t-distribution with y degrees of freedom, mcan ((1 + (va) and 2a variance:
a log St + (a+ ) log (Iv. - 7) log
r(a)) I(a + 5
ENLL log {/,
(26)
using 2,8(1 v). Akin the entropy regularizer for Dirichlet networks, Amini al. (2020) propose a regularization term that only allows for concentrating density on the correct prediction:
Lreg - (2v a)
(27)
Since E[T]= = 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 and a 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 cases where the prediction close 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 v, the model could learn to increase the predictive uncertainty by decreasing instead of improving its prediction. They propose to ameliorate this issue by using a third loss term of the form
(M) - 7)2
(y: - 7)2 < Una if (y: 7)2 2 Un
LMSE
(28)
2 2 Uv, 1/: - 71 Una
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where Unaa (Un Ua) denotes the minimum value for the uncertainty thresholds for v, given over mini-batch, which are themselves defined as
1( av
Ua 23( +1
exp 2b a (a(a+t))) 2) - v(a)) -
U.
(29)
These expression are obtained by taking the derivatives @LNLL/OV, @CNLL/0a and solving for the param- eters, thus giving us the values for v and a for which the loss gradients are maximal. In combination with Equation (28), Equation (29) ensures that, should the model error exceed Una, 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 generalization for exponential family distributions, it can be applied to regression well, using a Normal likelihood and Normal Inverse-Gamna prior. The Bayesian update rule in Equation (23) is adapted as follows: n is set to n = À = 2a, and X = no . + 2//nn 1- Feeding an input into the natural posterior network again first produces a latent encoding z, from which NF predicts n, NH p(z()), and an additional network produces X = (w(z), which are used in Equation (23) to produce x and npost, from which the parameters of the posterior Normal Inverso- Inverso-Gamma can be derived. The authors also produce a general exponential family form of the UCE loss by Bilos et al. (2019), consisting of expected log-likelihood and an entropy regularizer, which they derive for the regression parameterization. Again, this approach relics on the density estimation capabilities of the NF to produce an agnostic belicf about the right prediction for OOD examples (see Figure 5b).
Multivariate evidential regression There are also some works offering solutions for multivariate regres- sion problems: Malinin et al. (2020a) can be seen as a multivariate generalization of the work of Amini et al. (2020), where combined Normal-Wishart prior is formed to fit the now Multivariate Normal likelihood. Again, the prior paramcters 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 decrcasing the mismatch of its prediction. As solution, they simply propose to tic 3 and a via a hyperparameter.
5 Related Work
Other Approaches to Uncertainty Quantification The need for the quantification of uncertainty in order to carn the trust of end-users and stakcholders has been key driver for rescarch (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. 11 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.. to what extend the probability score of a class-also referred to as its confidence- cuffidence-correppondd 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 certain share of the total predictive mass (Kompa et al., 2021; Ulmer et al., 2022). By scoring held-out population of data points to calibrate these prediction sets, can
14 Pearce al. (2021) argue that some insights might partially be misled by low-dimensional intuitions, and that empirically OOD data higher dimensions tend be mapped into regions higher uncertainty.
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also obtain frequentist guarantees in a procedure referred to conformal prediction (Papadopoulos et al., 2002; Vork 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 & Ixmailov, 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; Tzmailov 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 thcoretical guarantees (Pearce et al., 2020; Ciosck 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 post-training calibration map. Hobbhahn et al. (2022) use the Laplace bridge, 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.; Elficin et al., 2021) or other hybrid architoctures (Lee et al., 2018; Mukhoti et al., 2021). Furthermore, there is a line of other single-pass uncertainly 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 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-tho-art? As apparent from Table 5, cvaluation mothods and datascts can vary tremen- dously 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 (Appendix 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 ensemble provide similar performance, but with the former displaying better uncertainty results. Again, this model requires training 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.
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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 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 el al., 2021) or using generalive 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 lend to learn spurious but highly predictive features (Myas el 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 network, 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 lics in the usage of OOD data as well, sometimes dubbed "outlier exposure" (Fort et al., 2021), but displaying the same shortcomings 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 ap- proaches 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 generaliza- tion 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 approx- imate Equation (2) via Monte Carlo estimates, remain stale-of-the-art on many uncertainly 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 gen- eralization, 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 uncortainty estimates (since ground truth estimates are not available), the surveyed litcrature 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-distributiin 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. 15 Indcod, Kopetzki et al. (2021) found OOD detection to deteriorate across a family of EDL models under adversarial pertur 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 NF' in posterior networks operate on the latent and not the feature space, they are also restricted to
15eThe same applies the artificial OOD data in Chen et. al. (2018); Shen et al. (2020); Sensny al. (2020).
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operate 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 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) Explicil epistemic uncerlainly eslimation; Since we often employ the point estimate in Equation (5) to avoid the posterior p(O[D), explicit estimation of the epistemic uncertainty is not possible, and some summary statistic of the concentration paramcters is used for classification problems instead (Section 3.3). Estimating model uncertainty through modelling the (approxinale) posterior p(0) 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 ct al. (2021) indicates that EDL classification models are not completely able robustly classify and detect OOD and adversarial inpuls. 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 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 affected by uncertainty themselves, impacting their usefulness.
7 Conclusion
This survey has given an overview over contemporary approaches for uncertainty estimation using neu- ral networks to parameterize conjugate priors or the corresponding posteriors instead of likelihoods, called Evidential Deep Learning. We highlighted their appcaling theoretical properties allowing for uncertainty es- timation 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 ofttof-distribbutin samples, the design of the model architecture and loss function have to be carefully considered. Based on 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 dis- criminators with generative models like normalizing flows as in Charpentier et al. (2020; 2022), embedded in sturdy Bayesian framework, also appcars as an exciting direction for practical applications. In summary, we believe that recent advances show promising results for Evidential Deep Learning, making it viable option in uncertainty estimation to improve safety and trustworthiness in Machine Learning systems.
Acknowledgements
We would like to thank Giovanni Cinå, Max MiillerEbersteinn, Daniel Varab and Mike Zhang for reading carly 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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A Code Appendix
A.1 Iris Example Training Details
The code used to produce Figure 2 is available online." 16 All models use three layers with 100 hidden units and ReLU activations each. We furthermore optimized all of the models with a learning rate of 0.001 using the Adam optimizer (Kingma & Ba, 2015) with its default paramcter settings. We also regularize the ensemble and MC Dropout model with a dropout probability of 0.1 each.
Prior Network specifics We choose the expected l2 loss by Sensoy et al. (2018) and regularize the network using the KL divergence w.r.t. to uniform Dirichlet as in Sensoy et al. (2018). In the regularization term, we do not use the original concentration parameters a, but a version in which the concentration of the parameter 0g corresponding to the correct class is removed using a one-hot label encoding y by ā = (1-x) o + y a where denotes point-wise multiplication. The regularization term added to the loss using a weighting factor of 0.05.
A.2 Code Availability
We list the available code repositories for surveyed works in Table 4. Works for which no official implemen- (ation could be found are not listed.
B Datasets & Evaluation Techniques Appendix
This section contains a discussion of the used datasets, methods evaluate the quality of uncertainty evaluation, well as a direct of available models based on the reported results to determine the most useful choices for practitioners. An overview over the differences between the surveyed works is given in Table 5.
Datasets Most models are applied to image classification problems, where popular choices involve the MNIST dataset (LeCun, 1998), using as OOD datasets Fashion-MNIST (Xiao et al., 2017), notMNIST (Bulatov, 2011) containing English letters, K-MNIST (Clanuwat et al., 2018) with ancient Japanese Kuzushiji characters, and the Omniglot dataset (Lake et al., 2015), featuring handwritten characters from more than 50 alphabets. Other choices involve different versions of the CTFAR-10 object recognition dataset (LeCun et al., 1998; Krizhevsky et al., 2009) for training purposes and SVHN (Goodfellow et al., 2014), iSUN (Xiao et al., 2010), LSUN (Yu et al., 2015), CelebA (Liu et al., 2015), ImageNet (Deng et al., 2009) and Tinylmagenee (Bastidas, 2017) for OOD samples. Regression image datasets include for instance the NYU Depth Estimation v2 dataset (Silberman et al., 2012), using ApolloScape (Huang et al., 2018) KITTI (Menze & Geiger, 2015) as an OOD dataset. Many authors also illustrate model uncertainty synthetic data, for instance by simulating clusters of data points using Gaussians (Malinin & Gales, 2018; 2019; Nandy et al., 2020; Zhao et al., 2019; Hu et al., 2020; Charpentier et al., 2020; 2022), spiral data (Malinin et al., 2020b) or polynomials for regression (Amini et al., 2020; Oh & Shin, 2022; Meinort & Lavin, 2021; Malinin et al., 2020a; Charpentier et al., 2022). Tabular datasets include the Segment dataset, predicting image segments based on pixel features (Dua et al., 2017), and the sensorless drive dataset (Dua et al., 2017; Paschke et al., 2013), describing the maintenance state of electric current drives as well popular regression datasets included in the UCI regression benchmark used by Hernández-Looato & Adams (2015); Gal & Ghahramani (2016): Boston house prices (Harrison Jr & Rubinfeld, 1978), concrete compression strength (Yeh, 1998), energy efficiency of buildings (Tssanas & Xifara, 2012), forward kinematics of an eight link robot arm (Corke, 1996), maintenance of naval propulsion systems (Coraddu et al., 2016), properties of protein tertiary stuctures, wine quality (Cortez et al., 2009), and yacht hydrodynamics (Gerritsma et al., 1981). Furthermore, Oh & Shin (2022) use a number of drug discovery datasots, such as Davis (Davis et al., 2011), Kiba. (Tang al., 2014), BindingDB (Liu et al., 2007) and PubChem (Kim al., 2019). Biloš et al. (2019) are the only authors working asynchronous time even prediction, and supply their own data in the form of processed stack exchange postings, smart home data, and car indicators.
16TThe code available under https///iitticcc/aaaaaaaooooroee deeplll-ddeeeeeeeee-eeeeeeee lear--eernnnngn survey.
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Table 4: Overview over code repositories of surveyed works.
Paper
Code Repository
Prior network
(Malinin & Gales, 2018)
Prior networks
(Malinin Gales, 2019)
Dirichlet via Function Decomposition
(Biloš et al., 2019)
Prior network with PAC Regularization (Haussmann et al., 2019)
http:///ggtauhcccccammmallllassa
Prior networks with representation gap (Nandy al., 2020)
Graph-based Kernel Dirichlet distribution
http:///stthbboco////3///ttttttt
estimation (GKDE) (Zhao et al., 2020)
Evidential Deep Learning
(Sensoy et al., 2018)
WGAN ENN
httpss//ggttuu..ccaaaooooottttt
(Hu et sl., 2021)
Belief Matching (Joo et al., 2020)
Pusterior Networks
(Charpentier et al., 2020)
Graph Posterior Networks (Stadler et al., 2021)
Generative Evidential Neural Networks (Sensoy al., 2020)
Deep Evidential Regression with Multi-task Learning (Oh & Shin, 2022)
http:://gtthbb.oo//eerrree///tt
Multivariate Deep Evidential Regression (Meinert & Lavin, 2021)
http:r//ggthabbcc//aawttsssa///
Regression Prior Network (Malinin al., 2020a)
Natural Posterior Network (Charpentier et al., 2022)
Shen al. (2020) provide the sole method on language data, and use three different concept learning datasets, i.e. Concept Learning (Jia et. al., 2017), Snips (Coucke el. al., 2018) and ATTS (Hemphill et al., 1990), which contains
OOD concepts to be learned by design. For graph neural networks, Zhao et al. (2020) and Stadler et al. (2021) seleet data from the co-purchasc datasots Amazon Computer, Amazon Photos (McAuley el al., 2015) and the CoraML. (MeCallum et. al., 2000), CileSeer (Giles et al., 1998) and PubMed (Namata et. al., 2012), Coauthors Physics (Shehur et al., 2018), CoauthorCS (Namata et al., 2012) and OGBN Arxiv (Hu et al., 2020) citation datasets. Lastly, Charpentier et (2022) use a sin- gle count prediction dataset concerned with predicting the number of bike rentals (Fanaco-T) T & Gama, 2014).
Uncertainty Evaluation Methods There usually are no gold labels for uncertainty estimates, which is why the efficacy of proposed solutions has be evaluated in different way. One such way used by almost all the surveyed works is using uncertainty estimates in proxy OOD detection task: Since the model is underspecified unseen samples from another distribution, it should be more uncertain. By labelling
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OOD samples the positive and ID imputs the negative class, measure the performance of uncertainty estimates using the area under the receiver-operator characteristic (AUROC) or the area under the precision-recall curve (AUPR). We can thereby characterize the usage of data from another dataset as a form of covariate shift, while using left-out classes for testing can be seen as a kind of concopt shift (Moreno-Torres et al., 2012). Instead of using OOD data, another approach is to use adversarial examples (Malinin Gales, 2019; Tsiligkaridis, 2019; Sensoy et al., 2018; Hu et al., 2021; Chen al., 2018; Amini et al., 2020), checking if they can be identified through uncertainty. In the case of Shen et al. (2020), OOD detection new concept extraction is the actual and not a proxy task, and thus can be evaluated using classical metries such as the F1 score. Another way is misclassification detection: In general, we would desire the model to be more uncertain about imputs it incurs a higher loss on, i.e., what it is more wrong about. For this purpose, some works (Malinin & Gales, 2018; Zhao et al., 2020; Charpentier et. al., 2020) measure whether let missclassified inputs be the positive class another binary proxy classification test, and again mcasure AUROC and AUPR. Alternatively, Malinin al. (2020b); Stadler et al. (2021); Amini al. (2020) show or measure the area under the prediction / rejection curve, graphing how lask performance varies predictions on increasingly uncertain inputs is suspended. Lastly, some authors look at model's calibration (Guo et al., 2017): While this does not allow to judge the quality of uncertainty estimates themselves, quantities like the expected calibration error quantify to what extent the output distribution of clas- sifier corresponds to the true label distribution, and thus whether aleatoric uncertainty is accurately reflected.
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Table 5: Overview over uncertainty evaluation techniques and datasets. (+) indicates that a datasct was used as an OOD dataset for evaluation purposes, while (o) signifies that it was used as an in-distribution or
out-of-distribution dataset. (+) means that a dataset was modified to create ID and OOD splits (for instance
by removing some classes for evaluation or corrupting samples with noisc).
Data Modality
Mothod
Uacetalitty Evticatia
images
Tabrlar
Other
Prior network
OOD Detection, Maclusiiiratioo Detection
MNIST, CIFAR-10, Omnightt SVHNN-) TSUN" TTM"
X
Chasters (Syathesi))
(Malinit & Gales, 2015)
Prior networks
OOD Detection, Adversuzial Attack Detection
MNIST, CHAAR-10/100 SVHNO LSUNOS, TIM
Crasters (Syatheste)
(Malinin & Gales, 2019)
MNIST. PobbsoMNNSTTO noMNNSST Ommigler!*) CIEAR 10, TIM SVHig
Information Robust Dirichler Neeworkss (Teligeersiie, 2019)
OOD Detection, Advcesaiial Attack Detcction
X
Eaiii-Ringi Graphs
Diriciket via Funelian Dnsoespoiiiio (Hilai er al.. 2019)
OOD Detection
X
(Syathutic). Slack Earlouge, Smatt Home, Cat Indicatoon
x
MNIST, FushioMMNSST
Priar mntwork PAC Regulacisation (Haussmamm al. 2019)
OOD Detection
Ensemble Desetburion Distilation (Malinin al., 2020b)
OOD Detection, Maclustfiaatio Detection Calibration
TMM", ISUN
CHEAR-10, CHAR-11019)
x
Sytnals (Syuthetic)
Diststicttion Distillatton (Fattralaah & Gales, 2022)
OOD Detection, Calibration
CIFAR-10 SVHNO ISUN9
Prior netwooks with represenzation gop (Nandy el., 2020)
OOD Detection
CITAR- CITAR CIFAB-1009! TMM, ImageNet
X
Craster (Srathette)
Concept Loaming! Stapef
Prlor RNN (Sben al., 2020)
New Concept Extraction
ATTS (Tanginge)
Counthoes Plgslcef
Craph-hased based Kernel Durchled distribation retimation (GKDR) (Zhno al., 2020)
OUD Detection Miclustflootion Detection
x
Amazoe Computer
Amazce Photo! (Grapla)
Evidcatial Deep Leataing (Sensoy ol., 2018)
0OD Delection, Adversuiial Attack Dotection
MNIST, miMNISTI-),
CITAR-10!t
x
Regularizd ENN Zhao (2019)
OOD Detection
CIFAR-190
x
Crates (Sratheste)
WCAN- ENN (Lto ol., 2021)
OOD Detection, Adversuiial Allack Detextion
MNIST, miMNNSSTI)), CITAD-1001
x
Cratts (Syattucie)
Varrattonal Dirichlet (Chem et al., 2018)
0OD Detection, Adversurtal Attack Dotection
MNIST, CIFAR-10/100 SUN ISUN" SVIN TIM
Dirichkt Micte-Modkl (Shen et
OUD Detection Miclustflottio Detection
MN CIFAR-106S)) CIFAR 1001 Omnicke!")
X
x
FodiiMMNIST"", K-MNISTO)) SVHN LSUN!O, TTM
Bebief Matching (Jon al., 2000)
OOD Detection, Calibention
CIEAR 100, SVHN(
OOD Detection, M Mocloodfiptim Detection, Calibration
MNIST, M K-MNISTIO CIFAR-10, SVIEN
Seymena OI, Sumsrrlsss Drivell)
Poscrior Networks (Clarpuntise el nl., 20201
Craasams (8yntheric)
Amazon Computee( Amnzon Photof
OOD Detection, Maclasifisstio Ditecctin, Callbration
ComBLL ClecSerCCeaML Pullded Cusullars Playsia!!))
Graph Posterior Networks (Stadker es al., 2021)
X
x
Counthrst OBGN Asdi (Gouph)
00D Detection, Maclosffiatim Detection, Adversurial Attack Detection Calibratton OOD Detection, Calibration
Depola
Deep Besdmnaa Regression (Amiii at al., 2030)
ApolloScage' (Tagah Retimation)
UCC Regression Deudimark
Unisuriste Regresion (Synthetis)
Daris Kiba"D,
DiddingDB PahChomf
Dosp Beikkntial Regression with Mulbi-task Learning (Ot & Shin, 2022)
(Drg discovery),
Uniouiste Regrssion (Syathutic)
UCT Regrossion Bondimark
Multivrriat Deep Fvidential Regresion Miineri& Lavin (2021)
Qualitation Fealustion
Multitarinte Regresson (Synthetic)
NYU Degpia v2",
Regresion Prior Network (Malinin et al., 2020m)
OOD Detection
KITTI
UCI Regression Bautiimack
Univoriste Regrosion (Synthetis)
(Tagghh Extimation)
NYU Dipila
KTTTI. I.SUN
UCI Rogression Bonehmockk!!
[Depth Eotimaatinn.
Clustees (Synthotic).
Natural Postactor Netwoek (Charpeatier al. 2022)
OUD Detection, Callbeution
MNIST, FishimMNNSTTs
Suaarlew lews Drise!t)
Univuriate Regression (Synlaatio)
K-NNIST CIFAR-100")
Tilon Sharing!!
SVHN GelesA
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C Fundamental Derivations Appendix
This appendix section walks the reader through generalized versions of recurring thooretical results using Dirichlet distributions in Machine Learning context, such as their expeclation in Appendix C.I, their entropy in Appendix C.2 and the Kullback-Leibler divergence between two Dirichlets in Appendix D.3.
C.1 Expectation of a Dirichlet
Here, we show results for the quantities IE [mx] and E log ma]. For the first, we follow the derivation by Miller (2011). Another proof is given by Lin (2016).
K II
E [
/ . T I [(ak))
((ao)
T
ay-
d
(30)
W=1
Moving " ---1 out of the product:
-..../ T(ao)
1+11+
"e
(31)
I [(aw)
For the next step, define set of Dirichlet parameters with Bn = a2 + 1 and Ak + A Bpr op. For those new parameters, So = E$ Be = =1+ + So by virtue of the Gamma function's property that r(3)) = I' + 1) = mol'(ao), replacing all terms in the normalization factor yields
O
r(8o)
II k=1
Bu -1 dm ... duK Ok T
(32)
00 1K= ((3w)
where in the last step we obtain the final result, since the Dirichlet with new parameters 3a must nevertheless integrate to 1, and the integrals do not regard ag or ag. For the expectation E[log mb), we first rephrase the Dirichlet distribution in terms of the exponential family (Kupperman, 1964). The exponential family encompasses many commonly-used distributions, such the normal, exponential, Beta Poisson, which all follow the form
p(x;n) = l(x) exp (n u(x) A(n))
333
with natural parometers n, sufficient statistic u(x), and log partition function A(n). For the Dirichlet distribution, Winn (2004) provides the sufficient statistic as u(T)) [log ......, πx]" and the log-partition function
A(a) Σ log I((x)) log I((.))
(34)
By Mao (2019), we also find that by the moment--generating function that for the sulficient statistic, its expectation can be derived by
0.(nn)
B[u(x)+]
(35)
Omk.
Therefore we can evaluate the expected value of log TH (i.e. the sufficient statistic) by inserting the definition of the log-partition function in Equation (34) into Equation (35):
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0
E loog ma.
log [((x)) log I((a)) (((x)) vb(a0)
(36)
which corresponds precisely the definition of the digamma function as vxxx) di log (()).
C.2
Entropy Dirichlet
The following derivation adapted from Lin (2016), with the result stated in Charpentier et al. (2020) as well.
H = E(logp(TT/c)
(37)
-E log
(38)
B(a))
E - log B(xx)
+
(on 1) log TT
(39)
K
log B(a))
(ak
[lo mR.
(40)
k=
Using Equation (36):
K
= log B(o))
Σ
(ox - ()(((a)) - (ba)))
(41)
k=
K
log B(aa)
Σ
(ax 1)bb(a0) - Σ (ax 1)b(au)
(42)
k=1
k=1
K
log B(a) + +av Kyp(ae)
Σ
[a 1)x(ax)
(43)
C.3 Kullback-Leible Divergence between two Dirichlets
The following result is presented using adapted derivation by Lin (2016) and appears in Chen et al. (2018) and Joo et al. (2020) as a starting point for their variational objective (see Appendix D.7). In the following we use Dir(T;; ax) to denote the optimizod distribution, and Dir(Tr:) the reference or target distribution.
KL (00TT()))/[(()))
=E
log
p( T (a)) pí π
E
log p( logp(
(T(x))
E log
(44)
17)
E
log B(xx)
+
(an 1) log TE
K
E
log B(y) +
Σ
(Yk - 1) log
(45)
km
Distributing and pulling out B(a) and B(y) out of the expoctation (they don't depend on TT):
B(>)
K
= log
E B(a)
+
(as 1) t - (7k - 1) log Ta
(46)
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B(r)
k
log
E
(a 7k. log
(47)
B(a)
k=1
Moving the expectation inward and using the identity E[ ((ax)) - v((o)) from Appendix C.1:
B(r)
K
lolog
(ax (xb((())) +bao))
(48)
((a)
R=1
The KL divergence is also used by somne works as regularizer by penalizing the distance to uniform Dirichlet with y 1 (Sensoy et al., 2018). In this case, the result above can be derived to be
KL p(TT)))|((TT1) = log
T(K) B(a)
K -(a - ()(((())) v((o)
(49)
k=1
where the log (K) term can also be omitted for optimization purposes, since it does not depend on a.
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D Additional Derivations Appendix
In this appendix we present relevant results in Machine Learning context, including from some of the surveyed works, featuring unified notation and annotated derivation steps. These include derivations of expected entropy (Appendix D.1) and mutual information (Appendix D.2) as uncertainty metrics for Dirichlet networks. Also, we derive multitude of loss functions, including the Ls norm loss of a Dirichlet w.r.l. one-hot encoded class label in Appendix D.3, the la norm loss in Appendix D.4, as well as the reverse KL loss by Malinin & Gales (2019), the UCE objective Biloš et al. (2019); Charpentier et al. (2020) and ELBO Shen et al. (2020); Chen et al. (2018) as training objectives (Appendices D.5 to D.7).
D.1 Derivation of Expected Entropy
The following derivation is adapted from Malinin & Gales (2018) appendix section C.4. In the following, we assume that Wk K ny
E p(
H P(w/m)] p(TT|xx, 0)
(-
(-Σ
Σ
Th log T dw
(50)
1=
--E p( 0) Thh log Th dr
(51)
Inserting the definition of p(TT/x, 0) so ID)
IT I((wx) /
I(ao)
K II k/
mE log Te
T
av
dT
(52)
Singling out the factor Tp:
T(ao)
I(ak) IN I(aw)
"
Th
log Tr
II
av
dn
(53)
Adjusting the normalizing constant (this is the same trick used in Appendix C.1):
ai
D(au + 1) T(ow) Th log TT
I(ao + 1)
II
av TM cf
dm
(54)
00
Using the identity E[log np]
(aa))) - ((a) (Equation (36)). Since the expectation here w.r. to a
Dirichlet with concentration parameters ag + 1, we obtain
(aa + 1) - $000 + +1)
(55)
aO
D.2 Derivation of Mutual Information
We start from the expression in Equation (13):
I y, x, ID H (p
P(ylT))
Eppr (x H p(ulT))
(56)
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Given that E[TTA] cxp
(Appendix C.1) and assuming that point estimate p( DD) = p( 0) is suffi-
cient (Malinin & Gales, 2018), we can identify the first term as the Shannon entropy E Th log Furthermore, the second part we already derived in Appendix D.1 and thus obtain:
log ft, Ck=l ag ap
K
O.k:
Ok +
O#k sba +) - 20(00 + +1)
log
(57)
ax
a
k=
log
al
vaa + 10 + +1)
(58)
au
ag
D.3 loc Norm Derivation
In this section we elaborate on the dcrivation of Tsiligkaridis (2019) deriving a generalized Ip loss, upper- bounding the Lxo loss. This in turn allows us to easily derive the ly loss used by Sensoy et al. (2018); Zhao et al. (2020). Here we assume the classification target y is provided in the form of one-hot encoded label y [1y=l..... 1==
1) - x xloo]] s p)) [lly T||p]
(59)
Using Jensen's inequality
(Ep(m/x.0)) [lly -x 1-(b]] 1/p
(60)
Evaluating the expression with Ak + y: = 0:
=(B((1 Ty)p1
Σ E 1 b
1/8
(61)
In order compute the expression above, first realize that all components of क are distributed according to Beta distribution Beta(a, 8) (since the Dirichlet is multivariate generalization of the beta distribution) for which the moment-generating function is given as follows:
T(a + (3) + r(a p) 3)
(62)
I(a p- 3)(
+
Given that the first term in Equation (59) is characterized by Beta( agy, ay) and the second one by Beta(ag, ag ag), we can evaluate the result in Equation (59) using the moment generating function:
#
E (p(mix,0) |lx |1|cc] <
M(ao 0yy + p)T - -96 + 96)
P(ax + p)T + ao - -as)
(63)
T(as 26 + I6 agy)
a'og + + ao - [(x)
(i(oo- ay + p))(
D(au + ((n)
Σ
(64)
T(ao + [ - a.)
T(p + ((ap))
kyy
Factoring out common terms:
I(ao))
I(an +p) +
I(an +p) I(ax)
(65)
l'aoo pp)
T'a0 a.))
kg
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Expressing au akk Enety Op:
T(ao) ((ao) ( +p)
$
r E<Eety ax +p)
P(o + p) T(as)
I(ao
+
(66)
Ekzzy 0sk)
D.4 l Norm Loss Derivation
Here we present an adapted derivation by Sensoy et al. (2018) for the 12-norm loss to train Dirichlet networks. Here we again use a one-hot vector for a label with y = [1g=1,.. 1 y x|"
[1ly
E
1 y= mmm?
(67)
E
1 y=k 2TT ly=k + 1
(68)
A==1
K
y=k [R([TR] 1 y + E(
(69)
k=1
Using the identity that [[x]]]
E[Tx]] + Var(
12 y=
2B(TR]] 1y=k + B[TTa] + Var(TT))
(70)
k=
1 yy=k [T(x]) Var(TT)
(71)
Finally, we use the result from Appendix C.1 and the result that Var(TT) = ax.c0 - -ax)
ag(as + (see Lin, 2016):
K
1
ak
2
+
ag(ap axk)
(72)
ag(as + 1)
ag
1=
D.5 Derivation of Reverse KL loss
Here we re-state and annotate the derivation of reverse KT loss by Malinin & Gales (2019) in more detail, starting from the forward KL loss by Malinin Gales (2018). Note that here, à contains a dependence on k since Malinin & Gales (2018) let dy ddāo with &g being hyperparameter and c 1x=y + (- 1k= K + 1)a and being small number.
K 1 y 1L KL (( (K](((((()
p( ]!|
E (x(x))
(73)
(T dm Σ ly=k / p( log
Ep(xa)
(74)
p( 0)
Writing the expectation explicitly:
K
K ply = k. x) Σ 1y=k / p(T((a)log log p(TT(()
=
Σ
dr
(75)
p(T|/x, 0)
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K / Σ - k/x)
K
p(ι ( Σ 1 y=b p((T(())lg log
=
p( | O)
dx
(76)
k=1
E = k/x) Σ yykk / p(TT(á) log
7 ()) p x 0) dm
p(x)
(77)
k=1
Adding factor in log, collapsing double sum:
K
P(y - k|x) / p( log
p(T [ Ek=1 Ply = k(x=)(xx
= E
p(x)
Σ
dT
(78)
p(T 0) > P(y = k:
Reordering, separating constant factor from log:
K Ply x(x)p(TT(a)
K Lk=1
E(xx
log
(79)
p(Txx,0 0)
K
- log
Ply = klx))
di
(80)
k=1
= Ep(x)
KL
Σ kx)p
||(T|x 0)
(81)
Mixture of K Dirichlets
where we can see that this objective actually trics to minimizes the divergence towards a mixture of K Dirichlet distributions. In the case of high data uncertainty, this is claimed incentivize the model to distribute mass around each of the corners of the simplex, instead of the desired behavior shown in Figure 4c. Therefore, Malinin & Gales (2019) propose swap the order of arguments in the KL-divergence, resulting in the following:
Ep(x)
Σ
Ply k/xx)
KL p( 0) p(T()a]]]
(82)
kml
K Σ ply = k/xx) / Oog log p(TT x, 0)
= Ep(x)
dn
(83)
Reordering:
f p( 0)
p 0) p(TT )+)
Exp(x)
Σ P(y k|x) log
dn
(84)
x=1
K
E Ep(mlx00) =Ep(x)
Σ
P(y= = k[x)
Σ Ply =
(85)
Ep(x))
/ p( 0)
log
0) () - log
K
II ((TT(()) P(o=k))
K
dn
(86)
k=l
k=
Ep(())
( |P 0)
log [p(T|xx 0) of P(o=h/x) )(
log
II B((x)
II
T
an
p(o=bix)
dr
(87)
k==
-Ep(x) [ / 0)
log (p(TT(x, 0)) - loog
K
B(x)
T&
-11
(()=
dn
(88)
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= Eppx))
/ p( 0)
log (p(T)(x,)) -log
K
K
k=
Plynkix
(89)
B(aa
"
k==1
K
Bp(x) KL p(TT/ p(TT/x.0)lllTTTl
where
Σ
ply k|x)aw
(90)
k=1
Therefore, instead of mixture of Dirichlet distribution, we obtain single distribution whose parameters are mirture of the concentrations of cach class.
D.6 Uncertainty-aware Cross- Entsopy Loss
The uncertainty cross-entropy loss in Biloš et al. (2019); Charpentier et al. (2020) has the form
LUCE = (x m]] = Ellog my) = vbaa)) - vb(a))
(91)
as ply /T) given by the true label form of a delta distribution, we can apply the result from Appendix C.I.
D.7 Evidence-Lower Bound For Dirichlet Posterior Estimation
The evidence lower bound a well-known objective to optimize the KL-divergence between an approximate proposal and target distribution (Jordan al., 1999; Kingma Welling, 2014). We derive based on Chen al. (2018) the following for the Dirichlet case with proposal distribution p(Tr x, 0) the target distribution p( For the first. part of the derivation, we omit. the dependence for clarity.
p(T x, 0) log p( π y)
p( p(T (TT(x,))(()) log
[l,[p(x|xx0) (| []][p([T|v]] Exorxx
xx) (x
p
(92)
p( y)
Factorizing p( P(y)T))(TF), pulling out p(u) it doesn't depend on
p(T 0)
log
p(y)
(93)
P(yl π (P(TT)
p( [x, 0)
ex(m(x00)
log
Ep(mix 0) [log P(y]=)] +p(v)
(94)
p(T))
[L|p((
0)
|p [TT]
Ep( x) [log P(y\T))
(95)
Now note that the second part of the result is the uncertainty-aware cross-entropy loss from Appendix D.6 and re-adding the dependence of p(T) on 7, re-use result regarding the KI-divergence between two Dirichlets in Appendix C.3 and thus obtain:
B(3) B(y)
LELBO - log
Σ [Bx (b)())) vBB))
(96)
which is exactly the solution obtained by both Chen et al. (2018) and Joo al. (2020).
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E Overview over Loss Functions Appendix
In Tables 6 and 7, we compare the forms of the loss function used by Evidential Deep Learning methods for classification, using the consistent notation from the paper. Most of the presented results can be found in the previous Appendix C and Appendix D. We refer to the original work for details about the objective of Nandy et al. (2020).
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Table 6: Overview over objectives used by prior networks for classification.
Method
Loss funetion
Regulacizer
Comment
Prior networles
Blok) log Blo)) + EE=i(a (x)(()((z)) -v(oo))
FCK) log ((())
+ -Emilon 1)((((a)) - tt(oo)
Target concentration paramnteess & are created using a label smoothing approach,
(Maliuin & Gales, 2018)
Le. R2. (K 1)e if
if of
Together setting de as hyperparameter, stydio
Prior networks
log Bla
Σx Lk= 62)(8((x)) v(aol)
log Bial EE -
Similar above, &e 11-
(Malinin & Gales, 2019)
in distribution (6)) 1eme +1
have hyperpammeterss set Ogn and Oout = 0. Then finally, á > Myy k/x) and a > 21-1 piy k/x]a..
Eme, mclu) Exer a)
Information Robust Dirichlet Networks (Telligkaridis, 2019)
Plmol Mmsspp)
Enet
Dostel Tioa
Ensslan (a)) (am))
pi is the polygammma fuuction defined as (()) (v()..
Dirichlet Function Decomposition (Biluš et al., 2019)
M(az)) v(as)
An So mu(r)arr bo (P()))
Factors and that treated hyperparametens that weigh first term pushing the for logit to zero, while pushing the variance in the first term
Prior uetwork with PAC Reg.
log [T
mx
kL mimtaa ((tris) -log
The expeciation in using parameter samples from weight distribution. 5 E [0, I].
function evaluated
(Haussmann et al., 2019)
Ensemble Distribation
The objective uses predictions from trained ensemble with parauetters o,...., ..t..........
Distillation
ply k/x, k/x,0/m)) 0lm))
(Malinin et al., 2020b)
Prior networks with
log tor Et o(as)
Enar * loggs Age R o(as)
The main objective is being optimized on in-distribution, the regulazizer out-of-dstzibution data. An and Aut weighing terms and denotes the sigmoid function.
representation gap (Nandy al., 2020)
Priur RNN (Shen et al., 2020)
Σn Jany
log B(a) (āy K)v((a)) - Ix 1)((a))
Here, the catropy regularizer operaess scaled version of the concentration parameters á = (Ix W)a, where W learned.
Graph-based Kernel
Eisl (1y
malor-os] (]oct]
Bia) log Flal + K 1021 (()(v(()))))) - )(ao))
& here corresponds to uniform prior including some information about the local graph strueture. The authors also use an additional knowlodge distillation objective, which was omitted
Dirichlet dist. est. (GKDE) (Zhao et al., 2020)
dlesn'' related the Dirichlet.

---

Published in Transaction tins on Machine Learning Research (04 (04 (2023)
Table 7: Overview over objectives used by posterior networks for classification.
Method
Loss function
Regularieve
Comment
Evidectial Deep Learning (Smesy al. 2018)
Exm (Im-
FIK log B(s)
(5, 11100))))) - v(56))
Variational Dirichlet (Chen et al. 2018)
log Biy)
>> (7)1(180)) v(Po1)
AePannnto EK n4 Eve ar Myrtel
Regulariod FNN Zhao al. (2019)
E (Na- #))
-A, (x..
The first term regreeenss vursityy Le. the lade of evtelmos and la optinalzed wsing ODD examples. The second term stands for dissonanor, and is computed using points with neighlorhoods with different classes from their own. An are hyperperancterr.
WGAN ENN (Hu ol., 2021)
> (Ayea #))
XB p0
Belief Matching (Joo et al, 2020)
((8))- -(0%)
log 813) + EE (84 - (196)
Ponterior networkor (Charpenter al., 2020)
((8,) vx(%)
- ((B) (16 K)0(0 E
1)9(8)
Graph Poeturior Networks (Sadke al, 2021)
((8,)) - v.(%)
- -log B(g))())+ + (36 - K(N0()))) 12
Generative Evidential
>x Bea [[og)1(/o(xx))]] Bp..its [log() [(55(x))))
- log (iR..) + Eegy 1)6e(())) ((
The main loss discrininative lose ID and OOD sumples, generated a VAB. The regalarizer is taken over clasees ezcluding the true class (also indicated by 9-.).
Neural Network
(Sensyy es aL, 2020)


Reviews:
[{"review_id": "4EfbXSzGIV", "reviewer": "Reviewer_bZNn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- The main strength of the paper is that it is an accessible introduction to a part of the literature that can be fairly confusing. As mentioned above, it contains often omitted details in derivations in the appendix that are of high pedagogical value. \n- The illustrations and examples that are included are helpful. \n- The paper contains a fairly comprehensive discussion of regularization strategies that are critical to getting evidential models to work. \n\n## Weaknesses\nThere are several (mostly small) weaknesses that should be addressed to strengthen the paper:\n\n- In the introduction, the authors write \"Bayesian model averaging involves the approximation of an otherwise infeasible integral using MC samples\". This is not true for several reasons. First, models that exploit conjugacy have been used. Second, other approximate methods such as moment matching have been used. \n- The authors motivate evidential learning models by claiming \"for OoD inputs, they fall back on a prior\". This is not inherent to the models, as is evident from the multitude of regularization strategies. This is a goal of these models, which is hopefully better achieved by architectural choices.\n- The discussion around distributional uncertainty and evidential methods is fairly sparse and not very clear, despite being the intellectual core of evidential deep learning. \n- As a minor point, the use of mu for model parameters is confusing as it is typically used to denote means. A variable with fewer connotations would clarify this discussion, such as phi. Moreover, relationship between variables is unclear and should be better defined. \n- The discussion of prior versus posterior parameterization was quite clear and should be introduced earlier and described in more detail.  More generally, the discussion of parameterizing the prior or posterior without any discussion of the role of the likelihood was confusing. \n- Relatedly, the presentation of the actual training procedure is quite limited, and an algorithm block would help clarify. \n- Terms are a bit strange in a few places; in particular, the use of \"foreign data\" to denote OoD data and \"gold label\" to denote the true label are both unusual and I would recommend changing the wording. \n- The discussion of future work/open questions is very limited for such a young field. I would recommend the authors try to highlight developing or important questions in evidential deep learning in the last section of their paper. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a review of work in evidential deep learning: broadly, models that aim to parameterize the prior or posterior of Bayesian conjugate models and thereby achieve better uncertainty estimation. The paper presents a gradual introduction to Bayesian inference, uncertainty in neural networks, and relevant fundamentals of conjugate Bayesian inference. It includes a particularly deep dive on Dirichlet distributions, inference in the Categorical-Dirichlet model, and Evidential methods that leverage Dirichlet models. The paper presents a comprehensive discussion of the methods used to produce good uncertainty estimates within evidential learning frameworks; moreover, the appendix contains many detailed derivations, which is a strong contribution. ", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Several requested changes are discussed above, under weaknesses. These are all writing changes. In particular, I think weakening the claims about what evidential deep learning _does_ achieve (versus what it aims to achieve) is necessary for acceptance. However, I would strongly encouraging addressing the previously-mentioned clarity issues. Improved clarify is of extremely high importance for a review paper like this, which aims entirely as a pedagogical resource. ", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "lHUGdBFr09", "reviewer": "Reviewer_qRpe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n1. The paper is generally well-written.\n2. Almost all published works in this space are included.\n3. The discussion is quite detailed.\n\n\n## Weaknesses\n\nAs I mentioned in my previous review, experimental comparisons between prior/posterior nets and other baselines are missing. Since the paper's goal is to encourage people to use EDL methods, not having hard numerical comparisons makes the paper a bit weaker.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "I thank the authors for incorporating most of the requested changes in the previous review round.\n\nThis paper reviews recent advances in single-model, single-pass networks for predictive uncertainty quantification. \nSpecifically, the models considered are networks with Dirichlet output---such as prior networks, posterior networks, and evidential deep learning (EDL)---which have a higher degree of freedom in terms of uncertainty quantification due to the fact that the Dirichlet can be seen as a distribution over softmax outputs of standard NNs.\nAdditionally, models for regression are also discussed.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Major\n\nThe current version of the paper is already quite good for the purpose of giving an overview of EDL methods. However, I think an additional section dedicated to benchmarking the UQ of EDL methods against non-EDL baselines would make this paper even stronger. \n\nAdditionally, providing an all-in-one EDL library would be very beneficial for practitioners. Though, I also understand that one could also argue that this endeavour is better suited for a separate paper.\n\n## Minor Comments\n\n* Footnote 1: Shouldn't it be $P(y \\mid x, D) = P(y \\mid x, \\hat\\theta)$?\n  * Related: Explaining the notation before Sec. 2.1 would be very helpful. E.g. what's the difference between $p$ and $P$.\n* Eq. 4: \n  * An explanation that $\\theta^{(k)} \\sim p(\\theta \\mid D)$ should be added.\n  * Missing a $1/K$ factor.\n* Page 4: Latin words/names (\"Iris versicolor\", \"Iris virginica\", etc) should be italicized.\n* Tables 1, 2, 3: I wonder whether the \"Model\" column is necessary---in general, prior/posterior/evidential framework can be applied to any type of neural network.\n* Related Work, page 17: Another \"Bayesian way\", which actually is quite similar to the evidential framework is to approximate the distribution of the softmax outputs (which is a logistic-Normal distribution) with a Dirichlet [1]. Note that, in this case, even if we have Dirichlet output just like prior nets, [1] still has a non-trivial approximate posterior over $\\theta$.\n\n## References\n\n1. Hobbhahn et al. \"Fast predictive uncertainty for classification with Bayesian deep networks.\" UAI 2022.\n", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "eTbz9gT7WD", "reviewer": "Reviewer_GNZE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I should note that I have not previously reviewed this paper, but have looked at the previous reviews and will try to take that into account. I also would like to let the record show that based on the TMLR author guidelines, it is still not entirely obvious to me that survey papers like this one are clearly within the scope of this journal, but that remains for the editors to decide.\n\nStrengths:\n- EDL is an important recent development and there is a shortage of good surveys on the subject\n- This version of the paper offers sufficient detail on the different approaches\n\nWeaknesses:\n- I find that considerations regarding the tradeoffs of the different methods and practical guidelines are generally lacking\n- I still find the discussion about the different kinds of uncertainty confusing\n- Some related work seems to still be insufficiently discussed\n\nMajor comments:\n- As a practitioner formerly unfamiliar with EDL, I feel that after reading the paper, I have a good overview of the different approaches. However, I don't really have any good guidance regarding which approach to use for a given problem at hand. While there is a small paragraph in the Appendix, I would appreciate more detail in the main text regarding the respective tradeoffs between the different approaches, e.g., in terms of performance, computational cost, scalability to larger problems, etc.\n- I appreciate the discussion of the three kinds of uncertainty, but I'm afraid I'm still confused. Especially since \"distributional\" uncertainty is not a common concept outside of EDL, I think it could be disambiguated more clearly. For instance, looking at Fig. 4, it seems to me that epistemic uncertainty is a subclass of distributional uncertainty. Surely, Fig 4e seems more \"spread\" than 4d, so should have a higher epistemic uncertainty? Are there cases where one can have low epistemic uncertainty and high distributional uncertainty or does high distributional uncertainty always imply high epistemic uncertainty? Likewise, can one have high epistemic and low distributional uncertainty? If so, I think this should be illustrated more clearly with examples to tell the two apart.\n- While it is mentioned somewhere that deep ensembles [1] are still state-of-the-art in most uncertainty estimation scenarios, they seem to not be mentioned at all in the related work. Especially since there is a recent line of work endowing their uncertainties with theoretical guarantees [e.g., 2,3,4], I think this should be discussed.\n- Similarly, there is a long line of work on single-pass uncertainty methods [e.g., 5,6,7,8], which seems to be entirely missing from the related work, even though it shares the motivation of EDL, namely to provide predictive uncertainties without having to perform model averages. This definitely seems to be a closely related approach that should be discussed.\n\nMinor comments:\n- Fig. 3: the entries in $\\beta$ seem to be shuffled a bit wrt. $N_k$\n- I find that \"Dirichlet Networks\" is a somewhat unexpected title for Sec. 3. Should it rather be called \"Evidential Deep Learning for Classification\" or similar?\n- Sec. 3.1: problem to from two -> problem from two\n- it seems like the spelling Categorical vs. categorical is not consistent in the paper\n- footnotes 3 and 4 somehow don't appear on their respective page\n- Eq. (11): is $\\alpha$ really directly given by $f_\\theta$ without an activation function? so can the $\\alpha$ entries be negative?\n- above Eq (15): see the full derivation is given -> see the full derivation given\n- below Eq. (19): to force the to model -> to force the model\n- above Eq. (20): negativ log-likelihood -> negative log-likelihood\n- p. 12 bottom: an thus is -> and thus is\n- p. 16, bottom: error is rescale -> error is rescaled\n- end of Sec. 4: instead of the decreasing -> instead of decreasing\n- footnote 12: mislead -> misled\n\n\n[1] https://arxiv.org/abs/1612.01474\n\n[2] https://openreview.net/forum?id=BJlahxHYDS\n\n[3] https://proceedings.neurips.cc/paper/2020/hash/0b1ec366924b26fc98fa7b71a9c249cf-Abstract.html\n\n[4] https://openreview.net/forum?id=LAKplpLMbP8\n\n[5] https://arxiv.org/abs/2003.02037\n\n[6] https://arxiv.org/abs/2102.11409\n\n[7] https://arxiv.org/abs/2006.10108\n\n[8] https://arxiv.org/abs/2110.02609\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper offers a survey of different recent methods that were proposed within the framework of evidential deep learning (EDL). It categorizes the different approaches based on whether they are used for classification or regression, parameterize the prior or posterior, and whether or not they need OOD data during training. They also describe EDL in general and provide some discussion of related approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See Major Comments above.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This is the second round of review for this paper.  The previous round essentially ran out of time, after the authors had made a significant revision.  This time around the reviewers were instructed to take the previous reviews into account and assess whether this revision addressed the previous reviews.  One reviewer overlapped, but unfortunately two previous reviewers were unavailable.  The previous meta review is pasted below.

The reviewers all voted for accept and found the paper relevant and interesting. The reviewers noted that they would have found more empirical comparison useful, particularly to help inform under which circumstances to use which method.  However, besides that it seems all the reviewers are satisfied with the paper.  Therefore, the recommendation is to accept as-is.

Previous Meta Review:
This paper presents a survey of evidential deep learning methods for uncertainty estimation in deep neural network classification.  This encompasses a collection of methods that develop the use of the Dirichlet distribution to model the outputs of the classifier.  The Dirichlet provides a natural formulation to balance new evidence with a prior over categorical outputs, and thus has advantages for uncertainty estimation and dealing with out-of-distribution data.

The reviewers found the paper interesting, of appropriate relevance to the community and timely, i.e. a well composed survey about evidential deep learning would be a useful contribution to the community.  They found that the survey seems to capture the breadth of the literature on this topic well.

There was substantial discussion relating to the paper.  In particular, the reviewers all found that the background and explanations of methods were too short and "terse" to serve as an adequate primer or introduction to the area.  The concern was essentially that in order to understand the survey, one had to already be intimately familiar with the methods, which of course is somewhat against the purpose of a survey paper.  Much of this issue seemed to stem from the authors' attempts to maintain a 12 page limit.  Reviewers asked authors to expand the explanations and welcomed going beyond 12 pages to do so.  At least one reviewer asked suggested adding open-source code would make the paper stronger and increase it's impact.

It seems that the authors agreed to expand on the paper and address the reviewers concerns.  They have indeed uploaded a new and significantly longer manuscript that has expanded explanations of the fundamental concepts and underlying methods and addressed other reviewer concerns.  However, this version was uploaded only shortly before reviewer decisions were due, and reviewers felt they did not have adequate time to re-review the new version of the paper.  As the reviewers felt uncomfortable accepting the paper without carefully reading the new version, the decision recommendation of the reviewers was reject, reject and leaning reject.

Therefore, I would recommend a reject.  However, I would strongly encourage that the authors resubmit, taking all the all the reviewer feedback into account, and ask for the same panel of action editor and reviewers (the reviewers have acknowledged that they would be willing to review the new manuscript in the context of these reviews).  The reviewers would like an opportunity to check the new version, but felt that it would have to reflect major revisions, and thus require more time than is allocated.

Decision: Accept (poster)