A Survey on Evidential Deep Learning For Single-Pass Uncertainty Estimation

1 INTRODUCTION Many existing methods for uncertainty estimation leverage the concept of Bayesian Model Averaging , that approaches such as Monte Carlo ( MC ) Dropout ( Gal & Ghahramani , 2016 ) , Bayes-by-backprop ( Blundell et al. , 2015 ) or ensembling ( Lakshminarayanan et al. , 2017 ) can be grouped under ( Wilson & Izmailov , 2020 ) . This involves the approximation of an otherwise infeasible to compute integral using Monte Carlo samples – for instance from an auxiliary distribution or in the form of ensemble members . This implies the following problems : Firstly , the qual- ity of the MC approximation depends on the veracity and diversity of samples from the weight posterior . Secondly , the approach often involves increasing the number of parameters in a model or training more model instances altogether . Recently , a new class of models has been proposed to side-step this conundrum by using a different factorization of the posterior predictive distribution . This allows to compute uncertainty in a single forward pass and set of weights . Furthermore , these models are grounded in a concept coined Evidential Deep Learning : For out-of-distribution ( OOD ) inputs , they fall back onto a prior , often expressed as knowing what they don ’ t know . Our contributions are as follows : We summarize the existing literature and group these approaches , critically reflecting on their advantages and shortcomings alike , as well as how they fare compared to other , related methods . This survey aims to both serve as an accessible introduction to this model family to the unfamiliar reader as well as an informative overview , in order to promote more applications outside the uncertainty estimation literature . We also provide a collection of the most important theoretical results for the Dirichlet distribution for Machine Learning , which plays a central role in many of the discussed approaches . We give an overview over all discussed work in Figure 1 . 2 BACKGROUND . We first familiarize the reader with the necessary prerequisites for the rest of the survey , including Bayesian Model Averaging and the alternative approach of Evidential Deep Learning in Section 2.2 along with a short introduction to the Dirichlet distribution in the next section . 2.1 THE DIRICHLET DISTRIBUTION . The Beta distribution is a commonly used prior for a Bernoulli likelihood , which can be used to formulate a binary classification problem . The Dirichlet distribution arises as a multivariate generalization of the Beta distribution for a multi-class classification problem and is defined as follows : Dir ( µ ; α ) = 1 B ( α ) K∏ k=1 µαk−1k ; B ( α ) = ∏K k=1 Γ ( αk ) Γ ( α0 ) ; α0 = K∑ k=1 αk ; αk ∈ R+ ( 1 ) where Γ ( · ) denotes the gamma function , a generalization of the factorial to the real numbers , K the number of categories or classes , and B ( · ) is called the beta function . For notational convenience , we also define K = { 1 , . . . , K } as the set of all classes . The distribution is characterized by its concentration parameters α , the sum of which , often denoted as α0 , is called the precision.1 The distribution becomes relevant for applications using neural networks , considering that most modern networks for classification use a softmax function after their last layer to produce a categorical distribution of classes , for which the Dirichlet is a conjugate prior . The class probabilities can be expressed using a vector µ ∈ [ 0 , 1 ] K s.t . µk ≡ P ( y = k|x ) . Then , using a Dirichlet prior for a categorical likelihood , due to its conjugacy , produces a Dirichlet posterior with parameters β , given a data set D = { ( xi , yi ) } Ni=1 of N observations with corresponding labels : 1The precision is analogous to the precision of a Gaussian , where a larger α0 signifies a sharper distribution . p ( µ |D , α ) ∝ p ( D|µ ) p ( µ |α ) = N∏ i=1 K∏ k=1 µ 1yi=k k 1 B ( α ) K∏ k=1 µαk−1k = K∏ k=1 µ ( ∑N i=1 1yi=k ) k 1 B ( α ) K∏ k=1 µαk−1k = 1 B ( α ) K∏ k=1 µNk+αk−1k = Dir ( µ ; β ) ( 2 ) where β is a vector with βk = αk + Nk , with Nk denoting the number of observations for class k and 1 being the indicator function . 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 . This distribution constitutes a distribution over categorical distributions over the K−1 probability simplex , multiple instances of which are shown in Figure 2 . Each point on the simplex corresponds to a categorical distribution , with the proximity to a corner indicating a high probability for the corresponding class . Figure 2a displays the predictions of an ensemble of classifiers as a point cloud on the simplex . Using a Dirichlet , this finite set of distributions can be extended to a continuous density over the whole simplex ( Figures 2b to 2f ) . 2.2 PREDICTIVE UNCERTAINTY IN NEURAL NETWORKS . In probabilistic modelling , uncertainty is commonly divided into aleatoric and epistemic uncertainty ( Der Kiureghian & Ditlevsen , 2009 ; Hüllermeier & Waegeman , 2021 ) . The former refers to the uncertainty that is induced by the data-generating process , and which e.g . might create an unresolvable overlap in class distributions.2 The latter describes the uncertainty about the optimal model parameters ( or even hypothesis class ) , reducible with an increasing amount of data , as less and less possible models become a plausible fit . These two notions resurface when formulating the posterior predictive distribution of a classifier for a new data point x:3 p ( y|x ) = ∫ P ( y|x , θ ) ︸ ︷︷ ︸ Aleatoric p ( θ |D ) ︸ ︷︷ ︸ Epistemic dθ ( 3 ) For a large number of real-valued parameters θ like in neural networks , this integral becomes intractable to evaluate , and thus is usually approximated using Monte Carlo samples – with the aforementioned problems of potential computational overhead and approximation errors . Malinin & Gales ( 2018 ) thus propose to factorize Equation ( 3 ) further : p ( y|x ) = ∫∫ P ( y|µ ) ︸ ︷︷ ︸ Aleatoric p ( µ |x , D ) ︸ ︷︷ ︸ Distributional p ( θ |D ) ︸ ︷︷ ︸ Epistemic dµ dθ = ∫ P ( y|µ ) p ( µ |x , θ̂ ) ︸ ︷︷ ︸ p ( θ |D ) =δ ( θ−θ̂ ) dµ ( 4 ) In the last step , we replace p ( θ |D ) by a point estimate θ̂ using the Dirac delta function , i.e . a single trained neural network , to get rid of the intractable integral . Although another integral remains , retrieving the uncertainty from this predictive distribution actually has a closed-form analytical solution for the Dirichlet ( see Section 3.2 ) . The advantage of this approach is further that it allows us to differentiate uncertainty about a data point because it lies in a region of considerable class overlap ( Figure 2c ) from it differing from the training distribution entirely ( Figure 2e ) . Assuming that a point estimate of the parameters suffices prevents one from estimating epistemic uncertainty like in earlier works , as discussed in the next section . However , there are works like Haussmann et al . ( 2019 ) ; Zhao et al . ( 2020 ) that combine both approaches . 2Unless additional features are added . 3Note that the predictive distribution in Equation ( 3 ) recovers the common case for a single network prediction where P ( y|x , θ ) ≈ P ( y|x , θ̂ ) . Mathematically , this is expressed by replacing the posterior p ( θ |D ) by a delta distribution like in Equation ( 4 ) , where all probability density rests on a single parameter configuration . 3 DIRICHLET NETWORKS . We will show in Section 3.1 how neural networks can parameterize Dirichlet distributions , while Section 3.2 reveals how such parameterization can be exploited for efficient uncertainty estimation . The remaining sections enumerate different examples from the literature parameterizing either a prior ( Section 3.3.1 ) or posterior Dirichlet distribution ( Section 3.3.2 ) according to Equations ( 1 ) and ( 2 ) . 3.1 PARAMETERIZATION . For a classification problem with K classes , a neural classifier is usually realized as a function fθ : RD → RK , mapping to logits for each class given an input x ∈ RD . Followed up by a softmax function , this then defines a categorical distribution over classes with a vector µ s.t . µk ≡ p ( y = k|x , θ ) . The same architecture can be used without any major modification to instead parameterize a Dirichlet distribution , as in Equation ( 1 ) .4 In order to classify a data point x , a categorical distribution is created from the predicted concentration parameters of the Dirichlet as follows ( this definition arises from the expected value , see Appendix A.1 ) : α = fθ ( x ) ; µk = αk α0 ; ŷ = arg max k∈K µ1 , . . . , µK ( 5 ) As discussed in Section 3.3.2 , this process is very similar when parameterizing a Dirichlet posterior distribution , except that in this case , a term corresponding to the class observation in Equation ( 2 ) is added to every concentration parameter as well . 3.2 UNCERTAINTY ESTIMATION WITH DIRICHLET NETWORKS . Let us now turn our attention on how to estimate the different notions of uncertainty laid out in Section 2.2 within the Dirichlet framework . Although stated for the prior parameters α , the following methods can also be applied to the posterior parameters β as well without loss of generality . Data ( aleatoric ) uncertainty . For the data uncertainty , we can evaluate the expected entropy of the data distribution p ( y|µ ) ( 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 all probability mass on a single class , while high entropy stands for a more uniform distribution – the model thus is undecided about the right prediction . For Dirichlet networks , this quantity has a closed-form solution ( for the full derivation , refer to Appendix B.1 ) : Ep ( µ |x , θ̂ ) [ H [ P ( y|µ ) ] ] = − K∑ k=1 αk α0 ( ψ ( αk + 1 ) − ψ ( α0 + 1 ) ) ( 6 ) where ψ denotes the digamma function , defined as ψ ( x ) = ddx log Γ ( x ) , andH the Shannon entropy . Model ( epistemic ) uncertainty . As we saw in Section 2.2 , computing the model uncertainty in the classical sense via the weight posterior p ( θ |D ) like in Blundell et al . ( 2015 ) ; Gal & Ghahramani ( 2016 ) ; Smith & Gal ( 2018 ) is usually not done in the Dirichlet framework ( with exceptions such as Haussmann et al. , 2019 ; Zhao et al. , 2020 ) . Nevertheless , the defining property of Dirichlet networks is that epistemic uncertainty is expressed in the spread of the Dirichlet distribution ( for instance in Figure 2 ( d ) and ( e ) ) . Therefore , the epistemic uncertainty can be quantified considering the concentration parameters α that shape this very same distribution : Charpentier et al . ( 2020 ) simply consider the maximum αk as a score akin to the maximum probability score by Hendrycks & Gimpel , while Sensoy et al . ( 2018 ) compute it by K/ ∑K k=1 ( αk + 1 ) or simply α0 ( Charpentier et al. , 2020 ) . In both cases , the underlying intuition is that larger αk produce a sharper density , and thus indicate increased confidence in a prediction . 4The only thing to note here is that the every αk has to be strictly positive , which can for instance be enforced by using an additional ReLU function ( and adding a small value , e.g . like in Sensoy et al. , 2020 ) on the output or predicting logαk instead ( Sensoy et al. , 2018 ; Malinin & Gales , 2018 ) . Distributional uncertainty . Another appealing property of this model family is to distinguish uncertainty due to model underspecification ( Figure 2d ) from uncertainty due to alien inputs ( Figure 2e ) . In the Dirichlet framework , the distributional uncertainty can be deduced by computing the difference between the total amount of uncertainty and the data uncertainty , which can be expressed in terms of the mutual information between the label y and its categorical distribution µ : I [ y , µ ∣∣∣x , D ] = H [ Ep ( µ |x , D ) [ P ( y|µ ) ] ] ︸ ︷︷ ︸ Total Uncertainty −Ep ( µ |x , D ) [ H [ P ( y|µ ) ] ] ︸ ︷︷ ︸ Data Uncertainty ( 7 ) Given that E [ µk ] = αkα0 ( Appendix A.1 ) and assuming the point estimate p ( µ |x , D ) ≈ p ( µ |x , θ̂ ) to be sufficient ( Malinin & Gales , 2018 ) , we obtain an expression very similar to Equation ( 6 ) : = − K∑ k=1 αk α0 ( log αk α0 − ψ ( αk + 1 ) + ψ ( α0 + 1 ) )

Summary: This paper surveys a collection of existing works that the author frames as evidential deep learning. For the classification case, evidential deep learning tries to train a network to output the parameters of a Dirichlet distribution, hoping the one could directly obtain the data uncertainty and model uncertainty from some functions of the output parameters. The authors provide a relatively comprehensive review of recent work in this direction. Besides classification, evidential deep learning for regression is also discussed briefly.