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: The paper presents a survey of evidential deep learning, a family of machine learning methods that suggest accounting for various forms of uncertainties via a hierarchical predictive model whose hyperprior parameters are a function of input observations. The paper classifies evidential deep learning approaches into categories and points out the strengths and weaknesses of each category. It also presents some theoretical properties of the Dirichlet distribution that could be relevant for machine learning tasks.