Evidential Turing Processes 
Authors: Melih Kandemir
Abdullah Akgül
Manuel Haussmann
Gozde Unal
Venue: ICLR 2022 Poster (2022)
Keywords: Evidential Deep Learning
Neural Processes
Attention
Neural Turing Machines

Abstract: A probabilistic classifier with reliable predictive uncertainties i) fits successfully to the target domain data, ii) provides calibrated class probabilities in difficult regions of the target domain (e.g. class overlap), and iii) accurately identifies queries coming out of the target domain and reject them. We introduce an original combination of Evidential Deep Learning, Neural Processes, and Neural Turing Machines capable of providing all three essential properties mentioned above for total uncertainty quantification. We observe our method on three image classification benchmarks to consistently improve the in-domain uncertainty quantification, out-of-domain detection, and robustness against input perturbations with one single model. Our unified solution delivers an implementation-friendly and computationally efficient recipe for safety clearance and provides intellectual economy to an investigation of algorithmic roots of epistemic awareness in deep neural nets.

Published as a conference paper at ICLR 2022
EVIDENTIAL TURING PROCESSES
Melih Kandemir
Abdullah Akgül
Dept of Math and Computer Science University of Southern Denmark
Department of Computer Engineering
Istanbul Technical University
Odense, Denmark
Istanbul, Turkey
kandemi r imada. sdu. ck
akgula150itu. edu. .r
Manuel Haussmann"
Gozde Unal
Department of Computer Science
Department Computer Engineering
Aalto University
Istanbul Technical University
Espoo, Finland
Istanbul, Turkey
manuel whaussmnn@alllto...
gozde. www.lunaaiiuu edu. edu.tz
ABSTRACT
A probabilistic classifier with reliable predictive uncertainties i) fits successfully to the target domain data, ii) provides calibrated class probabilities in difficult regions of the target domain (c.g. class overlap), and iii) accurately identifies queries coming out of the target domain and rejects them. We introduce original combination of Evidential Deep Learning, Neural Processes, and Neural Turing Machines capable of providing all three essential properties mentioned above for total uncertainty quantification. We observe our method on five classification tasks be the only one that can excel all three aspects of total calibration with single standalone predictor. Our unified solution delivers an pigllmmentationffienddee and compute efficient recipe for safety clearance and provides intellectual economy investigation of algorithmic roots of epistemic awareness in deep neural nets.
INTRODUCTION
The applicability of deep neural nets to safety-critical use such autonomous driving or medical diagnostics is an active matter of fundamental research (Schwalbe & Schels, 2020). Key challenges in the development of total safety in deep learning are at least three-fold: i) evaluation of model fit, ii) risk assessment in difficult regions of the target domain (e.g. class overlap) based on which aafety-preserving fallback functions can be deployed, and iii) rejection of inputs that do not belong to the target domain, such as an image of cat presented to a character recognition system. The attention of the machine learning community to each of these critical safety elements in steady increase (Naeini et al., 2015; Guo et al., 2017; Kuleshov et al., 2018). However, the developments often follow isolated directions and the proposed solutions are largely fragmented. Calibration of neural net probabilities focuses primarily
post-hoc adjustment algorithms trained on validation sets from in-domain data (Guo et al., 2017), ignoring the out-of-domain detection and model fit evaluation aspects. On the other hand, recent advances in oft-of-domain detection build strong penalization of divergence from the probability mass observed in the target domain (Sensoy et al., 2018; Malinin & Gales, 2018), distorting the quality of class probability scores. Such fragmentation of best practices not only hinders their accessibility by real-world applications but also complicates the scientific inquiry of the underlying reasons behind the sub-optimality of neural net uncertainties.
We aim to identify the guiding principles for Bayesian modeling that could deliver the most complete set of uncertainty quantification capabilities in one single model. We first characterize Bayesian models with respect to the relationship of their global and local variables: (a) Parametric Bayesian Models comprise likelihood function that maps inputs to outputs via probabilistic global param- eters, (b) Evidential Bayesian Models apply an uninformative prior distribution the parameters of the likelihood function and infer the prior hyperparameters by amortizing on an input observa- tion. Investigating the decomposition of the predictive distribution variance, find out that the Work done while Heidelherg University, Germany.

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Published as a conference paper at ICLR 2022
0
B
y
cr v) E
(1 v) E
(x, y)
(a) Parametric Bayesian Model
(b) Evidential Bayesian Model
(c) Complete Bayesian Model
Figure I: Plate diagrams of three main Bayesian modeling approaches. Shaded nodes are observed and unshaded ones are latent. The solid direct arrows denote conditional dependencies in the true model, while the bent dashed arrows denote amortization in variational inference.
uncertainty characteristics of these two approaches exhibit complementary strengths. We introduce a third approach that employs prior on the likelihood parameters that both conditions the input observations on the true model and amortizes on them during inference. The mapping from the input observations the prior hyperparameters is governed by random set of global parameters. We refer to the resulting approach as a (c) Complete Bayesian Model, since its predictive distribution variance combines the favorable properties of Parametric and Evidential Bayesian Models. Figure 1 gives an overview of the relationship between these three Bayesian modeling approaches.
The advantages of the Complete Bayesian Models come with the challenge of choosing input- dependent prior. We introduce a new stochastic process construction that addresses this problem by accumulating observations from context during training time into the parameters of global hyperprior variable by an explicitly designed update rule. We design the input-specific prior on the likelihood parameters by conditioning it on both the hyperprior and the input observation. As conditioning via explicit parameter updates amounts to maintaining an external memory that can be queried by the prior distribution, we refer to the eventual stochastic process as a Turing Process with inspiration from the earlier work on Neural Turing Machines (Graves et al., 2014) and Neural Processes (Garnelo et al., 2018b). We arrive our target Bayesian design that equipped with the complete set of uncertainty quantification capabilities by incorporating the Turing Process design into a Complete Bayesian Model. We refer to the resulting approach as an Evidential Turing Process. We observe on five real- world classification tasks that the Evidential Turing Process is the only model that excels simultaneously at model fit, class overlap quantification, and out-of of-mmmn detection.
2 PROBLEM STATEMENT: TOTAL CALIBRATION
Consider the following data generating process with K modes
ा Cat(yl=),
ar by
Σ kypapl = k),
where Car(yl-) is categorical distribution, I, is the indicator function for predicate and p(ælv) the conditionniiinn density function on input observation æ. For any prior class distribution n, the classification problem can be cast as identifying the class probabilities for observed patterns (z)) where firue (x) is a K -imeensional vector with the k-th element
Pr[y|a, pr, π|
K
pi.k (x) Tap(Zly k)) / Σ
T <xp(oly xt)
In many real-world cases fire is not known and should be identified from a hypothesis space hy E H. via inference given a set of samples D {E In In I....,, obtained from the true distribution. The risk of the Bayes classifier arg max, Carly) (x)) for a given input r is
(())
Σ n=1
Lxstarg max by (m) (
(3)
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For the whole data distribution, we have R(h+) Buj= [x))) The optimal case is when (x:) Jirue (3)), which brings about the irreducible Bayes risk resulting from class overlap:
R₂ (Utrue())) min {1 {1- max Strue (r), max Strue (z)
Total Calibration. Given a set of test samples Dio coming from the true data generating process in Eq. 1, we define Total Calibration as the capability of a discriminative predictor h, (z) quantify the following three types of uncertainty simultancously:
i) Model misfit evaluated by the similarity of (x) and firue (x) measurable directly via
(L(Cat(y)) (x))p(>) Car(y) ha (z)
Ep(s) [log Carlyl firue (x)] + Ep(s) |- log Carly)((n
where KL(-II-) denotes the Kullback-Leibler divergence. Since Ep(a) [log Cat(y) ftrue (x))] constant with respect to (x)), an unbiased estimator of the relative goodness of inferred model is the negative test log-likeling li-likkeiihod N (LL(hn(x)) = 1// (x,)))D. log hx
ii) Class overlap evaluated by Ra(Serue (x)). As there is way measure this quantity from D... is approximated indirectly via Expected Calibration Error (ECE)
M Σ m=
Bml N
= ECER
(ac(Bmm) con c (Bm)
where Bm = ((n()T((a))) E [(m - 1)/M, m/M) M]} are M bins that partition the test set Dts into DiS = B1 U... Bas and are characterized by their accuracy and confidence scores
acc(Bm)) =
1
Σ
Iy margarg max by (sr,))
con f =
Σ (xn)). mE B.m
(5)
[Bml
[m
ne neB.
tii) Domain mismatch defines the rarity of an input pattern xz for the target task, that is Príp(z= = z.) < for small e. This quantity cannot be measured since p(z) is unknown. It is approximated indirectly as the success of discriminating between samples x. coming from a different data distribution and those in Drs by calculating the Area Under ROC (AUROC) curve w.r.t. h (x).
UNCERTAINTY QUANTIFICATION WITH PARAMETRIC AND EVIDENTIAL BAYESIAN MODELS
Parametric Bayesian Models. A commonplace design choice is to build the hypothesis space by equipping the hypothesis function with a global parameter 0 that takes values from a fea- sible set O that is Hr [h (x)l0 E O). Parametric Bayesian Models (PBMs) employ a prior belief 9 ~ p(0), which updated via Bayesian inference on a training set Der as p())D.,)) I((.))), Ca(ly)l (z))p(0)/p((..,
Consider the update on the posterior belief
D U (I,, v+)) = Car(y-/h (x=)))(((())))))
(6)
after new observation (I., V-) where the normalizer be expressed
2 p(Drr) (a
with L(0)- (x.g)EDD Car(y [h (x)). (x)). This can be seen as an inner product between the functions
Car(y, (v (u (x.))) and (.()) on an embedding space modulated by p(0). As the sample size grows, the
high-density regions of Cat(y.Ihg (n..)/h)((x.)) (xx)) and L(0) will superpose with higher probability, causing Z
to increase, making p(0 Dr, (x.,3.)) increasingly more peaked, and consequently we will have
lim too
(0 = 0.
(8)
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This conjecture depends on the assumption that single random variable modulates all samples. The variance of a prediction in favor of a class &s decomposes according to the law of total variance
Varly. k| D] V Vare-p(e D:,) 1 (2r+) Ee-p(01D...) [11 ( (x.))) (ote)
(9)
Reducible model uncertainty
Data uncertainty
Due to the Eq. 8, the first term on the r.h.s. vanishes in the large sample regime and the second one coincides with the asymptotic solution of the maximum likelihood estimator dM In cases when H, contains (z), an ideal inference scheme has the potential to recover and we get
lim E N-+000
(1 (= (zx))
(1 h M (z.) h
OMLE (x+) R.
= -1- Sirue
The simple proposition below sheds light the meaning of this result. points the fact that the Bayes risk of a classifier (Eq. 4) and the second component of its predictive variance (Eq 9) are proportional. This gives a mechanistic explanation for why the second term on the r.h.s. of Eq. quantifies class overlap. Despite the commonplace allocation of the wording data uncertainty for this term, we are not aware of prior work that derives it formally from basic learning -arming-theoretic theoretic concepts.
Proposition. The following inequality holds for any π, x' E [0.5, 1) pair
min(1 n, x) min(1 nL, m²) (1 अ)) (1 x')''
Proof. Define π = 0.5 and n' 0.5+ f, > 0. As min(1 -, 2 min(1 - (1-=,,=))>=-
é, we get - π) -1- 0.5 (1-0.--))0-5++) 0.25 10.25-0...0 0.25c 1-55-- 2 0.25 0.25c' - (1-m')='l
The conclusion is that (1 - Strue (z))) x R. (firuc Hence, the second term on the r.h.s. of Eqq 9 caused by the class overlap and it can be used approximate the Bayes risk. Put together, the first term of the decomposition reflects the missing knowledge on the optimal value of 0 that can be reduced by increasing the training set size, while the second reflects the uncertainty stemming from the properties of the true data distribution. Hence we refer to the first term as the reducible model uncertainty and the second the data uncertainty.
Evidential Bayesian Models. In all the analysis above, we assumed fixed and unknown that modulates the whole process and cannot identified by the learning scheme. When we characterize the uncertainty on the class distribution by a prior belief π ~ p(r), we attain the joint The variance of the Bayes optimal classifier that averages over this Pr x]p(T]/)))))))))) decomposes
ply, a = Pr[y x a xlp(T(()) uncertainty Pr(y|æ]
Varly [|ar] = Var (=)["<((()))) [ . (1:) E (app(()))
(1
pT.kk Jtrue (x)) (rrle((r)) (x:)
Inreuciill model uncertainty
uncertainty
The main source of uncertainty p(T[a)) (z) is not a posterior this time but an empirical prior on a single observation æ, hence its variance will not shrink |Darl +00. Therefore, the first term on the r.h.s. reflects the irreducible model uncertainty caused by the missing knowledge about æ. The second term indicates data uncertainty in PBMs, but accounts for the local uncertainty at x caused by æ. Evidential Bayesian Models (EBMs) (Sensoy et al., 2018; Malinin & Gales, 2018) suggest
ply, [c) Pr(y (r, [|p(T//>) Pr(y (T/())
where qu (z(zz is a density function parameterized by vi and the dependency of the class-conditional on the input is dropped. Note that the resultant model employs uncertainty individual data points via æ. but it does not have any global random variables. The standard evidential model training is performed by maximizing the expected likelihood subject a regularizer that penalizes the divergence of qu (TT x)) from the prior belief
/ log ( + (KL(g. (p(T)).
arg min
Although presented in the original work as-hoc design choices, such training objective can alternatively be viewed as 3-regularized (Higgins et al., 2017) variational inference Pr m (T](()) with the approximate posterior qu (ar) (Chen et al., 2019).
Pr(y|
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4 COMPLETE BAYESIAN MODELS: BEST OF BOTH WORLDS
The uncertainty decomposition characteristics of PBMs and EBMs provide complementary strengths towards quantification of the three uncertainty types that constitute total calibration.
i) Model misfit: The PBM is favorable since it provides shrinking posterior variance with growing training set size (Eq. 8) and the recovery of 0 mme which inherits the consistency guarantees the frequentist approach. Since the uncertainty on the local variable л does not shrink for large samples, NLL would measure how well en can capture the regularities of heteroscedastic noise across individual samples, which is more difficult task than quantifying the fit of a parametric model.
ii) Class overlap: The EBM is favorable since its data uncertainty term Ex ha (nr)))nn, (z.) []..] is likely to have less estimator variance than Ee-p(o)D,,,) (... [11-in (2. ()x 2 (x,)] and calculating p(r( does not require approximate inference as for p( (((()r))
iii) Domain mismatch: The PBM is favorable since the effect of the modulation a global on the variance of Z applies also to the posterior predictive distribution with the only difference that t. is a test sample. When the test sample is away from the training set, i.e. millze I| x|| is large, then (2,,12 0) is less likely to superpose with those regions of where L(0) is pronounced, hence will be flattened out leading to a higher variance posterior predictive for samples coming from another domain. Since EBM has only local random variables, the variance of its posterior less likely to build a similar discriminative property for domain detection.
Main Hypothesis. A model that inherits the model uncertainty PBM and data uncertainty of EBM can simultaneously quantify: i) model misfit, ti) class overlap, and iii) domain mismatch.
Guided by our main hypothesis, we combine the PBM and EBM designs within unified framework that inherits the desirable uncertainty quantification properties of each individual approach
Pr pr x))(()
(10)
which we refer to as Complete Bayesian Model (CBM) hinting to its capability to solve the total calibration problem. The model simply introduces global random variable into the empirical prior p(z(0,z)) of the EBM. The variance of the posterior predictive of the resultant model decomposes as
V Varlyl [y|a]] = V ((rp( Ep(r [B[y] x|]
+ Ep(orD V Var p=((00== [[[y] mT]]
Reducible Model Uncertainty
Trreducible Model Unceraiitt
+ Epcord)
E Bp [r]l
(11)
Uncertainty
where the r.h.s. of the decomposition has the following desirable properties: i) The first term recovers the form of the reducible model uncertainty term of PBM when π is marginalized Varp(o))) (pp())200) [B[y]T, [ æI1l = V p) EEyl|, (0) æ]]; ii) The second term is the irreducible model D
uncertainty term of EBM averaged over the uncertainty on the newly introduced global variable f. quantifies the portion of uncertainty stemming from heteroscedastic noise, which is not explicit in the decomposition of PBM; iii) The third term recovers the data uncertainty term of EBM when marginalized: E)(erD)
(Bp(r,)) [[]()]))) [Var [r[y|=]]]
Ep ax) (V /arly/]]]
5 THE EVIDENTIAL TURING PROCESS: AN EFFECTIVE CBM REALIZATION
Equipping the empirical prior p(T|0,2) x) of CBM with the capability of learning to generate accurate prior beliefs on individual samples is the key for total calibration. We adopt the following two guiding principles to obtain highly expressive empirical priors: i) The existence of a global random variable 0 can be exploited to express complex reducible model uncertainties using the Neural Processes (Gamelo et al., 2018b) framework, ii) The conditioning of p( on global variable 0 and an individual observation I can be exploited with an attention mechanism where 0 is memory and x is query. Dependency context set during test time can be lifted using the Neural Turing Machine
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(Graves al., 2014) design, which maintains external memory that is updated by rule detached from the optimization scheme of the main loss function. We extend the stochastic process derivation of the Neural Processes by marginalizing out local variable from PBM with external memory that follows the Neural Turing Machine design and arrive at a new family of stochastic processes called Turing Process, formally defined as below.
Definition 1. A Turing Process is a collection of random variables Y = { E I) for an index ser = {I....,K) with arbitrary K N that satisfy the two properties
(Y) / II (m) (ii) PM (Y = / II p(y:1))Mr (0 Y')d,, WEY
YEYY
for another random variable collection Y' of arbitrary size that lives the same probability space as Y, a probability measure PM with free parameters M, and some function M' = ~rM,
The Turing Process can express all stochastic processes since property (i) is sufficient to satisfy Kolmogorov's 5 Extension Theorem (Øksendal, 1992) and choosing simply to be an identity mapping would revert us back to the generic definition a stochastic process. The Turing Process goes further by permitting to explicitly specify how information accumulates within the prior. This is a different approach from the Neural Process that performs conditioning p(Y by applying an amortized posterior ( Y") on a plain stochastic process that satisfies only Property (i), hence maintains data-agnostic prior p(0) and depends on a context set also at the prediction time.
Given disjoint partitioning of the data D De U DD into context set v') E De and target set (z. y) E Dr, we propose the data generating process on the target set below
ply. Dr, n. p(w) pM
II (2.
((ylT) p( Z, w,
(12)
External memory
speciisp
prior
We decouple the global parameters 0 = (w, 2} into a w that parameterizes a map from the input to the hyperparameter space and external memory Z = {=1,-..; .R) governed by the distribution Par (2) parameterized by M that embeds the context data during training. The memory variable Z updates its belief conditioned on the context set De by updating its parameters with an explicit rule PM (Z) + PE (r), (Z). The hyperpriors of p( Z, 2)) are then determined by querying the memory Z for each input observation using attention mechanism, for instance transformer network (Vaswani al., 2017). Choosing the variational distribution as gx (0, π T|/) = qx (w))ss (Z)))))(()9()))))))) w. z),, attain the variational free energy formula for target model
FFTP(A)
(13)
2 2, 2 x)) p( Z, w.
DA (w) p(w)
Eng (w0)
Σ 2,) (I)D))D
logp(y/>) + log
log
The learning procedure will then alternate between updating the variational parameters via gradient- descent on FA) and updating the parameters of the memory variable Z via an external update rule as summarized in Figure 2b. ETP can predict a test sample (I=, 3=) by only querying the input I. on the memory Z and evaluating the prior p(T. / IZ, w. xx) without need for any context set
p(y.|Dors xI)) = /f p(u.) IZ, w, x. )x (Z)qx (Z)axw))dZerl (w)dZdw.
(14)
6 MOST IMPORTANT SPECIAL CASES OF THE EVIDENTIAL TURING PROCESS
As we build the design of the Evidential Turing Process (ETP) on the CBM framework that over- arches the prevalent parametric and evidential approaches, its ablation amounts to recovering many state-of-the--rt modeling approaches as listed in Table 1 and detailed below.
Bayesian Neural Net (BNN) (Neal, 1995; MacKay, 1992; 1995) is the most characteristic PBM example that parameterize a likelihood p .r) with a neural network fo(.) whose weights follow assume as a prior 0 ~ (0 3 I) and infer its
distribution 0 ~ p(0). In the experiments posterior by mean-field variational posterior qı (0) applying the reparameterization trick (Kingma et al., 2015; Molchanov et al., 2017).

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while Model not converged do
Z
for Dbatch C DD do
Choose De C Diatch M f
Ez-pac [r(Z, Dc) A - (xFETr(A)
y'
w
y
end
(2,0)) DC
(r. v) By
end
(a) ETP Plate diagram
(b) ETP Training
Figure 2: The Evidential Turing Process: (left) The essential design choice is how p(T [2 w. x)) is constructed thanks to an external memory M that can be queried with respect to any input x without needing to store context data at test time. The dashed arrow indicates that can condition on context Dc by updates its parameters M with an explicit function r. (right) The ETP training routine.
Table Ablation Table. Deactivating the components of ETP one at a time equates to state-of-the- art methods. We curate ENP as a surrogate for the Attentive NP (ANP) of (Kim et al., 2019), which requires test-time context, and an improved NP variant with reduced estimator variance thanks to m.
Model
M
De Ditestt
(tost)
Compare
ETP (Target)
X
ANP (Kim et al., 2019)
No
ENP (Surrogate)
X
X
Yes
NP (Garnelo et al., 2018b)
X
X
No
EDL (Sensoy al., 2018)
Yes
BNN (Molchanov et al., 2017)
X
X
X
X
Yes
V: Component active
x: Component inactive
D(test), Demand for context data at test time
Evidential Deep Learning (EDL) (Sensoy et al., 2018) introduces the characteristic exam- ple of EBMs. EDL employs uninformative Dirichlet prior on the class probabilities, which then set the mean of a normal distribution on the one-hot-coded class labels y, π ~ Dir(T(I,... (2) 0.5lx). EDL links the input observations to the distribution of class probabilities by performing amortized variational inference with the approximate distribution (x(x)))) Dir(z)ax(x))
Neural Processes (NP) (Garnelo et al., 2018a;b) is PBM used to generate stochastic processes from neural networks by integrating out the global parameters f as in Property (i) of Definition 1. NPs differ from PBMs by amortizing an arbitrary sized context set De on the global parameters ( (e) via an aggregator network during inference. NPs can be viewed as Turing Processes with an identity map r. Follow-up work equipped NPs with attention (Kim al., 2019), translation equivariance (Gordon et al., 2020; Foong et al., 2020), and sequential prediction (Singh et al., 2019; Yoon et al., 2020). All of these NP variants assume prediction time access to context, making them suitable for interpolation tasks such as image in in-painting but not for generic prediction problems.
Evidential Neural Process (ENP) is the EBM variant of neural process we introduce to bring together best practices of EDL and NPs and make the strongest possible bascline for ETP, defined as ply. x, Z(Dc) = Car(y) (ir())
0²) = ((,(())))))) (25,8y) E Dcl)), (15)
with aggregation rule r(.), encoder net ha (:(.,),, and neural net ax(-) mapping the global variable Z to the class probability simplex. We further improve upon ENP with an attentive NP approach (Kim et al., 2019) by replacing the fixed aggregation rule with attention mechanism, thereby allowing the model to switch focus depending on the target. At the prediction time, we use 7 A( 1, x²1 for small induce uninformative prior in the absence of context set.
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7
CASE STUDY: CLASSIFICATION WITH EVIDENTIAL TURING PROCESSES
We demonstrate a realization of an Evidential Turing Process to a classification problem below
(w)) 8 1T y, n., w1 Z|M (()(y)))Dir(r) ())
II (Mz (mr, x² I).. (16)
The global variable Z is parameterized by the memory M (m1,..... mR) consisting of cells my RK! The input data is mapped to the same space via an encoding neural net un
parameterizing a Dirichlet distribution over the class probabilities и via an attention mechanism a(tw,> 2) = ΣvEZ (()(()) z²)z, where (x), z) = ((k Nwn (x)) / E 2)). The function (.) is the key generator network operating on the embedding space. We update the memory cells as weighted average between remembering and updating
(Vvn 2) (onehot(y) + soft(en (e))))
m 4 Ezzpp(ZMM)
tanh (ym + (1 (1
Σ (r.y)EDe
(17)
E E (0. 1) is fixed scaling factor controlling the relative importance. The second term adds new information to the cell as weighted sum over all pairs (ar, y). taking both the true label (via onchot(y)) as well as the uncertainty in the prediction (via soft( un (x))) into account. The final tanh(.) transformation ensures that the memory content remains in a fixed range across updates, as practised in LSTMs (Hochreiter & Schmidhuber, 1997).
where
8 EXPERIMENTS
We benchmark ETP against the state of the art as addressed in the ablation plan in Table according the total calibration criteria developed in Sec. 2 five real-world data sets. See the Appendix B for full details on the experimental setup in each case and for the hyperparameters used throughout. We provide a reference implementation of the proposed model and the experimental pipeline."
Total calibration. Table 2 reports the predic- tion error and the three performance scores intro- duced in Sec. 2 that constitute total calibration. We perform the out-of-domain detection task as in Malinin & Gales (2018) and classify the test split of the target domain and a data set from another domain based on the predictive entropy. ETP is the only method that consistently ranks among top performers in all data sets with re- spect to all performance scores, which supports our main hypothesis that CBM should outper- form PBM and EBM in total calibration.
FMNIST-C
CIFAR1O-C
SVHN-C
60
BNN
15
EDL
10.5
ENP
Error
13
ETP
40
9.5
11
30
8.5
(%)
10
17.5
10
12.5
6
7.5
3
5 Severity
3
Severity
Severity
Response gradual domain shift. In order to assess how well models can cope with grad- ual transition from their native domain, we eval- data perturbed
Figure 3: Results averaged across 19 types of cor- ruption (e.g. motion blur, fog, pixelation) applied on the test splits of FMNIST, CIFAR10 10, and SVHN at five severity levels.
uate their ECE performance by 19 types of corruption (Hendrycks & Diet- terich, 2019) at five different severity levels. Figure 3 depicts the performance of models averaged across all corruptions. ETP gives the best performance nearly in all data sets and distortion severity levels (see Appendix B.3 Table 4 for a tabular version of these results).
Computational cost. We measured the average wall-clock time per epoch in CIFAR10 to be 10.8 + 0.1 seconds for ETP, 8.2 0.1 seconds for BNN, 8.6 0.1 seconds for EDL, and 9.5 + 0.2 seconds for ENP. The relative standings of the models remain unchanged in all the other data sets.
In the experiments we fix x² = 0.1. Preliminary results showed stability as well for larger/smaller values. Pttpps: //gi / thub. com/ iuvi i on aab//v derriinnii.. a l'ruri a
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Table 2: Quantitative results on five data sets showing mean + standard deviation across 10 repetitions. Best performing models that overlap within three standard deviations are highlighted in bold.
Domain Data
IMDB
Fashion (LeNet5)
SVHN
CIFAR10 (LeNet5)
CIFARI00 (ResNet18)
(Architecture)
(LSTM)
(LeNet5)
Prediction accuracy as % test error
BNN
16.4 +
7.910.1
7.910.1
15.3
30.2 +0.3
EDL
38.3 133
8.6=0.1
7.3+0.1
18.5=0.2
45.2 +0.4
ENP
50.0 +0.0
7.9
6.7
14.8+0.2
39.0 =0.3
ETP (Target)
15.8 1
7.9
6.9
15.3
29.2 =0.3
In-domain calibration as % Expected Calibration Error (ECE)
BNN
14.4 10.4
6.7+0.
6.5+0.
5.5=0.3
15.2
EDL
41.1 12.6
3.7=0.2
4.0+0.1
9.0+0.2
5.3
ENP
0.8 1
6.0
10.7:0.2
7.2+0..3
39.7
ETP (Target)
3.1 =0.4
2.6::.22
2.6:1.11
2.7:0.1
6.6
Model
fit negative test log-likelihood
BNN
0.47 +o.0
0.65+1..0
0.71
0.50=0000
1.78+0.0
EDL
0.66 #0.1
0.37
0.34
0.72
2.24
ENP
0.69 =0.0
0.34
0.33+10.0
0.50
2.52+0.0
ETP (Target)
0.37 +o.00
0.29
0.26+10.
0.46
1.36
Out-of-domain Random
detection as % Area Under ROC Curve
OOD Data
MNIST
CIFAR100
SVHN
TinyImageNet
BNN
60.9 4
75.9+2.3
86.2
84.1+1.3
97.5
EDL
55.1 1
77.5+2.0
90.9+0.3
79.2
89.6#0.3
ENP
53.7 +
88.911.0
92.410.4
81.410.8
100.0
ETP (Target)
59.1 +
90.010.9
90.0
82.110.6
99.6
CONCLUSION
Summary. We initially develop the first formal definition of total calibration. Then we analyze how the design of two mainstream Bayesian modeling approaches affect their uncertainty charac- teristics. Next we introduce Complete Bayesian Models as a unifying framework that inherits the complementary strengths existing ones. We develop the Evidential Turing Process an optimal realization of Complete Bayesian Models. We derive an experiment setup from our formal definition of total calibration and a systematic ablation of target model into the strongest representatives of the state of the art. We observe in five real-world tasks that the Evidential Turing Process is the only model that can excel all three aspects of total calibration simulatenously.
Broad impact. Our work delivers evidence for the claim that epistemic awareness of a Bayesian model is indeed a capability learnable only from in-domain data, as appears in biological intelligence via closed-loop interactions of neuronal systems with stimuli. The implications of our work could follow up in interdisciplinary venues, with focus on the relation of associative memory and attention to the neuroscientific roots of epistemic awareness (Lorenz, 1978).
Limitations and ethical concerns. While we observed ETP to outperform BNN variants in our experiments, whether BNNs could bridge the gap after further improvements in the approximate inference remains an open question. The attention mechanism of ETP could also be improved, for instance to transformer nets (Vaswani et al., 2017). The explainability of the memory content of ETP deserves further investigation. The generalizeability of our results to very deep architectures, such as ResNet 152 DenseNet 161 could be topic of separate large-scale empirical study. ETP does not improve the explainability and fairness of the decisions made by the underlying design choices, such as the architecture of the used neural nets in the pipeline. Potential negative societal impacts of deep neural net classifiers stemming from these two factors need to circumvented separately before the real-world deployment of our work.

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Published as a conference paper at ICLR 2022
REFERENCES
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R. Neal. Bayesian Learning for Neural Networks. PhD thesis, 1995.
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APPENDIX
A FURTHER ANALYTICAL RESULTS AND DERIVATIONS
A.I KULLBACK LEIBLER BETWEEN DIRICHLET DISTRIBUTIONS
As both our ETP and and the EDL baseline use the Kullback-Leible divergence between two Dirichlet distributions, we give the full details of its calculation here. For two distributions over K ---imensional probability Dir(T(a) and Dir(m [3])), parameterized by a, B R the following identity holds
TE an) Ih T((+) & Ba) II I(an))
KL (Dir(Tla) ())(< log
- e ox)),
where r()) and (aa) d 1965).
log l(a) are the gamma and digamma functions (Abramowitz & Stegun,
da
A.2 THE EVIDENTIAL DEEP LEARNING OBJECTIVE
The analytical expression of the Evidential Deep Learning (Sensoy al., 2018) loss as defined in the main paper can be computed follows. For single K-dimensionnll observation y. dropping the n from the notation and instead using the index for the dimensionality throughout the following equations, we have
(o(( [lly 7. T
Fipoale) [(V-+)] TT m
K
Epporla) [(yk - )
K
Ep(r)a)) [(8x - Ep(e(a) [TT]] + Ep(rjo) [T xx) )3]
km
(yn Ep(rle) [mx])² + varp(=l>) [TT]]
with the tractable expectation and variance of Dirichlet distributed m. The Kullback-Leibler term is between two Dirichlet distributions and given (see the general form in A.I) as
KL (Dir(()(ass2)) Dir(T(I,...,.1)
T( (o()))) II, ((ao(T)))
K
= =log
((s(x)) - (oo(tr)x11(((aaaaa) ()(((((()))))) -b 26 Ex ao(r)x))
R=1
A.3 EVIDENTIAL DEEP LEARNING AS LATENT VARIABLE MODEL
As discussed in the main paper for a set N observations = ((11,9)) (NNN.NN)). the EDL objective minimizes is given as
pp.m).la.)) [lly> 1, + XKL (p(T,]/=,))))))))))))))){))))( LEDI. Σ
Dir(T, Di..,,//......11
where p( [xn) = (Hr(T(aa(Zx.)) We can instead assume the following generative model
Tu Dir(T,.I.....
Vn
W 1., 0.51x)
Vn,
ie. latent variables n, and K K-dimensional observations y.. following multivariate normal prior. Approximating the intractable posterior p(*ly) where (m1,-..: Fn), (y1,-., B.)) with an
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amortized variational posterior ((T, Dir(Tn (a where axp()) is the same architecture as in the EDL model, we have as the evidence lower bound (ELBO) to be maximized
N
(an [log ((pp((l(T))) KL (g(TT:;))) p(Tx))
NV.
log(2x) + K log(2) - (yhn Tn) (ynn - Tnd] E (TT,iir)
KL (q(TT: |l p(T,))
2
const
Σ (m [[yn
nn) (ynn nn]] KL (g(Tnian)) p(Tn))
const CEDL,
that is the negative ELBO is equal to the EDL loss up to an additive constant.
A.4 POSTERIOR PREDICTIVE
For new input observation xt, the corresponding posterior predictive distribution on the output y² can be computed
ply" = k|x", D) Ep(Z[M) (x 2) Ip(y" k|m)]] Ep(z))) hht/ Exe
where ht = he (x)( (x)); 2) That is, we can compute it analytically up to a sampling- based evaluation of the expectation over p))
B EXPERIMENTAL DETAILS
B. SYNTHETIC ID TWO-CLASS CLASSIFICATION
We illustrate the qualitative behaviour ETP on synthetic 1d two-class classifi- cation task. The data consist of observa- tions (20 per class) from two highly over- lapping Gaussian distributions. The neural net, simple one hidden layer architecture, can learn the task with high accuracy. Fig- ure shows the raw data and underlying distributions as well the learned mem- ory evidence. The model learns to place more evidence on the correct class further away from the decision boundary while also becoming more varied with increasing distance. Within the high-density region, the asymmetry in how the specific observa- tions are spread out leads to an asymmetry in the memory evidence. Below zero, the blue points are clearly separated, leading a quick emphasis on that class as we move towards the left. Above zero, how- ever, some blue observations in the orange region prevent the model from rapidly de- veloping over-confidence.
0.4
Density
0.0
-2
-1
Memory Evidence
15-
10-
-2
Input space
Figure 4: ID Classification Task. The upper plot shows the underlying distributions of each of the two classes, as well as the observed data. The lower shows the regularizing evidence the generative model places on each of the two classes depending on location in space, that is, mean + one standard deviation over ten samples from MM
This synthetic data set consisting of two classes with 20 observations each sampled from standard normal distributions centered at +1 respectively. The encoding function ii (-) consists a multi-layer perceptron with single hidden layer consisting of 32 neurons and ReLU activation function. It is optimized for 400 epochs with the Adam optimizer (Kingma Ba, 2015) using the Py PyTorch (Paszke al., 2019) default parameters and learning rate of 0.001. In order to keep the model simple
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Memory Exitence
10-
ffor
Figure 5: 2D Classification Task. The three upper plots visualize the regularizing evidence that the model places on the location in space as the average over ten samples from p( M). The color scales different across the three plots and vary in overall intensity. The lower plot visualizes horizontal cut through the middle of these plots, putting them on a common scale. The solid lines in each color indicate the mean memory evidence, and the corresponding shaded areas indicate one standard deviation. Around the separable blue class, the memory strongly emphasizes the correct class with large confidence and suppresses the other two classes depicted in orange and green. While always preferring the correct class, the memory regularizes against overconfidence around the overlapping orange and green classes.
possible, the key generator function kx(-) is constrained being the identity function, while he (v(z), a(v(r);; 2)) v(x) + tan)(a(v(z); Z). The memory update is simplified to dropping the tanh(-) rescaling.
B.2 IRTS 2D THREE-CLASS CLASSIFICATION
The data set used in this experiment is the classical Iris data set consisting of 150 samples of three iris species (50 per class), with four features measured per sample. We first map the data to the first two principal components for visualization and also this modified version the training data. This gives an interpretable toy learning setup where one class clearly separated from the other two overlapping classes. The encoding function (-(-) consists of MLP with two hidden layers of 32 neurons each and ReLU activation functions. We train for 400 epochs with the Adam optimizer using the PyTorch default parameters and a learning rate of 0.001. In order keep the model as simple as possible, we constrain the key generator function (-(-) to the identity function while setting hg (m(x), (((()); Z)) = v(x)= + tanh(a(v(r): Z). We simplify the memory update by dropping the tanh(-) rescaling. Figure shows the varying importance the model assigns to different parts input space depending the class under consideration. For the separate blue class, the model is significantly more confident in relative terms and absolute values.
B.3 EXPERIMENTS ON REAL DATA
All experiments are implemented in PyTorchh (Paszke et al., 2019) version 1.7.1 and trained on TITAN RTX. They have been replicated ten times over random initial seeds.
Fashion MNIST (FMNIST). We LeNet5-sized architecture (see Table 3). The out-of- distribution data is the MNIST (LeCun et al., 2010) data set. We z-score normalize all data sets with the in-domain mean and standard deviations. We train each model for 50 epochs with the Adam optimizer (Kingma & Ba, 2015) using a learning rate of 0.001.
Originally due to Fisher (1936) and Anderson (1935). We rely on the version provided by the 1earn. org/ stabl e/modu esa/cl asses. htm1
soikit 1earn library, see module- leare. datrrnddattssss...
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Table The neural network architectures used for the FMNIST, CIFARIO, and SVHN data set with ReLU activations between the layers.
FMNIST
CIFARIO/SVHN
Convolution (5 x 5) with 20 channels Convolution (5 5) with 192 channels
MaxPooling (2 x 2) with stride
Convolution (5 x 5) with 50 channels Convolution (5 5) with 192 channels
MaxPooling (2 x 2) with stride Linear with 1000 neurons
Linear with 500 neurons
Linear with 10 neurons
CIFARI00
Layer Name
ResNet-18
Conv1
7 x 7, 64, stride 2
3 max pool, stride 2
Conv2_x
x 6,64
2
3 x 3,64
3 x 3, 128
Conv3_x
X 2
x 3, 128
3 x 256
Conv4_x
2
3 3, 256
512
Conv5_x
2
3 x 3, 512
Average Pool
7 x average pool
Linear
100 neurons
IMDB
Layer Name
LSTM
Embedding LSTM
1001 - 64 2 layers with 256 hidden size neurons
Linear
CIFARIO (C10). We use LeNet5-sized architecture (see Table 3). The out-of-distribution data is the SVHN (Netzer et al., 1). We z-score normalize all data sets with the in-domain mean and standard deviations. We further rely on data augmentation for the in domain data using random crops with 32 pixels and four pixel padding as well as horizontal flips. We train each model for 100 epochs using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of 0.0001.
SVHN. We use a LeNet5-sized architecture (see Table 3). The out-of-distribution data is the CIFAR10 data set. We z-score normalize all data sets with the in-domain mean and standard deviations. We train each model for 100 epochs using the Adam optimizer (Kingma Ba, 2015) with learning rate of 0.0001.
IMDB Sentiment Classification. We use LSTM architecture with 64 embedding dimensions and 2 layers with 256 hidden dimensions. We generate the out-of-distributiin data by sampling values from the [1, 1000/interval uniformly random. We train each model for 20 epochs using the SGD optimizer (Robbins & Monro, 1951) with learning rate of 0.05 and 0.9 momentum. For data preparation, follow the source code of the original work
all experiments, we train the BNN models with cross- entropy loss using softmax as the squashing function to calculate class probabilities. The EDL model trained with the original loss (Sensoy et al., 2018). We make our predictions with the mean of the Dirichlet distribution as in the original work. For all models, we use entropy as the criterion for detecting dfist---itt instances.
Shared Details and Observations.
'https:/ http:///wwwwwg/leee.e/ wwt.://ummma/om0......33 sentiment lesllysssssss 1swiinggg-sstttppmmemeeeee wetmsssmmpppt. stm-pytorch
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Table 4: Quantitative results of models on the FMNIST-C, CIFARIO-C and SVHN-C test dataset using the LeNet5. The table below reports mean + three standard deviations.
FMNIST-C
CIFARIO-C
SVHN-C
Severity
Method
Em (%) (1)
ECE (%) (4)
(4)
ECE (%) (4)
(1)
ECE (%) (L)
BNN
9.4+2.8
8.042.4
21.9
7.6:2.9
8.8
7.3=10
EDL
9.942.6
4,3+10
25.1:44.
10.4:1.3
8.1:1.2
4.6=0.6
ENP
9.3:2.7
5.940.5
35.9+6.0
1.3
8.1
11.4=1.
ETP
9.542.8
3.0=1.0
21.91446
3.6:2.2
8.0#1.2
3,0=0.4
BNN 9.9+2.3
8.3+199
26. 36.1
10.0:3.7
8.9+1.2
7.3=0.9
EDL
10.5
4.5+0.8
29.145.1
10.1+1.6
8.2+1.1
4.6=0.6
ENP 9.9.2.3
6.1110
41.116.6
5.2
7.8+1.2
11.7111
ETP
10.1#2.4
3.1
26.1+5.3
5.2
7.7+1.1
3.0=0.5
BNN 10.7:22.
9,0+1.
30.017.6
12.3:5.3
9.2:1.
7.52:12
EDI 11.4:2.1
4.9=0.9
32.617.0
9.8
8.5
4.7=0.7
ENP 10.
6.4116
44.8
7.0
8.541.
1L.815
ETP 11.1
3.5:0.9
29.817.2
6.8:3.8
8.2+1.5
3.2=0.8
BNN 1..2
10.2#3.6
35. 35.19:109
15.9
9.942.5
8.0=1.
EDL 12.9:4.2
5.6=1.8
37.0
9.6+14
9.1124
5.1:100
ENP 12.4+4.3
6.5
48.9#11.2
9.2+5.44
9.4+2.
12.3+2.2
ETP 12.8:4.6
34.8:10.3
9.7:66.4
9.1+22
3.2:1.0
BNN 14.2:5.4
11.7
42.4
21.0=10.4
10.2
8.221.9
EDL 15.1:5.8
6.7
43.9#125
9.5+2.
9.412.4
5.1:11
ENP 14.8
6.9+3.1
54.6=12.7
12.5+7.0
9.5:2.
12.4+2.
ETP
15.3+7.1
5.2+3.4
42.1:13.1
14.1
9.0.2.5
3.3
Robustness against perturbations. We use the CIFARIO-C dataset in the same setup as original work (Hendrycks & Dietterich (2019). As FMNIST-C and SVHN-C are available, create them following the same procedure as in the original work. For FMNIST we adapt the procedure to gray-scale images by replicating the image three times for each channel. After applying the distortions, we map it back to the gray scale. Table 4 gives the numerical results used to create Figure 3 in the main paper.
16


Reviews:
[{"review_id": "pRSY3w-4aoY", "reviewer": "Reviewer_UsNy", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "Strengths: \n\n - The idea is interesting and well motivated.\n - The experimental results highlight the improvement over the baselines in terms of a series of metrics. \n - The paper is generally well-written.\n\nComments:\n \n - I believe the paper would benefit from experiments on a more diverse set of  (not necessarily very large) architectures (e.g., ResNet20).\n - It would be interesting to see results on real-world data beyond images, e.g.,  IMDB sentiment classification.\n - I appreciate the illustrative examples in 1-D and 2-D. It would be interesting to see a similar experiment on a small-scale image dataset (e.g., MNIST).", "summary_of_the_paper": "The paper introduces a “Complete Bayesian Model” that combines the benefits of Parametric Bayesian Models and Evidential Bayesian Models. To this end, the authors propose a technique to learn an input-dependent prior. The experiments showcase the efficacy of the method compared to the baselines.", "summary_of_the_review": "Overall, this is an interesting work. I lean towards acceptance, especially if the comments that were raised are addressed.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "mmty5iQ2Cq", "reviewer": "Reviewer_cnXy", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "I don't think I understood the paper well enough to form a strong opinion about this model. \nWhat lacks for me is a clear motivation for the following:\n1. Why the Complete Bayesian Model is constructed this way. Is this the only plausible way to combine parametric and evidential approaches? \n2. In the evidential DL papers that I've read, it seemed common to use entropies instead of variances to measure the spread of the distributions. Is there a reason for using variances here?\n3. I wish there was some more intuition on what the interaction term in Eq.10 is doing. Model uncertainty and data uncertainty are quite easy to grasp: in the evidential DL literature they are usually visualized on a simplex (for Dirichlet priors), which makes them readily understandable. Is there a similar illustration that could explain the interaction term?\n4. Also, how important is the interaction term?  I wish there were ablation studies that measure its effect in isolation.\n5. I disagree with the approach of verifying the Main Hypothesis by showing that experimentally ETPs work better. These are non-trivial models that probably depend on a number of hyperparameters, so tuning ETP slightly better can have a greater effect on the performance compared to the effect of having a genuinely better model. A proper ablation is thus needed to account for this.\n6. Definition 1, in my opinion, needs a better explanation. To me, it seems like any exchangeable process would have these properties. M', as I understand it, is a set of posterior parameters that can be computed from the prior parameters and the observations. What makes these properties so special that they define a Turing Process is unclear to me. \n7. If we take an NP and encode the whole training set as a context, which we then use during test time, would that work? I mean, is it really necessary to have a rather complex memory mechanism of the Neural Turing Process?\n8. I am underwhelmed by the experimental results: Table 1 is hard to understand, so it's not obvious why ETP is better. Also, if ECE is an important metric, which I assume is, then it would be best to explain it in the text.   Robustness experiments are pretty weird: I don't know how to interpet them or what was expected. If these experiments are included, then I think more than 4 lines of text are required to explain them. \n\nFinally, I was surprised by the abrupt ending of the paper: I wish there was some sort of discussion. I can only assume that the paper was submitted in a rather unfinished state. \n        ", "summary_of_the_paper": "The paper presents a complete Bayesian model that combines strengths of parametric and evidential Bayesian models.  An instance of this model is an Evidential Turing Process (ETP) -- a model in which the prior on the global variable that governs the likelihood function is input-dependent, and this dependence is established in a way similar to the Neural Turing Machines. Unlike Neural Processes (NPs), it doesn't need a context set during test time as the context is extracted from the training set and stored in a memory variable. Experiments show that  ETPs are somewhat better than BNNs and evidential NPs.", "summary_of_the_review": "Given the points above, I think that the paper lacks clarity. As I mentioned earlier, I cannot form a strong opinion before I understand the motivation for the model and the experiments.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "e3iClc14ZO1", "reviewer": "Reviewer_zTRM", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "This paper has several strengths to acknowledge:\n- I really appreciate the top-down approach of starting with general principles, which are progressively transformed into an actual implementation which obtains competitive performance. \n- The category of complete Bayesian models deserves its name of \"complete\" as it covers a very large range of models (see Section 4).\n- Creating a unified approach to uncertainty quantification is a valuable and ambitious goal.\n\nBut I am unfortunately not yet convinced by the validation of the main claims of the paper:\n\n*\"**Main Hypothesis.** A Bayesian model which is capable of quantifying both uncertainties in isolation, as well as their interaction explicitly is a guiding principle for predictors that can succeed in in-domain calibration and out-of-domain detection simultaneously.\"*\n\n*\"**Our Solution Proposal.** We conjecture that a prior can effectively depend on an input if it receives context information during training time, stores it into an external memory, and retrieves relevant parts of it using the conditioned input example as a query.\"*\n\n- The experimental validation is undermined by the very small models, self-described as \"LeNet5-size\", with only a few layers. The reported prediction error seem unreasonably high (15+% on CIFAR10 for instance). Please clarify any misunderstanding concerning this point. \n- There is no experimental validation for the choice of a Turing process. What about the alternative of using a simple neural network predicting $\\pi$ from input $x$ to amortize the prior $p(\\pi|\\theta, x)$?\n- Despite a lot of math in the paper, the theoretical justification that is actually relevant seems rather limited. A large part of the equations appear to be somewhat decorative. For instance:\n  - As far as I understand, the expressions of variational free energy (4) (5) (9) (13) (17) (19) don’t support any argument of the paper \n  - As a second example, Definition 1, (18) and (26) are not contributions of this paper, but describe already existing components (Turing processes, neural processes, the attention mechanism)\n\n", "summary_of_the_paper": "This paper introduces the category of complete Bayesian models, with the goal of providing both in-domain uncertainty quantification (that is, calibration) and and out-of-domain detection. Concretely, the authors develop the evidential Turing process as an implementation of complete Bayesian model. The approach is experimentally validated by experiments on FMNIST, CIFAR10 and SVHN, showing that the evidential Turing process is competitive on all criteria. ", "summary_of_the_review": "I really appreciate the top-down approach of the paper. Unfortunately, I am neither really convinced by the experimental validation which considers too small models, nor by the theoretical presentation that sometimes feel a bit decorative. Hence my weak reject recommendation, with confidence rated at 3 because I am not familiar with evidential Bayesian models. I am looking forward to discussion with the authors to clarify any misunderstanding. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "ZJrvd-HFlva", "reviewer": "Reviewer_L3yb", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "Even though I am far from an expert in the relevant literature, I find this paper very well written and was able to follow most of the presentation. It clearly introduces the relevant concepts and explains the underlying motivation for the proposed methods. \n\nI find the experimental section in the main text a bit brief. I personally find this justifiable as allows the manuscript to explain and motivate the proposed method in sufficient detail. I do want to encourage the authors to be a bit more restraint with marking the best method by bold letters in Table 1. If two or methods lead to the same results within one (or even three) standard deviations, one should either mark all of these methods as performing best or none of them. \n\nA few minor suggestions:\n-Equation (1) ends with both a full stop and comma.\n-Equation (4) misses a \"(\" in the subscript of the expectation value\n\n\n", "summary_of_the_paper": "The manuscript proposes Evidential Turing Process which is a combination of Neural Turing Machines and Neural Processes. The authors theoretically motivate and empirically demonstrate that their proposed method leads to better in-distribution uncertainty estimation as well as improved out-of-distribution detection.", "summary_of_the_review": "I find the paper very well written. The proposed method appears to be theoretically well-motivated and, empirically, leads to improved robustness. \n\nI want to stress however that I have only a very limited overview of the relevant field of research and I can thus not confidently state a definite opinion, especially concerning the novelty of the manuscript.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: It seems the reviewers are in an agreement that the work seems interesting, well motivated, and results are meaningful. The main complaints or issues that remain is the amount of rewriting involved, which might be hard for the reviewers to track, and maybe question regarding the results given for e.g. the choice of architectures for CIFAR-10 making numbers harder to interpret. 

However, I do see that the quality of the manuscript is quite good (and multiple reviewers commented on this), and the idea seems natural to me. I think the results, especially with the addition of CIFAR-100 and IMDB seem sufficient, and given the overall positive feeling of the reviewers over the work with no major concerns, I am happy to recommend this paper for acceptance.
Decision: Accept (Poster)