Original: U NCERTAINTY C ALIBRATION E RROR :
Revised: U NCERTAINTY C ALIBRATION E RROR :

Original: A N EW M ET -
Revised: A N EW M ET -

Original: RIC FOR M ULTI -C LASS C LASSIFICATION
Revised: RIC FOR M ULTI -C LASS C LASSIFICATION

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Various metrics have recently been proposed to measure uncertainty calibration of deep models for classiﬁcation.
Revised: Various metrics have recently been proposed to measure uncertainty calibration of deep models for classiﬁcation.

Original: However, these metrics either fail to capture miscalibration correctly or lack interpretability.
Revised: However, these metrics either fail to capture miscalibration correctly or lack interpretability.

Original: We propose to use the normalized entropy as a measure of uncertainty and derive the Uncertainty Calibration Error (UCE), a comprehensible calibration metric for multi-class classiﬁcation.
Revised: We propose to use the normalized entropy as a measure of uncertainty and derive the Uncertainty Calibration Error (UCE), a comprehensible calibration metric for multi-class classiﬁcation.

Original: In our experiments, we focus on uncertainty from variational Bayesian inference methods and compare UCE to established calibration errors on the task of multi-class image classiﬁcation.
Revised: In our experiments, we focus on uncertainty from variational Bayesian inference methods and compare UCE to established calibration errors on the task of multi-class image classiﬁcation.

Original: UCE avoids several pathologies of other metrics, but does not sacriﬁce interpretability.
Revised: UCE avoids several pathologies of other metrics, but does not sacriﬁce interpretability.

Original: It can be used for regularization to improve calibration during training without penalizing predictions with justiﬁed high conﬁdence.
Revised: It can be used for regularization to improve calibration during training without penalizing predictions with justiﬁed high conﬁdence.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Advances in deep learning have led to superior accuracy in classiﬁcation tasks, making deep learning classiﬁers an attractive choice for safety-critical applications like autonomous driving (Chen et al., 2015) or computer-aided diagnosis (Esteva et al., 2017).
Revised: Advances in deep learning have led to superior accuracy in classiﬁcation tasks, making deep learning classiﬁers an attractive choice for safety-critical applications like autonomous driving (Chen et al., 2015) or computer-aided diagnosis (Esteva et al., 2017).

Original: However, the high accuracy of recent deep learning models alone is not sufﬁcient for such applications.
Revised: However, the high accuracy of recent deep learning models alone is not sufﬁcient for such applications.

Original: In cases where serious decisions are made upon model’s predictions, it is essential to also consider the uncertainty of these predictions.
Revised: In cases where serious decisions are made upon model’s predictions, it is essential to also consider the uncertainty of these predictions.

Original: We need to know if a prediction is likely to be incorrect or if invalid input data is presented to a deep model, e.g. data that is far away from the training domain or obtained from a defective sensor.
Revised: We need to know if a prediction is likely to be incorrect or if invalid input data is presented to a deep model, e.g. data that is far away from the training domain or obtained from a defective sensor.

Original: The consequences of a false decision based on an uncertain prediction can be fatal.
Revised: The consequences of a false decision based on an uncertain prediction can be fatal.

Original: A natural expectation is that the certainty of a prediction should be directly correlated with the quality of the prediction.
Revised: A natural expectation is that the certainty of a prediction should be directly correlated with the quality of the prediction.

Original: In other words, predictions with high certainty are more likely to be accurate than uncertain predictions, which are more likely to be incorrect.
Revised: In other words, predictions with high certainty are more likely to be accurate than uncertain predictions, which are more likely to be incorrect.

Original: A common misconception is the assumption that the estimated softmax likelihood can be directly used as a conﬁdence measure for the predicted class.
Revised: A common misconception is the assumption that the estimated softmax likelihood can be directly used as a conﬁdence measure for the predicted class.

Original: This expectation is dangerous in the context of critical decision-making.
Revised: This expectation is dangerous in the context of critical decision-making.

Original: The estimated likelihood of models trained by minimizing the negative log-likelihood (i.e. cross entropy) is highly overconﬁdent; that is, the estimated likelihood is considerably higher than the observed frequency of accurate predictions with that likelihood (Guo et al., 2017).
Revised: The estimated likelihood of models trained by minimizing the negative log-likelihood (i.e. cross entropy) is highly overconﬁdent; that is, the estimated likelihood is considerably higher than the observed frequency of accurate predictions with that likelihood (Guo et al., 2017).

Original:  U NCERTAINTY E STIMATION
Revised:  U NCERTAINTY E STIMATION

Original: In this work, we focus on uncertainty from approximately Bayesian methods.
Revised: In this work, we focus on uncertainty from approximately Bayesian methods.

Original: We assume a general multi-class classiﬁcation task with C classes.
Revised: We assume a general multi-class classiﬁcation task with C classes.

Original: Let input x ∈
Revised: Let input x ∈

Original: X be a random variable with corresponding label y ∈ Y = { 1 , . .
Revised: X be a random variable with corresponding label y ∈ Y = { 1 , . .

Original: , C } .
Revised: , C } .

Original: Let f w ( x ) be the output (logits) of a neural network with weight matrices w , and with model likelihood p (
Revised: Let f w ( x ) be the output (logits) of a neural network with weight matrices w , and with model likelihood p (

Original: y = c | f w ( x )) for class c , which is sampled from a probability vector p = σ SM ( f w ( x )) , obtained by passing the model output through the softmax function σ SM ( · ) .
Revised: y = c | f w ( x )) for class c , which is sampled from a probability vector p = σ SM ( f w ( x )) , obtained by passing the model output through the softmax function σ SM ( · ) .

Original: From a frequentist perspective, the softmax likelihood is often interpreted as conﬁdence of prediction.
Revised: From a frequentist perspective, the softmax likelihood is often interpreted as conﬁdence of prediction.

Original: Throughout this paper, we follow this deﬁnition.
Revised: Throughout this paper, we follow this deﬁnition.

Original: The frequentist approach assumes a single best point estimate of the parameters (or weights) of a neural network.
Revised: The frequentist approach assumes a single best point estimate of the parameters (or weights) of a neural network.

Original: In frequentist inference, the weights of a deep model are obtained by maximum likelihood estimation (Bishop, 2006), and the normalized output likelihood for an unseen test input does not consider uncertainty in the weights (Kendall & Gal, 2017).
Revised: In frequentist inference, the weights of a deep model are obtained by maximum likelihood estimation (Bishop, 2006), and the normalized output likelihood for an unseen test input does not consider uncertainty in the weights (Kendall & Gal, 2017).

Original: Weight uncertainty (also referred to as model or epistemic uncertainty) is a considerable source of predictive uncertainty for models trained on data sets of limited size (Bishop, 2006; Kendall & Gal, 2017).
Revised: Weight uncertainty (also referred to as model or epistemic uncertainty) is a considerable source of predictive uncertainty for models trained on data sets of limited size (Bishop, 2006; Kendall & Gal, 2017).

Original: Bayesian neural networks and recent advances in their approximation provide valuable mathematical tools for quantiﬁcation of model uncertainty (Gal & Ghahramani, 2016; Kingma & Welling, 2014).
Revised: Bayesian neural networks and recent advances in their approximation provide valuable mathematical tools for quantiﬁcation of model uncertainty (Gal & Ghahramani, 2016; Kingma & Welling, 2014).

Original: Instead of assuming the existence of a single best parameter set, we place distributions over the parameters and want to consider all possible parameter conﬁgurations, weighted by their posterior.
Revised: Instead of assuming the existence of a single best parameter set, we place distributions over the parameters and want to consider all possible parameter conﬁgurations, weighted by their posterior.

Original: More speciﬁcally, given a training data set D and an unseen test sample x with class label y , we are interested in evaluating the predictive distribution p ( y | x , D ) =
Revised: More speciﬁcally, given a training data set D and an unseen test sample x with class label y , we are interested in evaluating the predictive distribution p ( y | x , D ) =

Original: (cid:82) p ( y | x , w ) p ( w |D )
Revised: (cid:82) p ( y | x , w ) p ( w |D )

Original: d w .
Revised: d w .

Original: This integral requires to evaluate the posterior p ( w |D ) , which involves the intractable marginal likelihood.
Revised: This integral requires to evaluate the posterior p ( w |D ) , which involves the intractable marginal likelihood.

Original: A possible solution to this is to approximate the posterior with a more simple, tractable distribution q ( w ) by optimization.
Revised: A possible solution to this is to approximate the posterior with a more simple, tractable distribution q ( w ) by optimization.

Original: In the following, we brieﬂy describe common approximately Bayesian methods which we use in our experiments to obtain weight uncertainty. Monte Carlo Dropout One practical approximation of the posterior is variational inference with Monte Carlo (MC) dropout (Gal & Ghahramani, 2016).
Revised: In this work, we incorporate the following approximately Bayesian methods which we use in our experiments to obtain weight uncertainty: Monte Carlo (MC) dropout (Gal & Ghahramani, 2016),

Original: To determine model uncertainty, dropout variational inference is performed by training the model f w with dropout (Srivastava et al., 2014) and using dropout at test time to sample from the approximate posterior distribution by performing

Original: N stochastic forward passes per test sample (Gal & Ghahramani, 2016; Kendall & Gal, 2017).

Original: This is also referred to as MC dropout.

Original: In MC dropout, the ﬁnal probability vector of the predictive distribution is computed by MC integration:

Original: [Equation]

Original: Gaussian Dropout Gaussian dropout was ﬁrst proposed by Wang & Manning (2013) and linked to variational inference by Kingma et al.

Original: Dropout introduces Bernoulli noise during optimization and reduces overﬁtting of the training data.

Original: The resulting output y of a layer with dropout is a weighted sum of Bernoulli random variables.

Original: Then, the central limit theorem states, that y is approximately normally distributed.

Original: Instead of sampling from the weights and computing the resulting output, we can directly sample from the implicit Gaussian distribution of dropout

Original: [Equation]

Original: with

Original: [Equation]

Original: using the reparameterization trick (Kingma et al., 2015)

Original: [Equation]

Original: Gaussian dropout is a continuous approximation to Bernoulli dropout, and in comparison it will better approximate the true posterior distribution and is expected to provide improved uncertainty estimates (Louizos & Welling, 2017).

Original: To obtain the ﬁnal probability vector p ( x ) , we again use MC integration with N stochastic forward passes.

Original: The dropout rate p is now a learnable parameter and does not need to be chosen carefully by hand.

Original: In fact, p could be optimized w.r.t. uncertainty calibration, scaling the variance of the implicit Gaussian of dropout. A similar approach was presented by Gal et al. (2017) using the Concrete distribution (Maddison et al., 2016; Jang et al., 2017).
Revised: Gaussian dropout (Wang & Manning, 2013; Kingma et al., 2015), Bayes by Backprop (Blundell et al., 2015), SWA-Gaussian (Maddox et al., 2019), and (although not Bayesian) deep ensembles (Lakshminarayanan et al., 2017).

Original: However, we focus on metrics for measuring calibration and therefore ﬁx p in our subsequent experiments.

Original: Bayes by Backprop Blundell et al.

Original: (2015) assume a Gaussian distribution with diagonal covariance matrix as variational posterior q ( w | θ ) , parameterized by mean µ and standard deviation σ , where θ = { µ , σ } .

Original: A sample of the weights can be obtained by sampling a multivariate unit Gaussian and shift it by µ and scale it by σ .

Original: Then, the network is trained by minimizing

Original: [Equation]

Original: In case of a zero mean Gaussian prior, the ﬁrst term can be implemented by weight decay.

Original: In contrast to Gaussian dropout, which operates on the implicit distribution of the activations, Bayes by Backprop (BBB) directly operates on the weights.

Original: This doubles the number of trainable parameters in practice.

Original: MC integration is used to obtain the ﬁnal probability vector p ( x ) .

Original: SWA-Gaussian Stochastic weight averaging (SWA) uses stochastic gradient descent steps around a local loss optimum of a trained network and averages the weights w SWA = 1 T (cid:80) Ti =1 w i ofthe model from each step i (Izmailov et al., 2018). This explores the loss landscape and averaging helps to ﬁnd a better weight estimate than converging to a single local optimum. SWA-Gaussian (SWAG) is closely related to Bayes by Backprop (Maddox et al., 2019).
Revised: A short review of each of the methods can be found in Appendix A.2.

Original: It assumes a Gaussian distribution with diagonal covariance matrix as approximate variational posterior.

Original: Instead of using backpropagation to directly optimize µ and σ , it ﬁts a Gaussian by using µ = w SWA and

Original: [Equation]

Original: This doubles the number of parameters at test time.

Original: The approximate Gaussian posterior results to

Original: N ( w SWA , Σ diag ) and MC integration with samples w i ∼ N ( w SWA , Σ diag ) is used to compute the ﬁnal probability vector p ( x ) .

Original: Deep Ensembles Training multiple randomly initialized copies of a deep network by performing maximum posterior estimation and ensembling them to get multiple predictions for a single input is not a variational inference method.

Original: However, they have been reported to produce surprisingly useful uncertainty estimates in practice that are better calibrated (Lakshminarayanan et al., 2017).

Original: Deep ensembles considerably increase the number of parameters at train and test time.

Original: We use deep ensembles as non Bayesian baseline for uncertainty estimation.

Original:  R ELATED C ALIBRATION M ETRICS
Revised:  R ELATED C ALIBRATION M ETRICS

Original: Expected Calibration Error
Revised: Expected Calibration Error

Original: The expected calibration error (ECE) is one of the most popular calibration error metrics and estimates model calibration by binning the predicted conﬁdences ˆ p = max c p ( y = c
Revised: The expected calibration error (ECE) is one of the most popular calibration error metrics and estimates model calibration by binning the predicted conﬁdences ˆ p = max c p ( y = c

Original: | x ) into M bins from equidistant intervals and comparing them to average accuracies per bin (Naeini et al., 2015; Guo et al., 2017):
Revised: | x ) into M bins from equidistant intervals and comparing them to average accuracies per bin (Naeini et al., 2015; Guo et al., 2017):

Original: [Equation]
Revised: [Equation]

Original: with number of test samples n and acc( B ) and conf( B ) denoting the accuracy and conﬁdence of bin
Revised: with number of test samples n and acc( B ) and conf( B ) denoting the accuracy and conﬁdence of bin

Original: B , respectively.
Revised: B , respectively.

Original: Several recent works have described severe pathologies of the ECE metric (Ashukha et al., 2020; Nixon et al., 2019; Kumar et al., 2019).
Revised: Several recent works have described severe pathologies of the ECE metric (Ashukha et al., 2020; Nixon et al., 2019; Kumar et al., 2019).

Original: Most notably, the ECE metric is minimized by a model constantly predicting the marginal distribution of the majority class which makes it impossible to directly optimize it (Kumar et al., 2018).
Revised: Most notably, the ECE metric is minimized by a model constantly predicting the marginal distribution of the majority class which makes it impossible to directly optimize it (Kumar et al., 2018).

Original: Additionally, the ECE only considers the maximum class probability and ignores the remaining entries of the probability vector p ( x ) .
Revised: Additionally, the ECE only considers the maximum class probability and ignores the remaining entries of the probability vector p ( x ) .

Original: Adaptive Calibration Error Nixon et al.
Revised: Adaptive Calibration Error Nixon et al.

Original: (2019) proposed the adaptive calibration error (ACE) to address the issue of ﬁxed bin widths of ECE-like metrics.
Revised: (2019) proposed the adaptive calibration error (ACE) to address the issue of ﬁxed bin widths of ECE-like metrics.

Original: For models with high accuracy or overconﬁdence, most of the predictions fall into the rightmost bins, whereas only very few predictions fall into the rest of the bins.
Revised: For models with high accuracy or overconﬁdence, most of the predictions fall into the rightmost bins, whereas only very few predictions fall into the rest of the bins.

Original: ACE spaces the bins such that an equal number of predictions contribute to each bin.
Revised: ACE spaces the bins such that an equal number of predictions contribute to each bin.

Original: The ﬁnal ACE is computed by averaging over per-class ACE values to address the issue raised by Kull et al.
Revised: The ﬁnal ACE is computed by averaging over per-class ACE values to address the issue raised by Kull et al.

Original: However, this makes the metric more sensitive to the manually selected number of bins M as the number of bins effectively becomes C · M , with number of classes C .
Revised: However, this makes the metric more sensitive to the manually selected number of bins M as the number of bins effectively becomes C · M , with number of classes C .

Original: Using ﬁxed bin widths, the numbers of samples in the sparsely populated bins is further reduced, which increases the variance of each measurement per bin.
Revised: Using ﬁxed bin widths, the numbers of samples in the sparsely populated bins is further reduced, which increases the variance of each measurement per bin.

Original: Using adaptive bins, this results in the lower conﬁdence bins spanning a wide range of values, which increases the bias of the bin’s measurement.
Revised: Using adaptive bins, this results in the lower conﬁdence bins spanning a wide range of values, which increases the bias of the bin’s measurement.

Original: Negative Log-Likelihood Deep models for classiﬁcation are usually trained by minimizing the average negative log-likelihood (NLL):
Revised: Negative Log-Likelihood Deep models for classiﬁcation are usually trained by minimizing the average negative log-likelihood (NLL):

Original: [Equation]
Revised: [Equation]

Original: The NLL is also commonly used as a metric for measuring the calibration of uncertainty.
Revised: The NLL is also commonly used as a metric for measuring the calibration of uncertainty.

Original: However, the NLL is minimized by increasing the conﬁdence max c p ( y = c | x ) , which favors over-conﬁdent models and models with higher accuracy (Ashukha et al., 2020).
Revised: However, the NLL is minimized by increasing the conﬁdence max c p ( y = c | x ) , which favors over-conﬁdent models and models with higher accuracy (Ashukha et al., 2020).

Original: This metric is therefore unable to compare the calibration of models with different accuracies and training a model by minimizing NLL does not necessarily lead to good calibration.
Revised: This metric is therefore unable to compare the calibration of models with different accuracies and training a model by minimizing NLL does not necessarily lead to good calibration.

Original: Brier Score The average Brier score is another popular metric for assessing the quality of predictive uncertainty and is deﬁned as (Brier, 1950; Lakshminarayanan et al., 2017)
Revised: Brier Score The average Brier score is another popular metric for assessing the quality of predictive uncertainty and is deﬁned as (Brier, 1950; Lakshminarayanan et al., 2017)

Original: [Equation]
Revised: [Equation]

Original: Similarly to the NLL, the Brier score favors high probabilities for correct predictions and low probabilities for incorrect predictions.
Revised: Similarly to the NLL, the Brier score favors high probabilities for correct predictions and low probabilities for incorrect predictions.

Original: Thus, models with higher accuracy tend to show a better Brier score, which makes the metric unsuitable for comparing the quality of uncertainty for models with different accuracies.
Revised: Thus, models with higher accuracy tend to show a better Brier score, which makes the metric unsuitable for comparing the quality of uncertainty for models with different accuracies.

Original: Maximum Mean Calibration Error Common recalibration methods are applied post-hoc,
Revised: Maximum Mean Calibration Error Common recalibration methods are applied post-hoc,

Original: e.g. temperature scaling on a separate calibration set.
Revised: e.g. temperature scaling on a separate calibration set.

Original: Kumar et al.
Revised: Kumar et al.

Original: (2018) proposed the maximum mean calibration error (MMCE), a trainable calibration surrogate for the calibration error.
Revised: (2018) proposed the maximum mean calibration error (MMCE), a trainable calibration surrogate for the calibration error.

Original: It is deﬁned as
Revised: It is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: over batch D ⊂ D with batch size m , matrix-valued universal kernel k and ˆ y = arg max c p ( y = c | x ) .
Revised: over batch D ⊂ D with batch size m , matrix-valued universal kernel k and ˆ y = arg max c p ( y = c | x ) .

Original: Trainable calibration metrics are used in joint optimization with the negative log-likelihood
Revised: Trainable calibration metrics are used in joint optimization with the negative log-likelihood

Original: [Equation]
Revised: [Equation]

Revised: min w (cid:88)

Revised: [Equation]

Original: Kumar et al.
Revised: Kumar et al.

Original: (2018) claim to have addressed the issue that the ECE is unsuitable for direct optimization due to its high discontinuity in w .
Revised: (2018) claim to have addressed the issue that the ECE is unsuitable for direct optimization due to its high discontinuity in w .

Original: However, MMCE is also minimized by a model constantly predicting the marginal distribution of the classes.
Revised: However, MMCE is also minimized by a model constantly predicting the marginal distribution of the classes.

Original: This leads to subpar logit temperature when training with MMCE and temperature scaling can further reduce miscalibration (Kumar et al., 2018).
Revised: This leads to subpar logit temperature when training with MMCE and temperature scaling can further reduce miscalibration (Kumar et al., 2018).

Original:  U NCERTAINTY C ALIBRATION E RROR
Revised:  U NCERTAINTY C ALIBRATION E RROR

Original: To give an insight into our general approach to measuring the calibration of uncertainty, we will ﬁrst revisit the deﬁnition of perfect calibration of conﬁdence (Guo et al., 2017) and show how this concept can be extended to calibration of our deﬁnition uncertainty.
Revised: To give an insight into our general approach to measuring the calibration of uncertainty, we will ﬁrst revisit the deﬁnition of perfect calibration of conﬁdence (Guo et al., 2017) and show how this concept can be extended to calibration of our deﬁnition uncertainty.

Original: Let ˆ y = arg max p be the most likely class prediction of input x with conﬁdence ˆ p = max p and true label y .
Revised: Let ˆ y = arg max p be the most likely class prediction of input x with conﬁdence ˆ p = max p and true label y .

Original: Then, following Guo et al.
Revised: Then, following Guo et al.

Original: (2017), perfect calibration of conﬁdence is deﬁned as
Revised: (2017), perfect calibration of conﬁdence is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: That is, the probability of a correct prediction ˆ y =
Revised: That is, the probability of a correct prediction ˆ y =

Original: y given the prediction conﬁdence ˆ p should exactly correspond to the prediction conﬁdence.
Revised: y given the prediction conﬁdence ˆ p should exactly correspond to the prediction conﬁdence.

Original: Instead of using only the probability of the predicted class, we use the entropy of p to express prediction uncertainty:
Revised: Instead of using only the probability of the predicted class, we use the entropy of p to express prediction uncertainty:

Original: [Equation]
Revised: [Equation]

Original: [Equation] Letq ( k
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: 15) be a probability vector of true marginal class probabilities for all inputs x predicted with class k .
Revised: ) be a probability vector of true marginal class probabilities for all inputs x predicted with class k .

Original: Consider the following example: Three i.i.d.
Revised: Consider the following example: Three i.i.d.

Original: inputs x 1:3 in a binary classiﬁcation task with ground truth labels { 1 , 1 , 2 } have all been predicted with arg max p ( x 1:3 ) = 1 .
Revised: inputs x 1:3 in a binary classiﬁcation task with ground truth labels { 1 , 1 , 2 } have all been predicted with arg max p ( x 1:3 ) = 1 .

Original: Then, q (1) = (cid:0) 23 , 13 (cid:1) .
Revised: Then, q (1) = (cid:0) 23 , 13 (cid:1) .

Original: With this, we deﬁne a model to be perfectly calibrated if
Revised: With this, we deﬁne a model to be perfectly calibrated if

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: From this, we derive an error metric for calibration of uncertainty:
Revised: From this, we derive an error metric for calibration of uncertainty:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: However, this metric and the use of the entropy as measure of uncertainty lacks interpretability, as the entropy scales with the number of classes C .
Revised: However, this metric and the use of the entropy as measure of uncertainty lacks interpretability, as the entropy scales with the number of classes C .

Original: This does not allow to compare the uncertainty or the calibration of models trained on different data sets.
Revised: This does not allow to compare the uncertainty or the calibration of models trained on different data sets.

Original: Therefore, we propose to use the normalized entropy to scale the values to a range between 0 and 1 :
Revised: Therefore, we propose to use the normalized entropy to scale the values to a range between 0 and 1 :

Original: [Equation]
Revised: [Equation]

Original: We further increase interpretability and argue, that the normalized entropy should correlate with the model error.
Revised: We further increase interpretability and argue, that the normalized entropy should correlate with the model error.

Original: From Eq.
Revised: From Eq.

Original: (13) and Eq.
Revised: (6) and Eq.

Original: (18), we deﬁne perfect calibration of uncertainty as
Revised: (11), we deﬁne perfect calibration of uncertainty as

Original: [Equation]
Revised: [Equation]

Original: That is, in a batch of inputs that are all predicted with uncertainty of e. g. 0 . 2 , a top-1 error of 20 % is expected.
Revised: That is, in a batch of inputs that are all predicted with uncertainty of e. g. 0 . 2 , a top-1 error of 20 % is expected.

Original: The conﬁdence is interpreted as the probability of belonging to a particular class, which should naturally correlate with the model error of that class.
Revised: The conﬁdence is interpreted as the probability of belonging to a particular class, which should naturally correlate with the model error of that class.

Original: This characteristic does not generally apply to entropy, and thus the question arises why entropy should correspond with the model error.
Revised: This characteristic does not generally apply to entropy, and thus the question arises why entropy should correspond with the model error.

Original: Proposition 1.
Revised: Proposition 1.

Original: The normalized entropy (uncertainty) ˜ H ( p ) approaches the top-1 error in the limit of number of classes C if the model p is well-calibrated.
Revised: The normalized entropy (uncertainty) ˜ H ( p ) approaches the top-1 error in the limit of number of classes C if the model p is well-calibrated.

Original: Proof.
Revised: Proof.

Original: [Equation]
Revised: [Equation]

Original: The top-1 error equals (1 − ˆ p ) if the model is perfectly calibrated in the sense of Eq.
Revised: The top-1 error equals (1 − ˆ p ) if the model is perfectly calibrated in the sense of Eq.

Original: For a detailed proof, see Appendix A.1.
Revised: For a detailed proof, see Appendix A.1.

Original: Thus, the normalized entropy gives us an intuitive and interpretable measure of uncertainty.
Revised: Thus, the normalized entropy gives us an intuitive and interpretable measure of uncertainty.

Original: If a model is perfectly calibrated, ˜ H corresponds to the top-1 error.
Revised: If a model is perfectly calibrated, ˜ H corresponds to the top-1 error.

Original: We propose the following notion to quantify miscalibration of uncertainty:
Revised: We propose the following notion to quantify miscalibration of uncertainty:

Original: [Equation]
Revised: [Equation]

Original: We refer to this as Expected Uncertainty Calibration Error (UCE) and approximate with
Revised: We refer to this as Expected Uncertainty Calibration Error (UCE) and approximate with

Original: [Equation]
Revised: [Equation]

Original: using the same binning scheme as in ECE estimation.
Revised: using the same binning scheme as in ECE estimation.

Original: The error per bin is deﬁned as
Revised: The error per bin is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where 1 (ˆ y i (cid:54) =
Revised: where 1 (ˆ y i (cid:54) =

Original: y )
Revised: y )

Original: = 1 and 1 (ˆ y i =
Revised: = 1 and 1 (ˆ y i =

Original: y )
Revised: y )

Original: = 0 .
Revised: = 0 .

Original: Uncertainty per bin is deﬁned asuncert
Revised: Uncertainty per bin is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: B

Original: [Equation]

Original: [Equation]

Original: Properties of UCE
Revised: Properties of UCE

Original: The proposed UCE metric solves several problems of other metrics.
Revised: The proposed UCE metric solves several problems of other metrics.

Original: First, the
Revised: First, the

Original: UCE is not zero for a model constantly predicting the marginal class distribution.
Revised: UCE is not zero for a model constantly predicting the marginal class distribution.

Original: Estimators of metrics with this pathology (e.g. ECE, MMCE) suffer from varying bias and therefore do not allow comparing miscalibration of different models (Ashukha et al., 2020; Vaicenavicius et al., 2019).
Revised: Estimators of metrics with this pathology (e.g. ECE, MMCE) suffer from varying bias and therefore do not allow comparing miscalibration of different models (Ashukha et al., 2020; Vaicenavicius et al., 2019).

Original: In contrast to ACE, UCE is not highly sensitive to the numbers of bins and provides a consistent ranking of different models for the same classiﬁcation task (see Fig. 2).
Revised: In contrast to ACE, UCE is not highly sensitive to the numbers of bins and provides a consistent ranking of different models for the same classiﬁcation task (see Fig. 1).

Original: Additionally, UCE can be used as a trainable regularizer in similar manner to MMCE.
Revised: Additionally, UCE can be used as a trainable regularizer in similar manner to MMCE.

Original: During training, we compute the UCE over mini-batches D ⊂ D and add it to the NLL training objective
Revised: During training, we compute the UCE over mini-batches D ⊂ D and add it to the NLL training objective

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: weighted by a factor λ .
Revised: weighted by a factor λ .

Original: UCE is zero for an optimal model and thus does not penalize high conﬁdent predictions for models with high accuracy, which is a major disadvantage of plain entropy regularization (Pereyra et al., 2017).
Revised: UCE is zero for an optimal model and thus does not penalize high conﬁdent predictions for models with high accuracy, which is a major disadvantage of plain entropy regularization (Pereyra et al., 2017).

Original: Predictions with low uncertainty, but high top-1 error are penalized whereas predictions with high accuracy are encouraged to have low uncertainty.
Revised: Predictions with low uncertainty, but high top-1 error are penalized whereas predictions with high accuracy are encouraged to have low uncertainty.

Original: [Table]
Revised: [Figure]

Original: We used M = 15 bins where necessary.

Original: Here, all metrics provide the same ranking of models.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We evaluate the proposed uncertainty calibration error on mutli-class image classiﬁcation on CIFAR10 with ResNet-34 and on CIFAR-100 with ResNet-50 (He et al., 2016; Krizhevsky & Hinton, 2009).
Revised: We evaluate the proposed uncertainty calibration error on mutli-class image classiﬁcation on CIFAR10 with ResNet-34 and on CIFAR-100 with ResNet-50 (He et al., 2016; Krizhevsky & Hinton, 2009).

Original: The feature extractor of ResNet is used as implemented in PyTorch 1.6 (Paszke et al., 2019) and the last linear layer is implemented using the different Bayesian approximations from § 2.
Revised: The feature extractor of ResNet is used as implemented in PyTorch 1.6 (Paszke et al., 2019) and the last linear layer is implemented using the different Bayesian approximations from § 2.

Original: All models were trained from random initialization.
Revised: All models were trained from random initialization.

Original: We employed early stopping at highest validation set accuracy.
Revised: We employed early stopping at highest validation set accuracy.

Original: More details on the training procedure and a link to our source code can be found in
Revised: More details on the training procedure and a link to our source code can be found in

Original: Appendix A.2.
Revised: Appendix A.3.

Original: First, we compute the accuracies and all calibration error metrics from § 3 and the UCE on the test sets of CIFAR-10/100 for all models.
Revised: First, we compute the accuracies and all calibration error metrics from § 3 and the UCE on the test sets of CIFAR-10/100 for all models.

Original: We investigate the effect of the number of bins in the estimators of the metrics involving binning and analyze the ranking of different models under varying softmax temperature τ , where p = σ SM ( τ − 1 f w ( x )) .
Revised: We investigate the effect of the number of bins in the estimators of the metrics involving binning and analyze the ranking of different models under varying softmax temperature τ , where p = σ SM ( τ − 1 f w ( x )) .

Original: Finally, we train a ResNet-50 on CIFAR-100 with added calibration error regularization as in Eq.
Revised: Finally, we train a ResNet on CIFAR-10/100, SVHN, and Fashion-MNIST with added calibration error regularization as in Eq.

Original: (12) and (25).
Revised: (5) and (18).

Original: We compare UCE regularization ( λ = 10 ) to regularization with MMCE ( λ = 10 ) and conﬁdence penalty arg min (cid:80) D NLL
Revised: We compare UCE regularization ( λ = 10 ) to regularization with MMCE ( λ = 10 ) and conﬁdence penalty arg min (cid:80) D NLL

Original: ( D, w ) + λ H ( D, w ) with λ = 0 .
Revised: ( D, w ) + λ H ( D, w ) with λ = 0 .

Original: 1 , which penalizes the entropy of the probability vector p of each prediction (Pereyra et al., 2017).
Revised: 1 , which penalizes the entropy of the probability vector p of each prediction (Pereyra et al., 2017).

Original: We combine the regularization experiments with post-hoc calibration using temperature scaling (Guo et al., 2017).
Revised: We combine the regularization experiments with post-hoc calibration using temperature scaling (Guo et al., 2017).

Original: Additionally, we analyze the utility of the normalized entropy as a measure of uncertainty and perform rejection and out-of-distribution (OoD) detection experiments using ˜ H .
Revised: Additionally, we analyze the utility of the normalized entropy as a measure of uncertainty and perform rejection and out-of-distribution (OoD) detection experiments using ˜ H .

Original: We deﬁne an uncertainty threshold H max and reject all predictions from the test set where ˜ H ( p ) > H max .
Revised: We deﬁne an uncertainty threshold H max and reject all predictions from the test set where ˜ H ( p ) > H max .

Original: A decrease in false predictions of the remaining test set is expected.
Revised: A decrease in false predictions of the remaining test set is expected.

Original: To demonstrate the OoD detection ability, we provide images from CIFAR-100 to a deep model trained on CIFAR-10 (note that both CIFAR data sets have no mutual classes).
Revised: To demonstrate the OoD detection ability, we provide images from CIFAR-100 to a deep model trained on CIFAR-10 (note that both CIFAR data sets have no mutual classes).

Original: In this experiment, we compose a batch of 100 random samples from the test set of the training domain and stepwise replace images with out-of-distribution data.
Revised: In this experiment, we compose a batch of 100 random samples from the test set of the training domain and stepwise replace images with out-of-distribution data.

Original: In practice, it is expected that models are applied to a mix of known and unknown classes.
Revised: In practice, it is expected that models are applied to a mix of known and unknown classes.

Original: After each step, we evaluate the mean batch uncertainty and expect, that the mean uncertainty monotonically increases as a function of the fraction of OoD data.
Revised: After each step, we evaluate the mean batch uncertainty and expect, that the mean uncertainty monotonically increases as a function of the fraction of OoD data.

Original: 5. R ESULTS
Revised: 5. R ESULTS

Original: In this section, the results of the above mentioned experimental setups are presented and discussed.
Revised: In this section, the results of the above mentioned experimental setups are presented and discussed.

Original: Comparison of Calibration Error Metrics Table 1 shows test set accuracy and all calibration error results for all model/data set conﬁgurations.
Revised: Comparison of Calibration Error Metrics Table 2 shows test set accuracy and all calibration error results for all model/data set conﬁgurations.

Original: Without any post-hoc calibration, such as temperature scaling, all metrics provide the same ranking of the models.
Revised: Without any post-hoc calibration, such as temperature scaling, all metrics provide the same ranking of the models.

Original: The deep ensemble and SWAG perform  best in terms of test set accuracy and calibration of uncertainty.
Revised: The deep ensemble and SWAG perform [Figure] best in terms of test set accuracy and calibration of uncertainty.

Original: Brier score and NLL are both highly sensitive to the model accuracy, which is especially apparent on CIFAR-10.
Revised: Brier score and NLL are both highly sensitive to the model accuracy, which is especially apparent on CIFAR-10.

Original: For the ﬁrst three models with similar accuracy, the Brier scores differ only marginally.
Revised: For the ﬁrst three models with similar accuracy, the Brier scores differ only marginally.

Original: Thus, both the Brier score and the NLL are unsuitable for comparing the calibration of different models.
Revised: Thus, both the Brier score and the NLL are unsuitable for comparing the calibration of different models.

Original: Ashukha et al.
Revised: Ashukha et al.

Original: (2020) propose to use the calibrated NLL at optimal temperature for model comparison.
Revised: (2020) propose to use the calibrated NLL at optimal temperature for model comparison.

Original: However, Fig.
Revised: However, Fig.

Original: [Figure] [Figure] (and Fig. 6–7 in the appendix) plot the metrics over varying softmax temperature and show, that the models with highest accuracy have lowest Brier and NLL, regardless of the temperature.
Revised: 5–6 plot the metrics over varying softmax temperature and show, that the models with highest accuracy have lowest Brier and NLL, regardless of the temperature.

Original: From this we deduce that both Brier and NLL should not be used for comparison of multi-class calibration, even at optimal temperature.
Revised: From this we deduce that both Brier and NLL should not be used for comparison of multi-class calibration, even at optimal temperature.

Original: The remaining metrics show consistent ranking before and after the point of optimal temperature.
Revised: The remaining metrics show consistent ranking before and after the point of optimal temperature.

Original: The metrics ECE, UCE and MMCE have a narrow region in which the optimal temperature for all models can be found.
Revised: The metrics ECE, UCE and MMCE have a narrow region in which the optimal temperature for all models can be found.

Original: This allows comparison of calibration of models if they are all over- or underconﬁdent.
Revised: This allows comparison of calibration of models if they are all over- or underconﬁdent.

Original: However, all metrics fail at comparing underconﬁdent models to overconﬁdent models (see model ranking left and right of optimal temperature in Fig. 1).
Revised: However, all metrics fail at comparing underconﬁdent models to overconﬁdent models (see model ranking left and right of optimal temperature in Fig. 6).

Original: Fig.
Revised: Fig.

Original: 2 shows the effect of the number of bins M in the estimators of ECE, ACE and UCE.
Revised: 1 shows the effect of the number of bins M in the estimators of ECE, ACE and UCE.

Original: Both ECE and ACE are more sensitive to the number of bins and do not provide a consistent ranking of models under varying bin count.
Revised: Both ECE and ACE are more sensitive to the number of bins and do not provide a consistent ranking of models under varying bin count.

Original: This is due to the fact that fewer bins are populated using ˜ H as uncertainty (cf.
Revised: This is due to the fact that fewer bins are populated using ˜ H as uncertainty (cf.

Original: Fig.
Revised: Fig.

Original: 10 in the appendix).
Revised: 10 in the appendix).

Original: This can be interpreted as possible downside of the UCE metric as the adaptive binning scheme of ACE explicitly addresses that.
Revised: This can be interpreted as possible downside of the UCE metric as the adaptive binning scheme of ACE explicitly addresses that.

Original: However, we argue that consistent ranking due to robustness against bin count results in a metric that is more useful in practice.[Table] [Figure]
Revised: However, we argue that consistent ranking due to robustness against bin count results in a metric that is more useful in practice.

Original: Uncertainty Regularization Tab.
Revised: Uncertainty Regularization Tab.

Original: 2 shows results of ResNet-50 with SWAG trained on CIFAR-100.
Revised: 1 shows results of ResNet-50 with SWAG trained on CIFAR-100.

Original: All regularization methods considerably reduce miscalibration compared to unregularized models from Tab. 1.
Revised: All regularization methods considerably reduce miscalibration compared to unregularized models.

Original: Plain entropy regularization is surprisingly effective on CIFAR-100; however, on CIFAR10 (see Tab. 4 in supplementary), it increased miscalibration and is generally outperformed by MMCE and UCE regularization at optimal temperature.
Revised: Plain entropy regularization is surprisingly effective on CIFAR-100; however, on CIFAR-10 (see Tab. 1), it increased miscalibration and is generally outperformed by MMCE and UCE regularization at optimal temperature.

Original: Therefore, when performing post-hoc temperature scaling, MMCE and UCE regularization is preferable to entropy regularization.
Revised: Therefore, when performing post-hoc temperature scaling, MMCE and UCE regularization is preferable to entropy regularization.

Original: UCE regularization can be interpreted as entropy penalization for predictions with low accuracy.
Revised: UCE regularization can be interpreted as entropy penalization for predictions with low accuracy.

Original: As UCE is zero for an optimal model, it encourages a model to reach high accuracy.
Revised: As UCE is zero for an optimal model, it encourages a model to reach high accuracy.

Original: Rejection & OoD Detection Fig.
Revised: Rejection & OoD Detection Fig.

Original: 3 (left) shows the top-1 error as a function of decreasing uncertainty threshold H max and (right) shows the mean batch uncertainty at increasing OoD data.
Revised: 2 (left) shows the top-1 error as a function of decreasing uncertainty threshold H max and (right) shows the mean batch uncertainty at increasing OoD data.

Original: Robust rejection of uncertain predictions and detection of OoD data based on the normalized entropy ˜ H ( p ) is possible and is generally more sensitive to OoD data than the conﬁdence max p .
Revised: Robust rejection of uncertain predictions and detection of OoD data based on the normalized entropy ˜ H ( p ) is possible and is generally more sensitive to OoD data than the conﬁdence max p .

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We have proposed to measure uncertainty based on the normalized entropy.
Revised: We have proposed to measure uncertainty based on the normalized entropy.

Original: From this, we derived the uncertainty calibration error; a new metric that avoids several pathologies of existing calibration errors.
Revised: From this, we derived the uncertainty calibration error; a new metric that avoids several pathologies of existing calibration errors.

Original: In our experimental evaluation, we focused on uncertainty from approximate Bayesian methods and deep ensembles.
Revised: In our experimental evaluation, we focused on uncertainty from approximate Bayesian methods and deep ensembles.

Original: The UCE does not only consider the class with the highest probability and is not minimized by a constant model predicting the marginal class distribution.
Revised: The UCE does not only consider the class with the highest probability and is not minimized by a constant model predicting the marginal class distribution.

Original: In contrast to the
Revised: In contrast to the

Original: Brier score and NLL, it allows comparison of models with different accuracy.
Revised: Brier score and NLL, it allows comparison of models with different accuracy.

Original: It is not sensitive to a varying number of bins and provides a consistent ranking of models.
Revised: It is not sensitive to a varying number of bins and provides a consistent ranking of models.

Original: However, we follow the suggestion of Ashukha et al.
Revised: However, we follow the suggestion of Ashukha et al.

Original: (2020) and state that comparison of calibration for different models should only be done at optimal softmax temperature.
Revised: (2020) and state that comparison of calibration for different models should only be done at optimal softmax temperature.

Original: Regularization with UCE during training reduces miscalibration and does not penalize high accuracy and predictions with justiﬁed high conﬁdence.
Revised: Regularization with UCE during training reduces miscalibration and does not penalize high accuracy and predictions with justiﬁed high conﬁdence.

Original: UCE regularization with temperature scaling often performed best in our experiments in terms of calibration.
Revised: UCE regularization with temperature scaling often performed best in our experiments in terms of calibration.

Original: The normalized entropy itself is a useful measure of uncertainty and allows for robust rejection of uncertain predictions and detection of OoD data.
Revised: The normalized entropy itself is a useful measure of uncertainty and allows for robust rejection of uncertain predictions and detection of OoD data.

Original: With this work, we hope to have provided a new useful metric for reliable evaluation of uncertainty estimation.
Revised: We hope to have provided a new useful metric for reliable evaluation of uncertainty estimation.

Original: The UCE is easy to implement and interpretable as it expresses the discrepancy of the uncertainty from the model error, which increases the chance of being accepted by deep learning practitioners.
Revised: UCE is easy to implement and interpretable as it expresses the discrepancy of the uncertainty from the model error, which increases the chance of being accepted by deep learning practitioners.

Original:  R EBUTTAL
Revised:  R EBUTTAL

Original: In this section, new or updated results during the rebuttal phase are presented.
Revised: In this section, new or updated results during the rebuttal phase are presented.

Original: We identiﬁed the main question of the reviewers as to why/where our calibration metric (UCE) is beneﬁcial compared to previous metrics (ECE, MMCE, ACE) and how normalized entropy is beneﬁcial over vanilla entropy.
Revised: We identiﬁed the main question of the reviewers as to why/where our calibration metric (UCE) is beneﬁcial compared to previous metrics (ECE, MMCE, ACE) and how normalized entropy is beneﬁcial over vanilla entropy.

Original: We project the following main changes for the ﬁnal manuscript:
Revised: We project the following main changes for the ﬁnal manuscript:

Original: • illustrate the advantage of normalized entropy over vanilla entropy • add toy experiments to show that UCE is able to capture miscalibration where ECE, ACE and MMCE fail (see results below) • add results from experiments on additional data sets (SVHN, Fashion-MNIST)
Revised: • illustrate the advantage of normalized entropy over vanilla entropy • add toy experiments to show that UCE is able to capture miscalibration where ECE, ACE and MMCE fail (see results below) • add results from experiments on additional data sets (SVHN, Fashion-MNIST)

Original: To make room for the following content, we propose the removal/reduction (approx.
Revised: To make room for the following content, we propose the removal/reduction (approx.

Original: -1 page) of
Revised: -1 page

Original:  • the review of Bayesian methods and refer to previous work instead, • experiments regarding temperature (remove Table 1 and 
Revised: of • the review of Bayesian methods and refer to previous work instead, • experiments regarding temperature (remove Table 1 and Figure 1)

Original: Figure
Revised: Additionally, we will include all minor comments of the reviewers in the ﬁnal manuscript.

Original: 7. U NCERTAINTY FROM N ORMALIZED E NTROPY VS . E NTROPY
Revised: 7. U NCERTAINTY FROM N ORMALIZED E NTROPY VS . E NTROPY

Original: The advantages of normalized entropy over vanilla entropy in our deﬁnition of UCE are twofold:
Revised: The advantages of normalized entropy over vanilla entropy in our deﬁnition of UCE are twofold:

Original: First, the domain of the metric does not scale with the number of classes C , which helps comparing the calibration of models trained on different data sets.
Revised: First, the domain of the metric does not scale with the number of classes C , which helps comparing the calibration of models trained on different data sets.

Original: Second, the value of the metric is more interpretable:
Revised: Second, the value of the metric is more interpretable:

Original: Rejecting test samples where ˜ H < 0 .
Revised: Rejecting test samples where ˜ H < 0 .

Original: 2 will result in a classiﬁcation error < 0 .
Revised: 2 will result in a classiﬁcation error < 0 .

Original: 2 for the remaining samples if the model is well-calibrated (see Fig. 3).
Revised: 2 for the remaining samples if the model is well-calibrated (see Fig. 2).

Original: We argue that using normalized entropy as uncertainty measure is as interpretable as max p , but avoids the pathologies of max p when used in a calibration metric.
Revised: We argue that using normalized entropy as uncertainty measure is as interpretable as max p , but avoids the pathologies of max p when used in a calibration metric.

Original: 7. UCE VS . O THER C ALIBRATION M ETRICS
Revised: 7. UCE VS . O THER C ALIBRATION M ETRICS

Original: UCE is more reliable than ECE and MMCE because it is based on normalized entropy and incorporates the predictions of all classes.
Revised: UCE is more reliable than ECE and MMCE because it is based on normalized entropy and incorporates the predictions of all classes.

Original: It is more robust than ACE, as it is signiﬁcantly less sensitive to binning.
Revised: It is more robust than ACE, as it is signiﬁcantly less sensitive to binning.

Original: Reliable Uncertainty Detection ECE and MMCE can be minimized by models with an uninformative, constant output (Ovadia et al., 2019).
Revised: Reliable Uncertainty Detection ECE and MMCE can be minimized by models with an uninformative, constant output (Ovadia et al., 2019).

Original: Given a data set of two classes, with 60 % class 1 and 40 % class 2, and a degenerated model that consistently predicts the marginal probabilities p
Revised: Given a data set of two classes, with 60 % class 1 and 40 % class 2, and a degenerated model that consistently predicts the marginal probabilities p

Original: =
Revised: =

Original: This leads to 60 % correctly classiﬁed samples for class 1 leading to perfect calibration scores for 
Revised: This leads to 60 % correctly classiﬁed samples for class 1 leading to perfect calibration scores for [Figure] uncertainty [Figure]

Revised: [Figure]

Original: ECE and MMCE.
Revised: ECE and MMCE.

Original: Class 2, however, is misclassiﬁed in 100% of the cases—and not 60 % as expected.
Revised: Class 2, however, is misclassiﬁed in 100% of the cases—and not 60 % as expected.

Original: The miscalibration of class 2 is not reﬂected in ECE or MMCE.
Revised: The miscalibration of class 2 is not reﬂected in ECE or MMCE.

Original: ACE is computed class wise and is able to capture this miscalibration.
Revised: ACE is computed class wise and is able to capture this miscalibration.

Original: UCE is based on the normalized entropy to determine uncertainty and therefore incorporates the predictions of all classes.
Revised: UCE is based on the normalized entropy to determine uncertainty and therefore incorporates the predictions of all classes.

Original: Fig.
Revised: Fig.

Original: 4 visualizes these results.
Revised: 3 visualizes these results.

Original: [Figure] [Figure] Robustness to Varying Number of Bins From a reliable calibration metric, we would expect a constant ranking of two models in comparison, independent of varying calibration parameters.
Revised: Robustness to Varying Number of Bins From a reliable calibration metric, we would expect a constant ranking of two models in comparison, independent of varying calibration parameters.

Original: However, ACE is highly sensitive to the number of bins and produces arbitrary rankings of models.
Revised: However, ACE is highly sensitive to the number of bins and produces arbitrary rankings of models.

Original: Fig.
Revised: Fig.

Original: 5 highlights this by comparing two differently calibrated random models.
Revised: 4 highlights this by comparing two differently calibrated random models.

Original: 7. UCE REGULARIZATION
Revised: 7. UCE REGULARIZATION

Original: At optimal temperature (as suggested by Ashukha et al.
Revised: At optimal temperature (as suggested by Ashukha et al.

Original: , UCE and MMCE regularization considerably reduce miscalibration for all employed calibration metric outperforming entropy regularization, with UCE achieving highest accuracy on CIFAR-100, SVHN and Fashion-MNIST.
Revised: , UCE and MMCE regularization considerably reduce miscalibration for all employed calibration metric outperforming entropy regularization, with UCE achieving highest accuracy on CIFAR-100, SVHN and Fashion-MNIST.

Original: (see
Revised: (see

Original: Tab. 3).
Revised: Tab. 1).

Original: We want to stress out that UCE, in contrast to MMCE, was not speciﬁcally designed for the use as a calibration regularizer (Kumar et al., 2018).
Revised: We want to stress out that UCE, in contrast to MMCE, was not speciﬁcally designed for the use as a calibration regularizer (Kumar et al., 2018).

Original: Why UCE Regularization Works UCE regularization works best when computed classwise (in similar manner to ACE): UCE = 1 C (cid:80) Cc =1 UCE( c ) , where UCE( c ) is computed for predictions of class c .
Revised: Why UCE Regularization Works UCE regularization works best when computed classwise (in similar manner to ACE): UCE = 1 C (cid:80) C c =1 UCE( c ) , where UCE( c ) is computed for predictions of class c .

Original: Consider the following binary classiﬁcation example: A batch with mainly samples from class 1 and few samples from class 2 are all predicted as class 1 with high conﬁdence.
Revised: Consider the following binary classiﬁcation example: A batch with mainly samples from class 1 and few samples from class 2 are all predicted as class 1 with high conﬁdence.

Original: NLL further pushes the conﬁdence of the predictions to 1.0, favoring overconﬁdence, whereas UCE is only reduced if the conﬁdence of the overconﬁdently false predictions is reduced.
Revised: NLL further pushes the conﬁdence of the predictions to 1.0, favoring overconﬁdence, whereas UCE is only reduced if the conﬁdence of the overconﬁdently false predictions is reduced.

Original: [Table]
Revised: [Table]

Original: A A PPENDIX A. P ROOFS
Revised: A A PPENDIX A. P ROOFS

Original: Proposition 1.
Revised: Proposition 1.

Original: The normalized entropy (uncertainty) ˜ H ( p ) approaches the top-1 error in the limit of number of classes C if the model p is well-calibrated.
Revised: The normalized entropy (uncertainty) ˜ H ( p ) approaches the top-1 error in the limit of number of classes C if the model p is well-calibrated.

Original: Proof.
Revised: Proof.

Original: With Lemma 1 and ˆ p = max p we rewrite the normalized entropy as
Revised: With Lemma 1 and ˆ p = max p we rewrite the normalized entropy as

Original: [Equation]
Revised: [Equation]

Original: Now, in the limit of number of classes C
Revised: Now, in the limit of number of classes C

Original: [Equation]
Revised: [Equation]

Original: The top-1 error equals (1 − ˆ p ) if the model is perfectly calibrated in the sense of Eq.
Revised: The top-1 error equals (1 − ˆ p ) if the model is perfectly calibrated in the sense of Eq.

Original: Lemma 1.
Revised: Lemma 1.

Original: Given a softmax output p with C entries and the most likely prediction ˆ y = arg max p with likelihood ˆ p = max p .
Revised: Given a softmax output p with C entries and the most likely prediction ˆ y = arg max p with likelihood ˆ p = max p .

Original: Then, the remaining entries p i,i (cid:54) =
Revised: Then, the remaining entries p i,i (cid:54) =

Original: ˆ y are approximately uniformly distributed with probability 1 − ˆ p C − 1 .
Revised: ˆ y are approximately uniformly distributed with probability 1 − ˆ p C − 1 .

Original: Proof.
Revised: Proof.

Original: This assumption is approximately correct (1) if ˆ p → 1 or (2) if C → ∞ .
Revised: This assumption is approximately correct (1) if ˆ p → 1 or (2) if C → ∞ .

Original: Let ˜ p
Revised: Let ˜ p

Original: j = p i ∀
Revised: j = p i ∀

Original: i (cid:54) =
Revised: i (cid:54) =

Original: ˆ y and ˜ q j = (1 − ˆ p )
Revised: ˆ y and ˜ q j = (1 − ˆ p )

Original: C − 1 .
Revised: C − 1 .

Original: Note that ˜ p and ˜ q are not proper probability distributions as (cid:80) ˜ p
Revised: Note that ˜ p and ˜ q are not proper probability distributions as (cid:80) ˜ p

Original: j = (cid:80) ˜
Revised: j = (cid:80) ˜

Original: q j = (1 − ˆ p ) .
Revised: q j = (1 − ˆ p ) .

Original: (1) Consider KL[˜ p (cid:107) ˜ q ] as ˆ p approaches 1 :
Revised: (1) Consider KL[˜ p (cid:107) ˜ q ] as ˆ p approaches 1 :

Original: [Equation]
Revised: [Equation]

Original: (2) Let z i be the logits of a model trained with L2 regularization.
Revised: (2) Let z i be the logits of a model trained with L2 regularization.

Original: The magnitude of the logits | z i | cannot become arbitrary large and due to the normalizing nature of softmax
Revised: The magnitude of the logits | z i | cannot become arbitrary large and due to the normalizing nature of softmax

Original: [Equation]
Revised: [Equation]

Original: Alternatively, let z ∈ A C and z (cid:48)
Revised: Alternatively, let z ∈ A C and z (cid:48)

Original: ∈ B K be two logit vectors with C < K .
Revised: ∈ B K be two logit vectors with C < K .

Original: If both models have been trained with L2 regularization, the magnitude of the logits | z i | , | z (cid:48) i | cannot become arbitrary large.
Revised: If both models have been trained with L2 regularization, the magnitude of the logits | z i | , | z (cid:48) i | cannot become arbitrary large.

Original: More speciﬁcally, A = B ⊂ R .
Revised: More speciﬁcally, A = B ⊂ R .

Original: Due to the normalizing nature of softmax, z (cid:48) corresponds to a lower softmax temperature and as the temperature decreases with increasing number of classes, softmax approaches a uniform distribution (Jang et al., 2017).
Revised: Due to the normalizing nature of softmax, z (cid:48) corresponds to a lower softmax temperature and as the temperature decreases with increasing number of classes, softmax approaches a uniform distribution (Jang et al., 2017).

Original: A. T RAINING S ETTING
Revised: A. B AYESIAN D EEP L EARNING M ETHODS

Revised: In the following, we brieﬂy describe common approximately Bayesian methods which we use in our experiments to obtain weight uncertainty.

Revised: Monte Carlo Dropout One practical approximation of the posterior is variational inference with

Revised: Monte Carlo (MC) dropout (Gal & Ghahramani, 2016).

Revised: To determine model uncertainty, dropout variational inference is performed by training the model f w with dropout (Srivastava et al., 2014) and using dropout at test time to sample from the approximate posterior distribution by performing

Revised: N stochastic forward passes per test sample (Gal & Ghahramani, 2016; Kendall & Gal, 2017).

Revised: This is also referred to as MC dropout.

Revised: In MC dropout, the ﬁnal probability vector of the predictive distribution is computed by MC integration:

Revised: [Equation]

Revised: [Equation]

Revised: Gaussian Dropout Gaussian dropout was ﬁrst proposed by Wang & Manning (2013) and linked to variational inference by Kingma et al.

Revised: Dropout introduces Bernoulli noise during optimization and reduces overﬁtting of the training data.

Revised: The resulting output y of a layer with dropout is a weighted sum of Bernoulli random variables.

Revised: Then, the central limit theorem states, that y is approximately normally distributed.

Revised: Instead of sampling from the weights and computing the resulting output, we can directly sample from the implicit Gaussian distribution of dropout

Revised: [Equation]

Revised: [Equation]

Revised: with

Revised: [Equation]

Revised: using the reparameterization trick (Kingma et al., 2015)

Revised: [Equation]

Revised: Gaussian dropout is a continuous approximation to Bernoulli dropout, and in comparison it will better approximate the true posterior distribution and is expected to provide improved uncertainty estimates (Louizos & Welling, 2017).

Revised: To obtain the ﬁnal probability vector p ( x ) , we again use MC integration with N stochastic forward passes.

Revised: The dropout rate p is now a learnable parameter and does not need to be chosen carefully by hand.

Revised: In fact, p could be optimized w.r.t.

Revised: uncertainty calibration, scaling the variance of the implicit Gaussian of dropout.

Revised: A similar approach was presented by Gal et al.

Revised: (2017) using the Concrete distribution (Maddison et al., 2016; Jang et al., 2017).

Revised: However, we focus on metrics for measuring calibration and therefore ﬁx p in our subsequent experiments.

Revised: Bayes by Backprop Blundell et al.

Revised: (2015) assume a Gaussian distribution with diagonal covariance matrix as variational posterior q ( w | θ ) , parameterized by mean µ and standard deviation σ , where θ = { µ , σ } .

Revised: A sample of the weights can be obtained by sampling a multivariate unit Gaussian and shift it by µ and scale it by σ .

Revised: Then, the network is trained by minimizing

Revised: [Equation]

Revised: In case of a zero mean Gaussian prior, the ﬁrst term can be implemented by weight decay.

Revised: In contrast to Gaussian dropout, which operates on the implicit distribution of the activations, Bayes by Backprop (BBB) directly operates on the weights.

Revised: This doubles the number of trainable parameters in practice.

Revised: MC integration is used to obtain the ﬁnal probability vector p ( x ) .

Revised: SWA-Gaussian Stochastic weight averaging (SWA) uses stochastic gradient descent steps around a local loss optimum of a trained network and averages the weights w SWA = 1 T (cid:80)

Revised: Ti =1 w i of the model from each step i (Izmailov et al., 2018).

Revised: This explores the loss landscape and averaging helps to ﬁnd a better weight estimate than converging to a single local optimum.

Revised: SWA-Gaussian (SWAG) is closely related to Bayes by Backprop (Maddox et al., 2019).

Revised: It assumes a Gaussian distribution with diagonal covariance matrix as approximate variational posterior.

Revised: Instead of using backpropagation to directly optimize µ and σ , it ﬁts a Gaussian by using µ = w SWA and

Revised: [Equation]

Revised: This doubles the number of parameters at test time.

Revised: The approximate Gaussian posterior results to

Revised: N ( w SWA , Σ diag ) and MC integration with samples w i ∼ N ( w SWA , Σ diag ) is used to compute the ﬁnal probability vector p ( x ) .

Revised: Deep Ensembles Training

Revised: multiple randomly initialized copies of a deep network by performing maximum posterior estimation and ensembling them to get multiple predictions for a single input is not a variational inference method.

Revised: However, they have been reported to produce surprisingly useful uncertainty estimates in practice that are better calibrated (Lakshminarayanan et al., 2017).

Revised: Deep ensembles considerably increase the number of parameters at train and test time.

Revised: We use deep ensembles as non Bayesian baseline for uncertainty estimation.

Revised: A. T RAINING S ETTING

Original: The base model implementations from PyTorch 1.6 (Paszke et al., 2019) are used and trained with following settings:
Revised: The base model implementations from PyTorch 1.6 (Paszke et al., 2019) are used and trained with following settings:

Original: • Adam optimizer with initial learn rate of 3e-4 and β 1 = 0 .
Revised: • Adam optimizer with initial learn rate of 3e-4 and β 1 = 0 .

Original: 9 , β 2 = 0 .
Revised: 9 , β 2 = 0 .

Original: 999 and mini-batch size of 256 (Kingma & Ba, 2015) • weight decay of 1e- • negative-log likelihood (cross entropy) loss • reduce-on-plateau learn rate scheduler (patience of 20 epochs) with factor of 0 .•
Revised: 999 and mini-batch size of 256 (Kingma & Ba, 2015) • weight decay of 1e- • negative-log likelihood (cross entropy) loss • reduce-on-plateau learn rate scheduler (patience of 20 epochs) with factor of 0 .•

Original: additional validation set is randomly extracted from the training set ( 5 , 000 samples) • ResNet-34 for CIFAR-10 and ResNet-50 for CIFAR-100 experiments • only the last linear layer is implemented in a Bayesian manner for MC dropout, Gaussian dropout, BayesByBackprop and SWAG • the deep ensemble comprises 3 fully individually trained networks • N = 25 forward passes were used Monte Carlo integration • in MC dropout and Gaussian dropout, a dropout rate of p = 0 .
Revised: additional validation set is randomly extracted from the training set ( 5 , 000 samples) • ResNet-34 for CIFAR-10 and ResNet-50 for CIFAR-100 experiments • only the last linear layer is implemented in a Bayesian manner for MC dropout, Gaussian dropout, BayesByBackprop and SWAG • the deep ensemble comprises 3 fully individually trained networks • N = 25 forward passes were used Monte Carlo integration • in MC dropout and Gaussian dropout, a dropout rate of p = 0 .

Original: 2 was used • in SWAG, a learn rate of 3e-6 was used during weight averaging
Revised: 2 was used • in SWAG, a learn rate of 3e-6 was used during weight averaging

Original: Our code is available at: https://github.com/link-withheld.
Revised: Our code is available at: https://github.com/link-withheld.

Original: A.A DDITIONAL R ESULTS Regularization
Revised: A.

Revised: [Table] A DDITIONAL R ESULTS

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: We used M = 15 bins where necessary. Here, all metrics provide the same ranking of models. [Table]

Original: [Figure]
Revised: [Figure]

Original: i
Revised: A DDITIONAL F

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure][Figure] [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: {+.uncertainty threshold 0 .0 .0 .
+}
Revised: {+.uncertainty threshold 0 .0 .0 .
+}

Original: t
Revised: .t

Original: [Figure]

Original: e rr o r
Revised: o pe rr o r

Original: [Figure]
Revised: [Figure]

Revised: [Figure]