Original: A STATISTICAL THEORY OF COLD POSTERIORS IN DEEP NEURAL NETWORKS
Revised: A STATISTICAL THEORY OF COLD POSTERIORS IN DEEP NEURAL NETWORKS

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: To get Bayesian neural networks to perform comparably to standard neural networks it is usually necessary to artiﬁcially reduce uncertainty using a “tempered” or “cold” posterior.
Revised: To get Bayesian neural networks to perform comparably to standard neural networks it is usually necessary to artiﬁcially reduce uncertainty using a “tempered” or “cold” posterior.

Original: This is extremely concerning: if the generative model is accurate,
Revised: This is extremely concerning: if the generative model is accurate,

Original: Bayesian inference/decision theory is optimal, and any artiﬁcial changes to the posterior should harm performance.
Revised: Bayesian inference/decision theory is optimal, and any artiﬁcial changes to the posterior should harm performance.

Original: While this suggests that the prior may be at fault, here we argue that in fact, BNNs for image classiﬁcation use the wrong likelihood.
Revised: While this suggests that the prior may be at fault, here we argue that in fact, BNNs for image classiﬁcation use the wrong likelihood.

Original: In particular, standard image benchmark datasets such as CIFAR-10 are carefully curated.
Revised: In particular, standard image benchmark datasets such as CIFAR-10 are carefully curated.

Original: We develop a generative model describing curation which gives a principled Bayesian account of cold posteriors, because the likelihood under this new generative model closely matches the tempered likelihoods used in past work.
Revised: We develop a generative model describing curation which gives a principled Bayesian account of cold posteriors, because the likelihood under this new generative model closely matches the tempered likelihoods used in past work.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Recent work has highlighted that Bayesian neural networks (BNNs) typically have better predictive performance when we “sharpen” the posterior (Wenzel et al., 2020).
Revised: Recent work has highlighted that Bayesian neural networks (BNNs) typically have better predictive performance when we “sharpen” the posterior (Wenzel et al., 2020).

Original: In stochastic gradient Langevin dynamics (SGLD) (Welling & Teh, 2011), this can be achieved by multiplying the log-posterior by 1 /T , where the “temperature”, T is smaller than 1 (Wenzel et al., 2020).
Revised: In stochastic gradient Langevin dynamics (SGLD) (Welling & Teh, 2011), this can be achieved by multiplying the log-posterior by 1 /T , where the “temperature”, T is smaller than 1 (Wenzel et al., 2020).

Original: Broadly the same effect can be achieved in variational inference by “tempering”, i.e. downweighting the KL term.
Revised: Broadly the same effect can be achieved in variational inference by “tempering”, i.e. downweighting the KL term.

Original: As noted in
Revised: As noted in

Original: Wenzel et al.
Revised: Wenzel et al.

Original: (2020), this approach has been used in many recent papers to obtain good performance, albeit without always emphasising the importance of this factor (Zhang et al., 2017; Bae et al., 2018;
Revised: (2020), this approach has been used in many recent papers to obtain good performance, albeit without always emphasising the importance of this factor (Zhang et al., 2017; Bae et al., 2018;

Original: Osawa et al., 2019; Ashukha et al., 2020).
Revised: Osawa et al., 2019; Ashukha et al., 2020).

Original: These results are puzzling if we take the usual Bayesian viewpoint, which says that the Bayesian posterior, used with the right prior, and in combination with Bayes decision theory should give optimal performance (Jaynes, 2003).
Revised: These results are puzzling if we take the usual Bayesian viewpoint, which says that the Bayesian posterior, used with the right prior, and in combination with Bayes decision theory should give optimal performance (Jaynes, 2003).

Original: Thus, these results may suggest we are using the wrong prior.
Revised: Thus, these results may suggest we are using the wrong prior.

Original: While new priors have been suggested (e.g. Ober & Aitchison, 2020), they give only minor improvements in performance — certainly nothing like enough to close the gap to carefully trained non-Bayesian networks.
Revised: While new priors have been suggested (e.g. Ober & Aitchison, 2020), they give only minor improvements in performance — certainly nothing like enough to close the gap to carefully trained non-Bayesian networks.

Original: In contrast, tempered posteriors directly give performance comparable to a carefully trained ﬁnite network.
Revised: In contrast, tempered posteriors directly give performance comparable to a carefully trained ﬁnite network.

Original: The failure to develop an effective prior suggests that we should consider alternative explanations for the effectiveness of tempering.
Revised: The failure to develop an effective prior suggests that we should consider alternative explanations for the effectiveness of tempering.

Original: Here, we consider the possibility that it is predominantly (but not entirely) the likelihood , and not the prior that is at fault.
Revised: Here, we consider the possibility that it is predominantly (but not entirely) the likelihood , and not the prior that is at fault.

Original: In particular, we note that standard image benchmark datasets such as ImageNet and CIFAR-10 are carefully curated, and that it is important to consider this curation as part of our generative model.
Revised: In particular, we note that standard image benchmark datasets such as ImageNet and CIFAR-10 are carefully curated, and that it is important to consider this curation as part of our generative model.

Original: We develop a simpliﬁed generative model describing dataset curation which assumes that a datapoint is included in the dataset only if there is unanimous agreement on the class amongst multiple labellers.
Revised: We develop a simpliﬁed generative model describing dataset curation which assumes that a datapoint is included in the dataset only if there is unanimous agreement on the class amongst multiple labellers.

Original: This model naturally multiplies the effect of each datapoint, and hence gives posteriors that closely match tempered or cold posteriors.
Revised: This model naturally multiplies the effect of each datapoint, and hence gives posteriors that closely match tempered or cold posteriors.

Original: We show that toy data drawn from our generative model of curation can give rise to optimal temperatures being smaller than 1 .
Revised: We show that toy data drawn from our generative model of curation can give rise to optimal temperatures being smaller than 1 .

Original: Our model predicts that cold posteriors will not be helpful when the original underlying labels from all labellers are available.
Revised: Our model predicts that cold posteriors will not be helpful when the original underlying labels from all labellers are available.

Original: While these are not available for standard datasets such as CIFAR-10, we found a good proxy: the CIFAR-10H dataset (Peterson et al., 2019), in which ∼ 50 humans annotators labelled the CIFAR-10 test-set (we use these as our training set, and use the standard CIFAR-10 training set for test-data).
Revised: While these are not available for standard datasets such as CIFAR-10, we found a good proxy: the CIFAR-10H dataset (Peterson et al., 2019), in which ∼ 50 humans annotators labelled the CIFAR-10 test-set (we use these as our training set, and use the standard CIFAR-10 training set for test-data).

Original: As expected, we ﬁnd strong cold-posterior effects when using the original single-label, which are almost entirely eliminated when using the 50 labels from CIFAR-10H.
Revised: As expected, we ﬁnd strong cold-posterior effects when using the original single-label, which are almost entirely eliminated when using the 50 labels from CIFAR-10H.

Original: In addition, curation implies that each label is almost certain to be correct, which is one way to understand the statistical patterns exploited by cold posteriors.
Revised: In addition, curation implies that each label is almost certain to be correct, which is one way to understand the statistical patterns exploited by cold posteriors.

Original: As such, if we destroy
Revised: As such, if we destroy

Original: [Equation]
Revised: [Equation]

Original: Labellers have the task of classifying images as cats or dogs.
Revised: Labellers have the task of classifying images as cats or dogs.

Original: The ﬁrst image is unambiguously a cat, so the three labellers agree, consensus is reached, and the image is included in the dataset.
Revised: The ﬁrst image is unambiguously a cat, so the three labellers agree, consensus is reached, and the image is included in the dataset.

Original: The second image is unambiguously a dog, the labellers agree, consensus is reached, and the image is included in the dataset.
Revised: The second image is unambiguously a dog, the labellers agree, consensus is reached, and the image is included in the dataset.

Original: However, the third image is ambiguous: the labellers disagree, consensus is not reached, and the image may be excluded from the dataset.
Revised: However, the third image is ambiguous: the labellers disagree, consensus is not reached, and the image may be excluded from the dataset.

Original: this pattern by adding noise to the labels, the cold posterior effect should disappear.
Revised: this pattern by adding noise to the labels, the cold posterior effect should disappear.

Original: We conﬁrmed that with increasing label noise, the cold posterior effect disappears and eventually reverses (giving better performance at temperatures close to 1).
Revised: We conﬁrmed that with increasing label noise, the cold posterior effect disappears and eventually reverses (giving better performance at temperatures close to 1).

Original:  B ACKGROUND : COLD AND TEMPERED POSTERIORS
Revised:  B ACKGROUND : COLD AND TEMPERED POSTERIORS

Original: Tempered (e.g. Zhang et al., 2017) and cold (Wenzel et al., 2020)
Revised: Tempered (e.g. Zhang et al., 2017) and cold (Wenzel et al., 2020)

Original: posteriors differ slightly in how they apply the temperature parameter.
Revised: posteriors differ slightly in how they apply the temperature parameter.

Original: For cold posteriors, we scale the whole posterior, whereas tempering is a method typically applied in variational inference, and corresponds to scaling the likelihood but not the prior,
Revised: For cold posteriors, we scale the whole posterior, whereas tempering is a method typically applied in variational inference, and corresponds to scaling the likelihood but not the prior,

Original: [Equation]
Revised: [Equation]

Original: While cold posteriors are typically used in SGLD, tempered posteriors are usually targeted by variational methods.
Revised: While cold posteriors are typically used in SGLD, tempered posteriors are usually targeted by variational methods.

Original: In particular, variational methods apply temperature scaling to the KL-divergence between the approximate posterior, Q ( θ ) and prior,
Revised: In particular, variational methods apply temperature scaling to the KL-divergence between the approximate posterior, Q ( θ ) and prior,

Original: [Equation]
Revised: [Equation]

Original: Note that the only difference between cold and tempered posteriors is whether we scale the prior, and if we have Gaussian priors over the parameters (the usual case in Bayesian neural networks), this scaling can be absorbed into the prior variance,
Revised: Note that the only difference between cold and tempered posteriors is whether we scale the prior, and if we have Gaussian priors over the parameters (the usual case in Bayesian neural networks), this scaling can be absorbed into the prior variance,

Original: [Equation]
Revised: [Equation]

Original: in which case, σ 2 cold =
Revised: in which case, σ 2 cold =

Original: σ 2 tempered /T , so the tempered posteriors we discuss are equivalent to cold posteriors with rescaled prior variances.
Revised: σ 2 tempered /T , so the tempered posteriors we discuss are equivalent to cold posteriors with rescaled prior variances.

Original:  M ETHODS : A GENERATIVE MODEL FOR CURATED DATASETS
Revised:  M ETHODS : A GENERATIVE MODEL FOR CURATED DATASETS

Original: Standard image datasets such as CIFAR-10 and ImageNet are carefully curated to include only unambiguous examples of each class.
Revised: Standard image datasets such as CIFAR-10 and ImageNet are carefully curated to include only unambiguous examples of each class.

Original: For instance, in CIFAR-10, student labellers were paid per hour (rather than per image), were instructed that “It’s worse to include one that shouldn’t be included than to exclude one”, and then Krizhevsky (2009) “personally veriﬁed every label submitted by the labellers”.
Revised: For instance, in CIFAR-10, student labellers were paid per hour (rather than per image), were instructed that “It’s worse to include one that shouldn’t be included than to exclude one”, and then Krizhevsky (2009) “personally veriﬁed every label submitted by the labellers”.

Original: For ImageNet, Deng et al.
Revised: For ImageNet, Deng et al.

Original: (2009) required the consensus of a number of Amazon
Revised: (2009) required the consensus of a number of Amazon

Original: Mechanical Turk labellers before including an image in the dataset.
Revised: Mechanical Turk labellers before including an image in the dataset.

Original: To understand the statistical patterns that might emerge in these curated datasets, we consider a highly simpliﬁed generative model of consensus-formation.
Revised: To understand the statistical patterns that might emerge in these curated datasets, we consider a highly simpliﬁed generative model of consensus-formation.

Original: In particular, we draw a random image X from some underlying distribution over images, P ( X ) , and ask S humans to assign a label, { Y s } Ss =1 (e.g. using Mechanical Turk).
Revised: In particular, we draw a random image X from some underlying distribution over images, P ( X ) , and ask S humans to assign a label, { Y s } Ss =1 (e.g. using Mechanical Turk).

Original: We force every labeller to label every image and if the image is ambiguous they are instructed to give a random label.
Revised: We force every labeller to label every image and if the image is ambiguous they are instructed to give a random label.

Original: If all the labellers agree, Y 1 = Y 2 = · · · = Y S , consensus is reached and we include the datapoint in the dataset.
Revised: If all the labellers agree, Y 1 = Y 2 = · · · = Y S , consensus is reached and we include the datapoint in the dataset.

Original: If any of the labellers disagree consensus is not reached, and we exclude the datapoint (Fig. 1), Formally, the observed random variable, Y , is taken to be the usual label if consensus was reached and
Revised: If any of the labellers disagree consensus is not reached, and we exclude the datapoint (Fig. 1), Formally, the observed random variable, Y , is taken to be the usual label if consensus was reached and

Original: None if consensus was not reached (Fig.
Revised: None if consensus was not reached (Fig.

Original: 2B),
Revised: 2B),

Original: [Equation]
Revised: [Equation]

Original: Taking the human labels, Y s , to come from the set Y , so Y s ∈ Y , the consensus label, Y , could be any of the underlying labels in Y , or
Revised: Taking the human labels, Y s , to come from the set Y , so Y s ∈ Y , the consensus label, Y , could be any of the underlying labels in Y , or

Original: None if no consensus is reached, so Y ∈ Y ∪ { None } .
Revised: None if no consensus is reached, so Y ∈ Y ∪ { None } .

Original: When consensus was reached, the likelihood is,
Revised: When consensus was reached, the likelihood is,

Original: [Equation]
Revised: [Equation]

Original: where we have assumed labellers are IID.
Revised: where we have assumed labellers are IID.

Original: This would appear to directly give an account of tempering, as we have taken the single-labeller likelihood to the power S , which is equivalent to setting λ = 1 /S .
Revised: This would appear to directly give an account of tempering, as we have taken the single-labeller likelihood to the power S , which is equivalent to setting λ = 1 /S .

Original: However, to see how the full generative model functions we need to go on to consider the case in which consensus was not reached,
Revised: However, to see how the full generative model functions we need to go on to consider the case in which consensus was not reached,

Original: [Equation]
Revised: [Equation]

Original: To understand the impact of our model of consensus formation, note that the probability of a particular class-label can be separated into two terms, a probability of consensus, and the probability of a particular class given that consensus was reached,
Revised: To understand the impact of our model of consensus formation, note that the probability of a particular class-label can be separated into two terms, a probability of consensus, and the probability of a particular class given that consensus was reached,

Original: [Equation]
Revised: [Equation]

Original: Remarkably, knowing there was consensus gives us information about the weights, even in the absence of the class label,
Revised: Remarkably, knowing there was consensus gives us information about the weights, even in the absence of the class label,

Original: [Equation]
Revised: [Equation]

Original: in essence, telling us that the output probability was close to 1 for one of the class labels, without telling us which one.
Revised: in essence, telling us that the output probability was close to 1 for one of the class labels, without telling us which one.

Original: Schematically (Fig. 3), we see that the datapoints with a consensus class-label, “cat” or “dog”, lie far from the decision boundary where the class is unambiguous, and consensus is easily reached.
Revised: Schematically (Fig. 3), we see that the datapoints with a consensus class-label, “cat” or “dog”, lie far from the decision boundary where the class is unambiguous, and consensus is easily reached.

Original: In contrast, in regions close to the decision boundary the inputs are ambiguous, which tends to produce disagreement in the labellers, leading to noconsensus.
Revised: In contrast, in regions close to the decision boundary the inputs are ambiguous, which tends to produce disagreement in the labellers, leading to noconsensus.

Original: Thus, the existence of one or more consensus points in a region implies that decision boundaries do not go through that region, giving us information about the decision boundary location, even if the label is not known.
Revised: Thus, the existence of one or more consensus points in a region implies that decision boundaries do not go through that region, giving us information about the decision boundary location, even if the label is not known.

Original: Concurrent work has shown that this likelihood can be used to explain classical semi-supervised likelihoods (Aitchison, 2020), so this term really does give information about the neural network parameters.
Revised: Concurrent work has shown that this likelihood can be used to explain classical semi-supervised likelihoods (Aitchison, 2020), so this term really does give information about the neural network parameters.

Original: Finally, the label probability, conditioned on consensus,
Revised: Finally, the label probability, conditioned on consensus,

Original: [Equation]
Revised: [Equation]

Original: simply represents a “reparameterised” softmax.
Revised: simply represents a “reparameterised” softmax.

Original: In datasets where the noconsensus inputs are known it is clear we should use the full likelihood, (Eq. 6 and 7).
Revised: In datasets where the noconsensus inputs are known it is clear we should use the full likelihood, (Eq. 6 and 7).

Original: The question is: in real-world datasets, where we only know the consensus inputs and the noconsensus inputs have been thrown away (Fig.
Revised: The question is: in real-world datasets, where we only know the consensus inputs and the noconsensus inputs have been thrown away (Fig.

Original: 2C), can we use Eq.
Revised: 2C), can we use Eq.

Original: (10), a reparameterisation of the softmax-categorical probabilities, for the known consensus points?
Revised: (10), a reparameterisation of the softmax-categorical probabilities, for the known consensus points?

Original: The answer is no because in
Revised: The answer is no because in

Original: Bayesian inference, we do not get to just pick a sensible-looking conditional probability distribution, such as Eq. 10, to use as the likelihood.
Revised: Bayesian inference, we do not get to just pick a sensible-looking conditional probability distribution, such as Eq. 10, to use as the likelihood.

Original: Instead, we need to write down the full generative model, and marginalise over unknown latents.
Revised: Instead, we need to write down the full generative model, and marginalise over unknown latents.

Original: To write down the generative model in the case of unknown noconsensus images (Fig.
Revised: To write down the generative model in the case of unknown noconsensus images (Fig.

Original: 2C), we need to take X to be an unobserved latent variable, and take Z to be an observed random variable,
Revised: 2C), we need to take X to be an unobserved latent variable, and take Z to be an observed random variable,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: which is the underlying image, X , if consensus is reached, and
Revised: which is the underlying image, X , if consensus is reached, and

Original: None otherwise.
Revised: None otherwise.

Original: As Z depends on θ (Fig.
Revised: As Z depends on θ (Fig.

Original: 2C), we cannot take the usual shortcut of using P ( Y | Z, θ ) ; we must instead use the full likelihood, P ( Y, Z | θ ) ,
Revised: 2C), we cannot take the usual shortcut of using P ( Y | Z, θ ) ; we must instead use the full likelihood, P ( Y, Z | θ ) ,

Original: [Equation]
Revised: [Equation]

Original: (12) for y not None ,
Revised: (12) for y not None ,

Original: P
Revised: P

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: = (cid:26)
Revised: = (cid:26)

Original: [Equation]
Revised: [Equation]

Original: if Y
Revised: if Y

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: y for all s ∈ { 1 , . .
Revised: y for all s ∈ { 1 , . .

Original: , S } 

Revised: , S }

Original: otherwise (13) and,
Revised: otherwise (13) and,

Original: P (cid:0) Z | X, { Y s } Ss =1 (cid:1) =
Revised: P (cid:0) Z | X, { Y s } Ss =1 (cid:1) =

Original: (cid:26) δ ( Z − X )
Revised: (cid:26) δ ( Z − X )

Original: if Y 1 = Y 2 = · · · = Y S
Revised: if Y 1 = Y 2 = · · · = Y S

Original: I Z
Revised: I Z

Original: =
Revised: =

Original: None otherwise (14) where δ is the Dirac-delta function, and where the indicator function, I Z
Revised: None otherwise (14) where δ is the Dirac-delta function, and where the indicator function, I Z

Original: =
Revised: =

Original: None is 1 if Z =
Revised: None is 1 if Z =

Original: None and 0 otherwise.
Revised: None and 0 otherwise.

Original: Substituting Eq.
Revised: Substituting Eq.

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

Original: (14) into Eq.
Revised: (14) into Eq.

Original: [Equation]
Revised: [Equation]

Original: where the proportionality arises because P ( X ) does not depend on the parameters of interest, θ .
Revised: where the proportionality arises because P ( X ) does not depend on the parameters of interest, θ .

Original: Note that this is proportional to Eq.
Revised: Note that this is proportional to Eq.

Original: (6) above, and not Eq.
Revised: (6) above, and not Eq.

Original: (10), so this does not just represent a reparameterisation of the softmax.
Revised: (10), so this does not just represent a reparameterisation of the softmax.

Original: Finally, this explanation for the cold posterior effect would suggest that we would see cold posteriors even in maximum a-posteriori (MAP) inference, as we conﬁrm in Appendix A.
Revised: Finally, this explanation for the cold posterior effect would suggest that we would see cold posteriors even in maximum a-posteriori (MAP) inference, as we conﬁrm in Appendix A.

Original: This is expected as it is “common knowledge” (but we do not know of a good reference) that while weight-decay is closely related to MAP inference with Gaussian priors, the best performing value of the weight decay coefﬁcient tend to be lower than those suggested by untempered MAP inference.
Revised: This is expected as it is “common knowledge” (but we do not know of a good reference) that while weight-decay is closely related to MAP inference with Gaussian priors, the best performing value of the weight decay coefﬁcient tend to be lower than those suggested by untempered MAP inference.

Original:  D IFFERENCES BETWEEN COLD POSTERIOR AND DATASET CURATION SETUPS
Revised:  D IFFERENCES BETWEEN COLD POSTERIOR AND DATASET CURATION SETUPS

Original: Our model of data curation provides a direct explanation for the effectiveness of cold posteriors, as
Revised: Our model of data curation provides a direct explanation for the effectiveness of cold posteriors, as

Original: Eq.
Revised: Eq.

Original: (6) takes the underlying likelihoods, P ( Y s | X, θ ) to the power S , which has exactly the same effect as 1 /λ in tempered posteriors (Eq. 2).
Revised: (6) takes the underlying likelihoods, P ( Y s | X, θ ) to the power S , which has exactly the same effect as 1 /λ in tempered posteriors (Eq. 2).

Original: However, it is important to note two differences between the cold posterior setup and the ideal setup for our generative model.
Revised: However, it is important to note two differences between the cold posterior setup and the ideal setup for our generative model.

Original: First, our generative model assumes that the noconsensus points are known, or if they are unknown, we can compute the integral,
Revised: First, our generative model assumes that the noconsensus points are known, or if they are unknown, we can compute the integral,

Original: [Equation]
Revised: [Equation]

Original: which is obtained by evaluating Eq.
Revised: which is obtained by evaluating Eq.

Original: (12) in the case where
Revised: (12) in the case where

Original: Y =
Revised: Y =

Original: None , and substituting Eq.
Revised: None , and substituting Eq.

Original: Of course, in reality it is not possible to compute this integral because we do not know the underlying
Revised: Of course, in reality it is not possible to compute this integral because we do not know the underlying

Original: P ( X ) (and usually, we do not even have samples from this distribution).
Revised: P ( X ) (and usually, we do not even have samples from this distribution).

Original: In contrast, the standard cold-posterior setup entirely ignores these terms.
Revised: In contrast, the standard cold-posterior setup entirely ignores these terms.

Original: Second, in the usual setting, the test data is also subject to the same consensus-formation process, in which case, we should use Eq.
Revised: Second, in the usual setting, the test data is also subject to the same consensus-formation process, in which case, we should use Eq.

Original: (10) for prediction.
Revised: (10) for prediction.

Original: In contrast, in the standard cold-posterior setting, we use the single-labeller distribution, P ( Y s | X, θ ) .
Revised: In contrast, in the standard cold-posterior setting, we use the single-labeller distribution, P ( Y s | X, θ ) .

Original: (Note that if the test-set was drawn “from the wild”, without dataset curation, and labelled by a single labeller, then we should use P ( Y s | X, θ ) for prediction).
Revised: (Note that if the test-set was drawn “from the wild”, without dataset curation, and labelled by a single labeller, then we should use P ( Y s | X, θ ) for prediction).

Original: Overall then, while our model of data curation offers a potential explanation of the beneﬁts of tempering, the differences in setup imply that we cannot expect λ ∗ = 1 /S , to hold exactly, where λ ∗ is the optimal temperature, and this is conﬁrmed in our results on toy data below.
Revised: Overall then, while our model of data curation offers a potential explanation of the beneﬁts of tempering, the differences in setup imply that we cannot expect λ ∗ = 1 /S , to hold exactly, where λ ∗ is the optimal temperature, and this is conﬁrmed in our results on toy data below.

Original:  R ESULTS 5. D ATA SAMPLED FROM A KNOWN G AUSSIAN PROCESS MODEL
Revised:  R ESULTS 5. D ATA SAMPLED FROM A KNOWN G AUSSIAN PROCESS MODEL

Original: In the introduction, we took it as given that if the prior is correct, the optimal approach is to use the true Bayesian posterior, avoiding either tempering or cold posteriors.
Revised: In the introduction, we took it as given that if the prior is correct, the optimal approach is to use the true Bayesian posterior, avoiding either tempering or cold posteriors.

Original: To check that this is indeed the case, we tested the performance, measured as test-log-likelihood for tempered posteriors using data generated from a known Gaussian process model.
Revised: To check that this is indeed the case, we tested the performance, measured as test-log-likelihood for tempered posteriors using data generated from a known Gaussian process model.

Original: We uniformly generated 50 input points on the 1D interval [-10, 10], and used a Gaussian process with squared exponential kernel with a standard deviation of 4 , and kernel bandwidth of 1 .
Revised: We uniformly generated 50 input points on the 1D interval [-10, 10], and used a Gaussian process with squared exponential kernel with a standard deviation of 4 , and kernel bandwidth of 1 .

Original: For inference, we used reparameterised VI, with an approximate posterior given by multiplying the prior by a single Gaussian factor for each datapoint (Ober & Aitchison, 2020).
Revised: For inference, we used reparameterised VI, with an approximate posterior given by multiplying the prior by a single Gaussian factor for each datapoint (Ober & Aitchison, 2020).

Original: In particular, we used a Gaussian process prior for the function values, u , at the training inputs (Williams & Rasmussen, 2006),
Revised: In particular, we used a Gaussian process prior for the function values, u , at the training inputs (Williams & Rasmussen, 2006),

Original: [Equation]
Revised: [Equation]

Original: 17) and we use an approximate posterior (Hensman et al., 2015; Matthews et al., 2016; Ober & Aitchison, 2020) deﬁned by multiplying the prior by a Gaussian with diagonal covariance, where we treat v and
Revised: 17) and we use an approximate posterior (Hensman et al., 2015; Matthews et al., 2016; Ober & Aitchison, 2020) deﬁned by multiplying the prior by a Gaussian with diagonal covariance, where we treat v and

Original: [Figure]
Revised: [Figure]

Original: λ −
Revised: λ −

Original: [Figure]
Revised: [Figure]

Original: −
Revised: −

Original: [Figure]
Revised: [Figure]

Original: D
Revised: D

Original: [Figure]
Revised: [Figure]

Original: 1 / λ ∗
Revised: 1 / λ ∗

Original: E
Revised: E

Original: [Figure]
Revised: [Figure]

Original: Λ − 1 as variational parameters.
Revised: Λ − 1 as variational parameters.

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Note that this captures the true posterior in the case of regression (where v is set to y and Λ − 1 is the output noise covariance; Ober & Aitchison, 2020).
Revised: Note that this captures the true posterior in the case of regression (where v is set to y and Λ − 1 is the output noise covariance; Ober & Aitchison, 2020).

Original: We then optimize the tempered ELBO (e.g.
Revised: We then optimize the tempered ELBO (e.g.

Original: Zhang et al., 2017),
Revised: Zhang et al., 2017),

Original: [Equation]
Revised: [Equation]

Original: And we used the standard GP approach for prediction at test points (Williams & Rasmussen, 2006).
Revised: And we used the standard GP approach for prediction at test points (Williams & Rasmussen, 2006).

Original: Initially, we tried using standard regression and classiﬁcation generative models, without including our model of consensus formation.
Revised: Initially, we tried using standard regression and classiﬁcation generative models, without including our model of consensus formation.

Original: Unsurprisingly, λ = 1 , corresponding to the Bayesian posterior, is optimal for GP regression (Fig. 4A) and classiﬁcation (Fig. 4B).
Revised: Unsurprisingly, λ = 1 , corresponding to the Bayesian posterior, is optimal for GP regression (Fig. 4A) and classiﬁcation (Fig. 4B).

Original: For GP regression, we use a
Revised: For GP regression, we use a

Original: Gaussian likelihood with standard deviation 1, and for classiﬁcation, we use a standard sigmoid probability with a Bernoulli likelihood.
Revised: Gaussian likelihood with standard deviation 1, and for classiﬁcation, we use a standard sigmoid probability with a Bernoulli likelihood.

Original: Next, we conﬁrmed that even under our new, more complex generative model of dataset curation λ = 1 was still optimal (Fig. 4C), if we trained and tested using the correct form for the loglikelihood.
Revised: Next, we conﬁrmed that even under our new, more complex generative model of dataset curation λ = 1 was still optimal (Fig. 4C), if we trained and tested using the correct form for the loglikelihood.

Original: In particular, we considered no-consensus inputs as known and used Eq.
Revised: In particular, we considered no-consensus inputs as known and used Eq.

Original: 7 to incorporate their likelihood.
Revised: 7 to incorporate their likelihood.

Original: However, in standard benchmarks, only the consensus inputs are known.
Revised: However, in standard benchmarks, only the consensus inputs are known.

Original: As such, next we generated curated data, trained using P ( Y s | X, θ ) and ignoring noconsensus points, but tested on the correct likelihood, including noconsensus points (Fig. 4D); the optimal temperature remains around 1 .
Revised: As such, next we generated curated data, trained using P ( Y s | X, θ ) and ignoring noconsensus points, but tested on the correct likelihood, including noconsensus points (Fig. 4D); the optimal temperature remains around 1 .

Original: Finally, we considered the standard cold/tempered posterior setup, where we use P ( Y s | X, θ ) , excluding noconsensus points, for training and testing.
Revised: Finally, we considered the standard cold/tempered posterior setup, where we use P ( Y s | X, θ ) , excluding noconsensus points, for training and testing.

Original: The optimal λ indeed fell with S (Fig. 4EF), giving a potential explanation for the cold posterior effect.
Revised: The optimal λ indeed fell with S (Fig. 4EF), giving a potential explanation for the cold posterior effect.

Original: As expected, we have
Revised: As expected, we have

Original: S ≈ 1 /λ ∗ , but the relation does not appear to hold exactly due to the mismatches discussed in Sec. 4.
Revised: S ≈ 1 /λ ∗ , but the relation does not appear to hold exactly due to the mismatches discussed in Sec. 4.

Original: Finally, we show in Appendix B that the same patterns emerge in a Bayesian neural network, using
Revised: Finally, we show in Appendix B that the same patterns emerge in a Bayesian neural network, using

Original: Langevin sampling for inference.
Revised: Langevin sampling for inference.

Original: [Figure] [Figure] f r equen cy A 10−10 −10 −10λ − 0.
Revised: 10 −10λ − 0.

Original: − 0.
Revised: − 0.

Original: − 0.
Revised: − 0.

Revised: t e s t l og - li k e li hood

Original: − 0. − 0.
Revised: A 10 −10λ 0.

Original:  t e s t l og - li k e li hood
Revised: ∆ t e s t l og - li k e li hood

Original: B 
Revised: B 10 −10λ     t e s t a cc u r a cy ( % )

Original: CIFAR-10 CIFAR-10H 10−10 −10 −10λ 
Revised: C 10 −10λ − 1. − 1. − 0.

Original:  t e s t a cc u r a cy ( % )
Revised: ∆ t e s t a cc u r a cy ( % )

Revised: D uncurated curated

Revised: 5. C URATED AND UNCURATED G ALAXY Z OO DATASETS

Revised: The most direct test of our theory is to evaluate the cold-posterior effect on real-world curated and uncurated datasets.

Revised: To this end, we used the Galazy Zoo 2 dataset (Willett et al., 2013), as the original dataset is not “curated” in our sense (the criteria for inclusion were e.g. brightness) and the dataset gives classiﬁcations from ∼ 50 labellers.

Original: C
Revised: We used a reduced label set for simplicity (see Appendix C). For the uncurated dataset, we selected 12500 points at random from the full dataset, and for the curated dataset, we selected the 12500 most conﬁdent points (deﬁned in terms of labeller agreement), such that the correct class-balance was maintained. These datasets were split at random intotraining points and 10000 test points. Note that using our generative model directly would lead to drastically different class-balance in the curated and uncurated datasets, due to different levels of certainty for different classes.

Revised: To perform probabilistic inference, we used SGLD with code adapted from (Zhang et al., 2019).

Revised: In particular, we used their standard settings of a ResNet18, momentum of 0 . 9 , and a cyclic cosine learning rate schedule.

Revised: Due to the smaller size of our training set, we used longer cycles (600 epochs rather than 50), and more cycles (8 rather than 4).

Revised: As expected, we found that performance on curated data was far better than that on uncurated data, both in terms of test log-likelihood and test accuracy (Fig.

Revised: 5AC).

Revised: In agreement with our theory, we found strong cold-posterior effects for curated data (Fig.

Revised: 5BD) especially when considering test log-likelihood (Fig.

Revised: 5B).

Revised: In contrast, for uncurated data, test log-likelihood remained roughly similar (Fig 5B) and test accuracy decreased dramatically (Fig.

Revised: 5D) as temperature fell.

Revised: It is important to remember that while the absolute difference in performance in Fig.

Revised: 5D appears small, this is in large part because performance is already so good (Fig.

Revised: 5C) that there is very little room for improvement.

Original: 5. CIFAR-10H
Revised: 5. CIFAR-10H

Original: One prediction made by our framework is that if we had access to the original underlying human labels on the full dataset, including images that were rejected because consensus was not reached, then tempering should not be necessary.
Revised: One prediction made by our framework is that if we had access to the original underlying human labels on the full dataset, including images that were rejected because consensus was not reached, then tempering should not be necessary.

Original: Obviously, all this additional information is not available for standard datasets such as CIFAR-10.
Revised: Obviously, all this additional information is not available for standard datasets such as CIFAR-10.

Original: However, we are able to get close by considering the
Revised: However, we are able to get close by considering the

Original: CIFAR-10H dataset (Peterson et al., 2019).
Revised: CIFAR-10H dataset (Peterson et al., 2019).

Original: The authors of this work asked around 50 human labellers to label the CIFAR-10 test-set.
Revised: The authors of this work asked around 50 human labellers to label the CIFAR-10 test-set.

Original: As we might expect given the careful curation that went in to creating the original CIFAR-10 dataset (Krizhevsky, 2009), for almost half of the datapoints, all ∼ 50 labellers agreed, and more than three-quarters of images had 2 or fewer disagreements (corresponding to 4%) of labellers (Fig. 5A).
Revised: As we might expect given the careful curation that went in to creating the original CIFAR-10 dataset (Krizhevsky, 2009), for almost half of the datapoints, all ∼ 50 labellers agreed, and more than three-quarters of images had 2 or fewer disagreements (corresponding to 4%) of labellers (Fig. 6A).

Original: While it is not possible to estimate S without having information about the unknown image distribution, the high level of agreement would indicate that the effective value of S is large — potentially even larger than 10 .
Revised: While it is not possible to estimate S without having information about the unknown image distribution, the high level of agreement would indicate that the effective value of S is large — potentially even larger than 10 .

Original: Next, we trained a neural network on the ∼ 50 labels provided by the CIFAR-10H dataset.
Revised: Next, we trained a neural network on the ∼ 50 labels provided by the CIFAR-10H dataset.

Original: As these labels are provided for the test-set with 10,000 images, this required us to swap the identities of the test and training sets (so that our training set consisted of 10,000 points, each with around 50 labels, and the test set consists of 50,000 points, with the single label from the original CIFAR-10 dataset).
Revised: As these labels are provided for the test-set with 10,000 images, this required us to swap the identities of the test and training sets (so that our training set consisted of 10,000 points, each with around 50 labels, and the test set consists of 50,000 points, with the single label from the original CIFAR-10 dataset).

Original: We compared against training with the standard CIFAR-10 test set, containing the same images but only with a single label.
Revised: We compared against training with the standard CIFAR-10 test set, containing the same images but only with a single label.

Original: To perform probabilistic inference, we used SGLD with code adapted from (Zhang et al., 2019).
Revised: To perform probabilistic inference, we used SGLD with code adapted from (Zhang et al., 2019).

Original: In particular, we used their standard settings of a ResNet18, momentum of 0 . 9 , and a cyclic cosine learning rate schedule.
Revised: In particular, we used their standard settings of a ResNet18, momentum of 0 . 9 , [Figure] f r equen cy A 10 −10 −10 −10λ − 0.

Revised: − 0.

Original: Due to the smaller size of our training set, we used longer cycles (150 epochs rather than 50), and more cycles (8 rather than 4).
Revised: − 0. − 0. − 0. t e s t l og - li k e li hood

Revised: B

Original: Importantly, we kept all the parameters of the learning algorithm the same for CIFAR-10H and our CIFAR-10 comparison.
Revised: CIFAR-10 CIFAR-10H 10 −10 − [Figure] 10 −10λ   t e s t a cc u r a cy ( % )

Original:  The only complication was that to keep the same effective learning rate for CIFAR-10H we need to take into account the number of labellers per datapoint (if we have 50 labels for a datapoint, the loss and the gradients of the loss are 50 times larger, resulting in step-sizes that are also 50 times larger).
Revised: C and a cyclic cosine learning rate schedule. Due to the smaller size of our training set, we used longer cycles (150 epochs rather than 50), and more cycles (8 rather than 4). Importantly, we kept all the parameters of the learning algorithm the same for CIFAR-10H and our CIFAR-10 comparison. The only complication was that to keep the same effective learning rate for CIFAR-10H we need to take into account the number of labellers per datapoint (if we have 50 labels for a datapoint, the loss and the gradients of the loss are 50 times larger, resulting in step-sizes that are also 50 times larger).

Original: As such, to keep the same step-sizes, we divided the actual learning rate by 50.
Revised: As such, to keep the same step-sizes, we divided the actual learning rate by 50.

Original: An alternative way to look at this is that we use the average log-likelihood per labeller (rather than per-image).
Revised: An alternative way to look at this is that we use the average log-likelihood per labeller (rather than per-image).

Original: Importantly, this change in learning rate, leaves the stationary distribution was unchanged, as changing the learning rate is not equivalent to tempering.
Revised: Importantly, this change in learning rate, leaves the stationary distribution was unchanged, as changing the learning rate is not equivalent to tempering.

Original: All other parameters were left at the values speciﬁed by Zhang et al.
Revised: All other parameters were left at the values speciﬁed by Zhang et al.

Original: As expected, when training on the single-label from the original CIFAR-10 testset (blue lines
Revised: As expected, when training on the single-label from the original CIFAR-10 testset (blue lines

Original: Fig.
Revised: Fig.

Original: 5BC), there are very large tempering effects.
Revised: 6BC), there are very large tempering effects.

Original: In contrast, when training on the ∼ 50 labels provided CIFAR-10H (orange), the effects of tempering are far smaller.
Revised: In contrast, when training on the ∼ 50 labels provided CIFAR-10H (orange), the effects of tempering are far smaller.

Revised: In a sense, this is not surprising — using 50 labellers in effect makes the likelihood 50 times stronger, which is very similar to applying tempering.

Revised: But this simplicity is the point: in the Bayesian setting we need to condition on all data — in our case all the labels, and once we do that, the cold posterior effect disappears.

Original: Note that while tempering-effects are dramatically reduced, they have not been eliminated entirely.
Revised: Note that while tempering-effects are dramatically reduced, they have not been eliminated entirely.

Original: This is expected in our setting, both because of the mismatch between this setup and the exact setup (Sec. 4), but also because the inference method, SGLD, becomes more accurate as the minibatch-size increases, but we can never use full-batch in practical settings due to the large size of datasets such as[Figure] t e s t l og - li k e [Figure]
Revised: This is expected in our setting, both because of the mismatch between this setup and the exact setup (Sec. 4), but also because the inference method, SGLD, becomes more accurate as the minibatch-size increases, but we can never use full-batch in practical settings due to the large size of datasets such as

Original: hood

Original: [Figure]

Original: t

Original: [Figure]

Original: og

Original: [Figure] hood [Figure] CIFAR-10 (Welling & Teh, 2011).
Revised: CIFAR-10 (Welling & Teh, 2011).

Original: In SGLD minibatch gradients are used as a proxy for the gradient for the full dataset, but these minibatch gradients contain additional noise, and there is a potential that reducing the temperature may partially compensate for this additional noise.
Revised: In SGLD minibatch gradients are used as a proxy for the gradient for the full dataset, but these minibatch gradients contain additional noise, and there is a potential that reducing the temperature may partially compensate for this additional noise.

Original: This is particularly evident if we consider (Wenzel et al., 2020, Fig. 6), which showed that cold-posterior effects can be ampliﬁed by using very small minibatch sizes — smaller minibatches imply larger variance in gradient estimates, and hence more potential for lower temperatures to compensate for that additional noise.
Revised: This is particularly evident if we consider (Wenzel et al., 2020, Fig. 6), which showed that cold-posterior effects can be ampliﬁed by using very small minibatch sizes — smaller minibatches imply larger variance in gradient estimates, and hence more potential for lower temperatures to compensate for that additional noise.

Original: That said, these effects do not appear to be signiﬁcant for CIFAR-10 (Wenzel et al., 2020,
Revised: That said, these effects do not appear to be signiﬁcant for CIFAR-10 (Wenzel et al., 2020,

Original: Fig. 5), so we are in agreement with Wenzel et al.
Revised: Fig. 5), so we are in agreement with Wenzel et al.

Original: (2020) that minibatch noise are unlikely to be the primary source of tempering effects.
Revised: (2020) that minibatch noise are unlikely to be the primary source of tempering effects.

Original: 5. CIFAR-10 WITH NOISY LABELS
Revised: 5. CIFAR-10 WITH NOISY LABELS

Original: Curated labels are almost certain to represent the true class, and this is one way to understand the statistical patterns induced by curation that are exploited by tempered posteriors.
Revised: Curated labels are almost certain to represent the true class, and this is one way to understand the statistical patterns induced by curation that are exploited by tempered posteriors.

Original: We therefore considered disrupting this property by adding noise to the labels.
Revised: We therefore considered disrupting this property by adding noise to the labels.

Original: In the standard classiﬁcation setting, this should have very little effect (except to shrink the logit outputs somewhat to give more uncertain outputs).
Revised: In the standard classiﬁcation setting, this should have very little effect (except to shrink the logit outputs somewhat to give more uncertain outputs).

Original: However, and as expected under our theory, we ﬁnd that adding noise destroys and eventually reverses the cold-posterior effect (Fig. 6).
Revised: However, and as expected under our theory, we ﬁnd that adding noise destroys and eventually reverses the cold-posterior effect (Fig. 7). This conﬁrms that cold-posteriors are exploiting

Revised: 10 −10λ −−t e s t l og - li k e li hood

Revised: A 10 −10λ − 0.

Revised: ∆ t e s t l og - li k e li hood

Original: This conﬁrms that cold-posteriors are exploiting speciﬁc properties of the labels that are destroyed by adding noise.
Revised: B 10 −10λ  t e s t a cc u r a cy ( % ) C 10 −10λ −  ∆ t e s t a cc u r a cy ( % ) D p 0. speciﬁc properties of the labels that are destroyed by adding noise.

Original: As such, cold posteriors are not likely to arise e.g. from failing to capture the prior over neural network weights (in which case adding noise to the outputs should have little effect).
Revised: As such, cold posteriors are not likely to arise e.g. from failing to capture the prior over neural network weights (in which case adding noise to the outputs should have little effect).

Original: We suspect that the small improvement in performance at the ﬁrst value of λ below 1 might be due to partial compensation for additional noise introduced by minibatch estimates of the gradient.
Revised: We suspect that the small improvement in performance at the ﬁrst value of λ below 1 might be due to partial compensation for additional noise introduced by minibatch estimates of the gradient.

Original:  R ELATED WORK
Revised:  R ELATED WORK

Original: Wenzel et al.
Revised: Wenzel et al.

Original: (2020) introduced the cold-posterior effect in the context of neural networks, and proposed and dismissed multiple potential explanations (although none like the one we propose here).
Revised: (2020) introduced the cold-posterior effect in the context of neural networks, and proposed and dismissed multiple potential explanations (although none like the one we propose here).

Original: Other past work, though not in the neural network context, argues that tempering may be important in the context of model misspeciﬁcation (Grünwald, 2012; Grünwald et al., 2017).
Revised: Other past work, though not in the neural network context, argues that tempering may be important in the context of model misspeciﬁcation (Grünwald, 2012; Grünwald et al., 2017).

Original: Critically, we believe that there may be many causes of cold-posterior like effects, including but not limited to curation, model misspeciﬁcation and artifacts from SGLD.
Revised: Critically, we believe that there may be many causes of cold-posterior like effects, including but not limited to curation, model misspeciﬁcation and artifacts from SGLD.

Original: Ultimately, the contribution of each of these factors in any given setting will depend on the exact dataset and inference method in question.
Revised: Ultimately, the contribution of each of these factors in any given setting will depend on the exact dataset and inference method in question.

Original: Importantly, this also means we do not necessarily expect there to be no tempering in uncurated data, merely that we should see less tempering in the case of uncurated data.
Revised: Importantly, this also means we do not necessarily expect there to be no tempering in uncurated data, merely that we should see less tempering in the case of uncurated data.

Original: Finally, the closest paper is concurrent work (Adlam et al., 2020) raising the possibility that BNNs overestimate aleatoric uncertainty, in part because of high-quality labels available in benchmark datasets.
Revised: Finally, the closest paper is concurrent work (Adlam et al., 2020) raising the possibility that BNNs overestimate aleatoric uncertainty, in part because of high-quality labels available in benchmark datasets.

Original: However, they concluded that while cold posteriors might help us to capture our priors, they do not correspond to an exact inference procedure.
Revised: However, they concluded that while cold posteriors might help us to capture our priors, they do not correspond to an exact inference procedure.

Original: In contrast, here we give a generative model of dataset curation in which tempered likelihoods emerge naturally even under exact inference methods.
Revised: In contrast, here we give a generative model of dataset curation in which tempered likelihoods emerge naturally even under exact inference methods.

Original:  C ONCLUSIONS
Revised:  D ISCUSSION AND CONCLUSIONS

Original: We showed that modelling the process of data-curation can explain the improved performance of tempering or cold posteriors in Bayesian neural networks.
Revised: We showed that modelling the process of data-curation can explain the improved performance of tempering or cold posteriors in Bayesian neural networks.

Original: That said, we expect that both the misspeciﬁcation of priors over neural networks weights and inexact inference makes additional contributions to  the cold-posterior effect.
Revised: While this work does provide a justiﬁcation for future practitioners to use tempering in their Bayesian neural networks, we would urge caution. Importantly, the tempering may give beneﬁts not just due to data curation, but because inference is inaccurate (e.g. compensating for additional noise added in SGLD), or may compensate for issues arising from model mispeciﬁcation (Grünwald, 2012; Grünwald et al., 2017). As such, we urge practitioners to regard tempering with caution: if a very large amount of tempering is necessary to achieve good performance, it may indicate issues with either inference or the prior, and ﬁxing these issues is of the utmost importance to obtaining accurate uncertainty estimation.

Original: We hope that our work will prompt more careful dataset  and  study of  data curation.
Revised: More importantly, we hope that our work will prompt more careful dataset design, and further study of how data curation might impact downstream analyses in machine learning.

Original: Finally, the same likelihood has much broader applicability, as it also provides an explanation for the effectiveness of common semi-supervised learning methods  (Aitchison, 2020)
Revised: Indeed, there are initial suggestions that semi-supervised learning methods are also exploiting the artiﬁcial clustering (Fig. 3) induced by data curation (Aitchison, 2020).

Original: A C OLD POSTERIORS IN MAXIMUM A - POSTERIORI INFERENCE
Revised: A C OLD POSTERIORS IN MAXIMUM A - POSTERIORI INFERENCE

Original: Our explanation for the cold posterior effect suggests that it is not just due to the stochasticity of the posterior, but should also arise in e.g. maximum a-posteriori (MAP) inference.
Revised: Our explanation for the cold posterior effect suggests that it is not just due to the stochasticity of the posterior, but should also arise in e.g. maximum a-posteriori (MAP) inference.

Original: However, MAP inference causes serious issues in networks with batchnorm.
Revised: However, MAP inference causes serious issues in networks with batchnorm.

Original: In particular, with batchnorm, the outputs become invariant to the scale of the weights, so the weights can continually decay towards zero (Van Laarhoven, 2017).
Revised: In particular, with batchnorm, the outputs become invariant to the scale of the weights, so the weights can continually decay towards zero (Van Laarhoven, 2017).

Original: Note that these considerations are not relevant when we are doing full approximate inference, as the product of the prior and likelihood is a static quantity with a well-deﬁned scale that cannot e.g. decay towards zero.
Revised: Note that these considerations are not relevant when we are doing full approximate inference, as the product of the prior and likelihood is a static quantity with a well-deﬁned scale that cannot e.g. decay towards zero.

Original: As such, we considered tempering in our usual ResNet18, but where we have deleted the batchnorm layers.
Revised: As such, we considered tempering in our usual ResNet18, but where we have deleted the batchnorm layers.

Original: We used a standard protocol for these types of network:
Revised: We used a standard protocol for these types of network:

Original: SGD with momentum of 0.9 and an initial learning rate of 0.1, followed by decays to 0.01 and 0.001 at epochs 150 and 200.
Revised: SGD with momentum of 0.9 and an initial learning rate of 0.1, followed by decays to 0.01 and 0.001 at epochs 150 and 200.

Original: We read off performance at the ﬁnal epoch (250) and used a batch size of 128.
Revised: We read off performance at the ﬁnal epoch (250) and used a batch size of 128.

Original: We use a standard prior over the weights, with variance 2 / fan-in, to keep the scale of the inputs and outputs similar.
Revised: We use a standard prior over the weights, with variance 2 / fan-in, to keep the scale of the inputs and outputs similar.

Original: Remarkably, we found extremely strong cold-posterior effects (Fig.7).
Revised: Remarkably, we found extremely strong cold-posterior effects (Fig.

Original: These are perhaps expected as it is common-knowledge (but difﬁcult to ﬁnd a reference for) that while weight-decay is closely related to MAP inference with a Gaussian prior, good settings for the weight-decay coefﬁcient are much smaller than those suggested by MAP inference with a sensible prior.
Revised: These are perhaps expected as it is common-knowledge (but difﬁcult to ﬁnd a reference for) that while weight-decay is closely related to MAP inference with a Gaussian prior, good settings for the weight-decay coefﬁcient are much smaller than those suggested by MAP inference with a sensible prior.

Original: Interestingly, these results suggest that the robustness of standard models to very low temperatures might arise due to batchnorm, and not be a fundamental property (e.g. having very large amounts of data). 
Revised: Interestingly, these results suggest that the robustness of standard models to very low temperatures might arise due to batchnorm, and not be a fundamental property (e.g. having very large amounts of data). [Figure] −

Original: [Figure]
Revised: [Figure]

Original: −
Revised: −

Original: [Figure]
Revised: [Figure]

Original: t e s t l og - li k e li hood
Revised: t e s t l og - li k e li hood

Original: A
Revised: A

Original: S = 6 S =
Revised: S = 6 S =

Original: S = 4 S =
Revised: S = 4 S =

Original: S = 2 S =
Revised: S = 2 S =

Original: S      1 / λ ∗
Revised: S      1 / λ ∗

Original: B
Revised: B

Original: B T OY B AYESIAN NEURAL NETWORK EXAMPLE
Revised: B T OY B AYESIAN NEURAL NETWORK EXAMPLE

Original: To conﬁrm the results in the Gaussian processes toy example in the main text, we did the same exercise with a toy Bayesian neural network.
Revised: To conﬁrm the results in the Gaussian processes toy example in the main text, we did the same exercise with a toy Bayesian neural network.

Original: In particular, we considered 100 training examples, drawn IID from standard Gaussians, each with 5 input dimensions and 2 class outputs.
Revised: In particular, we considered 100 training examples, drawn IID from standard Gaussians, each with 5 input dimensions and 2 class outputs.

Original: We generated the “true” or target outputs from a one hidden layer ReLU network, with 30 hidden units, drawn from the prior.
Revised: We generated the “true” or target outputs from a one hidden layer ReLU network, with 30 hidden units, drawn from the prior.

Original: We trained the same network, using Langevin dynamics (full batch, but no rejection step; initialized randomly from the prior).
Revised: We trained the same network, using Langevin dynamics (full batch, but no rejection step; initialized randomly from the prior).

Original: As in the GP example, we applied our generative model of dataset curation with different settings for S in the generative model.
Revised: As in the GP example, we applied our generative model of dataset curation with different settings for S in the generative model.

Original: For test and training, we used the standard single-labeller log-likelihood, which ignores rejected noconsensus inputs.
Revised: For test and training, we used the standard single-labeller log-likelihood, which ignores rejected noconsensus inputs.

Original: We implemented a highly parallelised Langevin sampling algorithm, which allowed us to drawdifferent datasets, and run one chain for each dataset (and each setting of λ in parallel).
Revised: We implemented a highly parallelised Langevin sampling algorithm, which allowed us to drawdifferent datasets, and run one chain for each dataset (and each setting of λ in parallel).

Original: The large number of chains allowed us to obtain very accurate results, but did mean that it was not possible to plot lines for individual datasets.
Revised: The large number of chains allowed us to obtain very accurate results, but did mean that it was not possible to plot lines for individual datasets.

Original: We used a learning rate of 0 . 01 and burn in of 4,000 steps; to compute performance, we averaged over 50 samples of neural-network parameters, which were taken from a chain that was 20,000 steps long in total.
Revised: We used a learning rate of 0 . 01 and burn in of 4,000 steps; to compute performance, we averaged over 50 samples of neural-network parameters, which were taken from a chain that was 20,000 steps long in total.

Original: In Fig.
Revised: In Fig.

Original: 8A, we plot the test-log-likelihood for different settings of λ .
Revised: 9A, we plot the test-log-likelihood for different settings of λ .

Original: There is a clear trend that as S increases, the optimal temperature falls, and at the same time, the curves become ﬂatter.
Revised: There is a clear trend that as S increases, the optimal temperature falls, and at the same time, the curves become ﬂatter.

Original: Notably, these curves become ﬂatter at the top for larger settings of S , making it difﬁcult to identify the optimal setting for λ at higher values of S .
Revised: Notably, these curves become ﬂatter at the top for larger settings of S , making it difﬁcult to identify the optimal setting for λ at higher values of S .

Original: We then plotted the optimal setting of λ against S (Fig 8B).
Revised: We then plotted the optimal setting of λ against S (Fig 9B).

Original: Initially, S = 1 is equivalent to the standard classiﬁcation setup, as consensus is always reached, and as such, we ﬁnd that the optimal temperature was 1 .
Revised: Initially, S = 1 is equivalent to the standard classiﬁcation setup, as consensus is always reached, and as such, we ﬁnd that the optimal temperature was 1 .

Original: As in Fig.
Revised: As in Fig.

Original: 4E, we found that λ ∗ = 1 /S held only approximately.
Revised: 4E, we found that λ ∗ = 1 /S held only approximately.

Original: Nonetheless, these results suggest that we might need more tempering than suggested by λ ∗ = 1 /S , which supports our proposed explanation of the cold posterior effect.
Revised: Nonetheless, these results suggest that we might need more tempering than suggested by λ ∗ = 1 /S , which supports our proposed explanation of the cold posterior effect.

Original: Finally, it should be noted that these results depend sensitively on the accuracy of the inference algorithm, so it is difﬁcult to say as yet the degree to which mismatch between λ ∗ and 1 /S arises due to inaccuracy in inference or would emerge in the intractable true-posterior. 
Revised: Finally, it should be noted that these results depend sensitively on the accuracy of the inference algorithm, so it is difﬁcult to say as yet the degree to which mismatch between λ ∗ and 1 /S arises due to inaccuracy in inference or would emerge in the intractable true-posterior. C G ALAXY Z OO 2 DETAILS The Galaxy Zoo 2 data set (Willett et al., 2013) was constructed by presenting images of galaxies to volunteers, who were asked to answer a questionnaire about the morphological features of the presented galaxy. These questions are not independent from each other but are part of a decision tree.

Revised: As such, the questions a volunteer gets asked depends on their answers to earlier questions.

Revised: For each galaxy, answers from multiple volunteers were collect.

Revised: The raw data is recorded as counts of how many times each answer was picked across all volunteers.

Revised: From this data we wanted to derive classes in such a way that the classiﬁcation performed by each volunteer was equivalent to a vote towards one of our classes.

Revised: The most straightforward approach is to consider each unique path in the decision tree to be a unique class.

Revised: Given that a volunteer ﬁlls in the questionnaire only once, their classiﬁcation is equivalent to a single vote towards a speciﬁc class.

Revised: The problem with this approach is that there are 1265 possible paths in the original decision tree.

Revised: The number of volunteers per galaxy is less than a hundred, which means that the majority of these paths would receive 0 votes.

Revised: In order to address this, we simpliﬁed the tree to one where there are only 9 possible paths.

Revised: The simpliﬁcations was performed by manually pruning branches of the decision tree.

Revised: This way each path in the original tree could be mapped to a path in the simpliﬁed one, while maintaining speciﬁc aspects of its meaning.

Revised: The simpliﬁed tree can be seen in (modiﬁed_graph.png).

Revised: The following pruning operations were performed in order to create this simpliﬁed tree.

Revised: Here we refer to each question using its task number, as given in the original paper.

Revised: • Task 02, Answer ’no’: leads directly to Task 05, skipping Task 03, Task 04, Task 10 and Task• Task 05, Answers ’dominant’ and ’obvious’: merged together becoming a combined category labelled ’obvious’ • Task 05, all answers: lead directly to the end of the graph, becoming exit points • Task 07, all answers: lead directly to the end of the graph, becoming exit points • Task 09, Answers ’rounded’ and ’boxy’: merged together becoming a combined category labelled ’with bulge’ • Task 09, all answers: lead directly to the end of the graph, becoming exit points • As a consequence of the above pathing changes Task 03, Task 04, Task 06, Task 08, Taskand Task 11 are no longer reachable and thus removed from the graph

Revised: If we are to take each of the 9 paths in the simpliﬁed tree to form a class, then the resulting 9 classes would have the following meanings:

Revised: • smooth galaxy, completely round shape • smooth galaxy, in-between round and cigar shaped • smooth galaxy, cigar shaped • edge-on disk galaxy, with central bulge • edge-on disk galaxy, no central bulge • face-on disk galaxy, obvious central bulge • face-on disk galaxy, just noticeable central bulge • face-on disk galaxy, no central bulge • star or artifact

Revised: Using the simpliﬁed decision tree, we can decompose the data into votes for each path.

Revised: The Galaxy

Revised: Zoo data was recorded as counts for each speciﬁc answer, thus losing the information about which route a speciﬁc volunteer took in the tree.

Revised: However, with the help of network ﬂow algorithms it is possible to translate the data into counts of how many volunteers took a speciﬁc path.

Revised: This way, by taking each path to be a class and the number of volunteers that followed a path to be the number of votes toward a class, we have a data set where each data point has received multiple classiﬁcation over a ﬁxed range of classes.

Revised: For each galaxy in the Galaxy Zoo 2 data set we also have the image that was presented to the volunteers.

Revised: As such, we can use our derived classes in an image classiﬁcation task.

Revised: In order to do this, we performed certain pre-processing.

Revised: The original images have a resolution of 424x424 with the galaxy being at the center of the image.

Revised: Thus, we performed a center crop of size 212x212 and then resized the images down to 32x32.

Revised: For data augmentation, we used full 360 degree continuous rotations.