Generative Posterior Networks for Approximately Bayesian Epistemic Uncertainty Estimation
Authors: Melrose Roderick
Felix Berkenkamp
Fatemeh Sheikholeslami
J Zico Kolter
Venue: ICLR 2022 Submitted (2022)
Keywords: Uncertainty
Bayesian
Neural Networks
Generative Models

Abstract: Ensembles of neural networks are often used to estimate epistemic uncertainty in high-dimensional problems because of their scalability and ease of use. These methods, however, are expensive to sample from as each sample requires a new neural network to be trained from scratch. We propose a new method, Generative Posterior Networks (GPNs), a generative model that, given a prior distribution over functions, approximates the posterior distribution directly by regularizing the network towards samples from the prior. This allows our method to quickly sample from the posterior and construct confidence bounds. We prove theoretically that our method indeed approximates the Bayesian posterior and show empirically that it improves epistemic uncertainty estimation over competing methods.

Under review as a conference paper at ICLR 2022
GENERATIVE POSTERIOR NETWORKS FOR APPROXI- MATELY BAYESIAN EPISTEMIC UNCERTAINTY ESTI- MATION
Anonymous authors Paper under double-blind review
ABSTRACT
Ensembles of neural networks are often used to estimate epistemic uncertainty high-dimensional problems because of their scalability and ease of use. These methods, however, are expensive to sample from as each sample requires a new neural network be trained from scratch. We propose new method, Generative Posterior Networks (GPNs), a generative model that, given a prior distribution over functions, approximates the posterior distribution directly by regularizing the network towards samples from the prior. This allows our method to quickly sam- ple from the posterior and construct confidence bounds. We prove theoretically that our method indeed approximates the Bayesian posterior and show empirically that improves epistemic uncertainty estimation over competing methods.
INTRODUCTION
Deep learning systems are becoming ubiquitous in many industries, specifically in applications with large datasets and where prediction mistakes are inexpensive. However, for many other industries, such as autonomous manufacturing, where datasets are smaller and mistakes can be catastrophic, the adoption of deep learning techniques has been slow. One of the critical challenges to applying deep learning models in these safety-critical domains is the lack of uncertainty estimation. If model makes a prediction in safety critical environment, we would like that model be highly confident that prediction is correct. Uncertainty prediction also provide significant benefit in data-poor domains by guiding exploration towards areas of uncertainty. Most deep learning models, however, not able to estimate epistemic uncertainty uncertainty deriving from lack of data samples. In this work, we introduce a new method for modeling epistemic uncertainty in neural networks.
We formulate this problem Bayesian Inference problem: given prior distribution over functions and a set of training data, we want to model the posterior function distribution. There are many existing approaches constructing such posterior, such Gaussian Processes (GPs) Variational Inference (VI) methods, but these methods can require significant engineering to work in practice and can be expensive in nigh-dimensional settings. For this reason, many practitioners turn using
(a) 2D embedding samples
(b) Corresponding posterior samples
Figure 1: Samples from Generative Posterior Network using 2D embedding trained on simple sine-function. On the left are samples from the embedding with corresponding posterior samples on the right. Black &3 represent observed data points.

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ensembles of neural networks to approximate epistemic uncertainty for their ease of use in high- dimensional settings. Pearce et al. (2020) illustrated that, given the correct regularization, these ensemble members approximate samples from the posterior distribution of functions. However, using this technique, every new sample of the posterior requires training a new neural network from seratch, which can be prohibitively expensive for large problems.
To address this challenge, we introduce Generative Posterior Networks (GPNs), a generative neural network model that directly approximates the posterior distribution by regularizing the output of the network towards samples from the prior distribution. By learning a low-dimensional latent representation of the posterior, our method can quickly sample from the posterior and construct confidence intervals. We prove that our method approximates the Bayesian posterior over functions and show empirically that our method can improves uncertainty estimation over competing methods.
2 RELATED WORK
There are two categories of uncertainty in the context of probabilistic modeling: aleatoric uncertainty and epistemic uncertainty. Aleatoric uncertainty refers uncertainty inherent in system, even when the true parameters of the system are known, while epistemic uncertainty refers to modeling uncertainty that can be reduced through collecting more data (Der Kiureghian & Ditlevsen) 2009)] Gal 2016). Deep learning models have shown impressive performance in quantifying alcatoric un- certainty when given enough data. On the other hand, despite many methods seeking to estimate epistemic uncertainty due its well-motivated applications in Bayesian Optimization (Snoek et al.) 2012 Springenberg et al. 2016), Reinforcement Learning (Curi" et al. 2020). and adversarial ro- bustness (Stutz et al. 2020). quantifying it remains a challenging problem for deep learning models.
Gaussian Processes (GPs) remain one of the most effective methods for epistemic uncertainty es- timation in low-dimensional, low-data settings. While scalability can be improved with interpola- tion methods (Hensman et al., 2013) Titsias 2009 Wilson & Nickisch| (2015) and kernel learning methods (Wilson et al. (0166). GPs tend perform worse than neural network-based methods on high-dimensional problems.
Another way to formulate epistemic uncertainty estimation as Bayesian Inference problem. Specifically, given prior distribution over functions and some labeled data, the goal is to con- struct posterior distribution in the Bayesian sense. Because this posterior distribution is usually intractable to compute exactly, approximate sampling methods are used instead. Markov-Chain Monte Carlo (MCMC) (Tanner & Wong, 1987 and Variational Inference (VI) (Blei et al.. 2017) are some of the most common methods for posterior sampling. These methods, however, tend to struggle in high dimensional problems. The computational complexity of MCMC scales poorly with the dimension of the data and VI methods tend to require restrictive function families with strong independence assumptions. Bayesian Neural Networks (BNNs) (Blundell et al. 2013) are another way to perform Bayesian Inference neural networks by modeling each parameter in the network with an independent Gaussian distribution. While BNNs perform well in low dimensional settings, their performance tends to degrade in high-dimensional settings that require large networks.
Gal & Ghahramani (2016) proved that dropout can also be used as a form of Bayesian Inference. Specifically, if a network is trained using dropout and L2 regularization, posterior samples can be collected by applying dropout at test-time. This method has surprisingly good out-of-distribution detection and can be applied to any network. However, it can be hard tune the dropout parameter to have good uncertainty prediction without significantly degrading performance.
Neural network ensembling is another commonly used epistemic uncertainty estimation technique, which is closely related to our proposed method in this work. This technique involves training a number of completely separate neural networks on the same data to form an ensemble. Because of the stochasticity of initialization and the training process, each member of the ensemble should be a slightly different function. Specifically, the ensemble members should produce similar out- puts in data-dense regions of the state-space and dissimilar outputs in data-sparse regions. Pearre et al. (2020) proved that, by using the correct regularization, each ensemble member becomes an approximate sample from the function posterior. Ensembling methods particularly attractive to practitioners because they require very little fine-tuning and work surprisingly well in practice. However, these methods have one main drawback, namely that each new sample from the posterior
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requires training network from scratch. Our method, on the other hand, seeks to construct generative posterior model using similar regularization techniques to Pearce et al. (2020). allowing for quick sampling from the posterior.
3 PROBLEM STATEMENT AND BACKGROUND
We will focus on the problem of Bayesian Inference on the parameters of 9 of some function f(: 0). We assume have some known prior over parameters Pprioe( (0) N (Mprioor Eptorr well as
access to noisy dataset (Nobs Yohs) {z Mohs)... ...., (a Vohs }. where Vons = f 0) +
c c (O, o.) for some unknown parameter 0. We can then define the Lata-likelihood as
Pike (Yobs 0, Xobs) = II [I [ o),
(1)
The goal is to approximate samples from the posterior,
pusa( (f) Yobs, Xuts) O Pike (Y 00 Xobs prior (nioo(O)) (0).
For ease of notation, we will drop the Xobs from the likelihood.
For our analysis. it is convenient if the posterior distribution is Gaussian. To show this, will use the fact that the product of two Gaussian PDFs is proportional to a Gaussian PDF. We make a distinction here between the data-likelihoodd likelihood, a function of the parameters 0, defined as follows:
function of the observed data yots, and the parameter-
P Param like (0; Yohs) = (ykee(yan ().
The garameter-likllhoodd PDF is equivalent to the Kata-likelihood PDF for all values of Yobs and 0, but is notably a function of the parameters O. For a more detailed explanation of the distinction between the parameter and data-likelihoods, see Definition from Pearce et al. (2020).
When the function f is linear, it can be shown that the parameter-likklihood is also Gaussian, P Ppraan-liks like (0; yobs) o N (Hiiee )
While this is not true in general. Pearce et al. (2020) assume that the parameter likelihood is Gaussian so that the posterior also becomes Gaussian,
Ppou (0 Yots) x P param-like (0: Yobs) Pptoor (0) x N ((iike Emse Na Eptior = N Σpos)
(5)
where the posterior covariance and mean are given by
Epost
like
) prior
Σ
Mpost Epsst D
like
Mike
Epost Uprice.
(6)
Note, however, that after the reformulation of the problem in Section we longer require this assumption for our theoretical analysis.
3.1 RANDOMIZED MAP SAMPLING (RMS)
Since the values plike and Diie are intractible compute in practice, the challenge of computing the posterior distribution remains. However, Pearce el al. (2020) showed how we can use a method called Randomized MAP sampling (RMS) approximately sample from the posterior distribution. With RMS we first assume access to a mechanism to compute the maximum a posteriori (MAP) solution for our posterior. Then, by randomly shifting the prior distribution we use our MAP solver, we can estimate samples from the posterior.
Specifically, RMS samples "anchor points" 0amc from some distribution Panc N (Mane Eunc). ). Each anchor point defines a shifted prior distribution Pa (0me) = (O Eprior. Now we define the MAP function which finds the MAP solution using this shifted prior:
MA = arg max Pikke (yotss
0) Panc (0).
(7)
By sampling different anchor points, we can then construct probability distribution over MAP solutions (AP (0) which can shown to Gaussian. Using moment matching, Pearce al. (2200) proved, in Theorem that if we choose Fanc = pprior and Ennc = Eprior + Eprice D Eprirr then P(OMAP(Om..) posterior, Opost can collected by sampling Oune ~ (M Eane) and optimizing the following:
(p Yoks). Using the approximation Σane Epriors sample from
0 ppsst = arg max log Paraam-like (0; Yobs) + log (anc(0).
(8)
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4 GENERATIVE POSTERIOR NETWORKS
While RMS can be used to construct samples from the posterior, every new sample requires solving a new optimization problem. Instead, we propose learning the MAP function itself.
Let be a neural network parameterized by some vector, é, that takes as input a sample from the anchor distribution, Oune Panc (Onnc), and outputs (MAP)Onnc) If we assume that g has enough expressive power to represent the MAP function, then g( Φ) = @MAP(0anc) if:
arg max P.. log (ikeeyoo (aacc Φ)) + log ancii 6)).
In practice, however, this a very hard optimization problem for two main reasons. (1) Optimizing in parameter space tends to be challenging for gradient-descent optimization. (2) The space of all parameters Oane is not only very large, but there are also many parameters Onc that map to the same outputs for all inputs. Moreover, we would expect the posterior parameters to be highly correlated and, therefore, need less expressive power to be represented.
To address these two problems, we will first change the problem definition slightly to find the poste- next section that, by changing our loss function, we can still use RMS to approximate the posterior output space. Next, we show practically how we construct a low-dimensional representation of the anchor distribution.
rior in output space as opposed to parameter space. We prove in the
4.1 THEORETICAL RESULTS
As mentioned above, optimizing a generative model to output parameters can be challenging. In- stead, we reformulate the Bayesian Inference problem slightly and consider the posterior in output space. Concretely, consider some discretization of the input space, Xsample {>samplee ... x sample }-
Let Y. [ sampee 0) be a set of sample points and Y .F....., "y..,,Yyr) be transformation of the random variable 0 where each element is the output of the function f parameterized by evaluated at points sample- The prior and likelihood for this transformed random variable Y can expressed in terms of the prior and likelihood of A:
I 1 =f oyr b Preeec (0)d0,
p(
(10)
Plyons Y)) 1 oyv Pinke (yan priar (0)d0.
(11)
Note that samples from the prior of Y can be easily obtained by sampling 0 ~ Pise and com- puting Y, (rrsmmpo θ) for all i. The goal in this reformulated problem is to find the posterior P(Ý |oou) x P(y PÝ)(.)
As before, we estimate the posterior using RMS and define a new MAP function in the output space:
9MAP anc) = arg max P(yots Y) Panc anc)
(12)
for some anchor distribution Y anc ~ Panc ann) = N (( Enne). We now need to show that, with the correct choice of Mane and Eunc, P(YMAP( (Yane)) = P( Yous).
To simplify the analysis, we assume that the output prior is normally distributed p(Ý) N(wy, Ex). Since the parameters are normally distributed, this approximation becomes exact when either f is linear the final layer is infinitely wide.
Next, we need to show that the likelihood function P(y Y) is roughly normally distributed with respect to the vector Y. Recall that each observed data point, V'nss is equal to a noisy output of for some parameter 0, Vites fI i 0) + In the case where Xobs C Xsample Vons is evaluated on the same points as Y, so the likelihood of P same reasoning, if we choose Xsample to contain all the points in the domain X, Pyy () becomes equivalent to P(y Y).
obs. 0) becomes equivalent to Pyyotn Y). By the
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Lemma 1. Let f be some neural network parameterized by 9. Let Nobs correspond to vectors of observed labeled data. Let Ssample be some discretization over the input space of M points such that for any I E X there exists an x' E Xuample where || - ll2 S M for some constant l. Define Y == (f( 0) '/ Xsample be vector of evaluations of f all points in Xsample for given parameter vector 0.
Then, if we take the limit as + oo, P(yobs Y)- = -P(Yob: (
Proof. See Appendix A.I
turns out that, since the observation noise is additive, the likelihood in parameter space P(yots 0) also Gaussian as a function of the output vector Y. Thus, if we take |xamplel - oc and apply Lemma 1 we get that the output likelihood P(yobs Y is Gaussian as function of Y.
Now, since we have that both the prior and likelihood of Ý can be expressed as Gaussian functions of Y, we also have that the posterior is Gaussian. As before, we just need to show that the distribution of MAP solutions P(YMAP(Y am))) ) is also Gaussian, then apply moment matching to find the right = P(Ý yobs).
values of Mane and Enne such that P P(YMAP(Y
Theorem 1. Let I be some neural network parameterized by 0. Let xsample be some vector of inputs sampled uniformly from XM, Define Ý == ()(1;)).... (x 0) as transformation of the random variable 0. Assume the prior of Y follows multivariate normal distribution, P(Y) N N(pg Ex)
We assume access to some function which takes as imput the center of a shifted distribution and outputs the MAP estimate of Y, YMAP (Y anc).
If we choose the distribution over & anc to be P( anc) = -N (Honer Eane), where Mane po and Eune Eg + E Eg. then limos+00 P(YMAP(E anc)) = P(Y yobs).
Proof. See Appendix A.2
4.2 PRACTICAL IMPLEMENTATION
In practice, as was done Pearce et al, 2020 we will instead use the approximation Eane R Σy to make it tractable to sample from the anchor distribution. Pearce et al. (2020) argue that this approximation, in general, causes RMS to over-estimate the posterior variance.
With this assumption, the anchor distribution Pane (Yaan)) anc) becomes approximately equal to the prior distribution P(Ý). Recall that, to sample from the output prior, we simply sample a set of param- eters 0 Pprior(( (0) and evaluate the function f all evaluation points Xample- Now, if we want to learn the MAP function in the output space, we can construct a neural network g. parameterized by some vector o, that takes a sample from the output anchor distribution, Ya NN [Mann: Eanc), ) along with the evaluation points Xsample: and outputs (6 ann)) If we assume that g has enough expressive power to represent the MAP function, then gi Yanc Φ) = (MAP((Y.nc)) if:
Φ = arg max Eg.- Ppen log Po(yons l g[Xsamplee Omnci x)) + log Panc Emn(g((e) (g(xampple Oanci ()) (13)
This can be re-written as:
Φ arg min Eg Ppon( Wiies
g(attsss Ounc: 6)11 + oZ
(14)
2 where 8, sample amci ; ②)
f(x (aapppe Ounc). See Appendix A.3 for step-by- derivation.
1 of 8 Σ 5 as |xsmplle| - oo. In practice, of course, we cannot compute the regularization term, N Instead, we approximate this term using a finite number of sample points from the input space X. Because of this, for high-dimensional problems, the off-diagonal terms of Eg near zero almost
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MCMC- Ground Truth
CP
Bayasian Drapout
Ensemble 10
GPN (Durs)
-55
-1.0
-15-
Figure 2: Predicted posterior distributions of different methods using the same observed data. Em- pirical distributions are constructed using 100 samples for all methods except for the Ensemble-10 method, which only over 10 ensemble members.
always. However, we found that, in practice, assuming that the samples of Y are independent did not significantly degrade performance. Using this approximation, our optimization problem becomes:
N
arg min Eo... Pare
(ma a.00. comp (mil(
Ve g amce +)
(15)
where 5 = g sumple amnc ②)
f sample amnc) and 3 is hyper- srper-parammter.
In order for to have enough expressive power to represent the full MAP function, it may require the size of the parameters Φ be much larger than O. However, after applying the low-dimensional approximation of Oanc described below, in our experiments, we only used roughly double the number of parameters compared each member of the ensemble method. And since the ensemble we used has 10 members, the total number of parameters needed for our method is significantly smaller.
4.3 NOW-DIMENSIONAL EMBEDDING
As mentioned above, optimizing the generative function g over the the space of all anchor parameters amc did not yield great results because the space was too large. Instead, we sought to learn a low- dimensional representation of the anchor parameters. There expect to be able to decrease the representational power of the anchor distribution and maintain the same of fidelity in the posterior estimation. (1) Because of the nature of neural networks, there are many settings of parameters 0 that would result in the same output vector Y. Because we are focusing on the posterior of Y, we need less representational power model Y. (2) And, maybe more importantly, because the posterior parameters are highly correlated, we would expect to need less expressive power to represent the posterior distribution than the prior distribution.
two key reasons why would
Thus, we would like to construct simpler estimate of the prior space using a low-dimensional embedding vector, z. That is, we want our generative model to take as input the embedding vector 2 (0, 7), instead of 0anc- in order to estimate the MAP function. To do this, we need to learn a mapping from anchor parameters Oanc to embedding vectors z. For our experiments, we used 1-1 embedding scheme where sample k parameters from the true prior 0,,... Ox Pprioo(O) and & independent samples from our embedding Z1...., Zt- We then jointly optimize over & and Zi,..., , 2% as follows:
arg min E EjE(I....A Σ Vl g( 2j
2)1 318112 L (z) ..... zn), (16)
where 8y = g( sample+ Zj + E: Φ) fz smmppe 02), E~ (0(,,) is a noise injection vector that improves the smoothness of interpolations between embedding vectors, and Lreg (. Zạ.) is regularizer to keep Z1... 2ş roughly normally distributed, which allows us to easily sample from the embedding space.. For our experiments distribution. Lreg (z)..... 2ς) = DxI. (N,(2, sz). (O, 1)). Figure] illustrates how we can sample from this embedding space construct posterior functions.
the KL divergence between and the normal

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Nermaleed RC Pertomance
Out Dist. Prediction ROC Curve
Out of Dist. Prediction ROC Curve
Posthe
Deep
True
Rayezian Dropout
Beyastan Dropout
Thsemble
Enwember
633
(Oun)
GPW (Oum)
Po Pos live faze
talue Positive Rute
Figure 3: Best normalized sample value for each method throughout training averaged across 5 BO problems and 5 tri- als for each problem.
(a) MNIST
(b) CIFAR-10
Figure ROC curves for out of distribution prediction based sample variance from 100 samples.
4.4 CLASSIFICATION
Our method uses the assumption that the prior over Y is approximately normally distributed. While this approximation is exact, assuming the prior parameters are normally distributed, only when either the network is linear the final layer infinitely wide and does not have activation function, for most deep networks, this is good approximation. However, if we add a soft-max to the final output, is usually done for classification tasks, this assumption is violated. To get around this, for classification tasks, we add the anchor loss to the pre-softmax outputs.
5 EXPERIMENTS
The goal of our experiments is to illustrate the ability of our method to accurately model epistemic uncertainty for out of distribution data while retaining high performance in distribution data. We will test our method on three problems: (1) a small-scale regression task, (2) Bayesian Optimization (BO), and (3) Classification with out of distribution examples.
We compare our method against competing Bayesian methods: Gaussian Processes (GPs), Bayesian Dropout (Gal & Ghahramani] 2016), and anchor-regularized neural-network ensembles (Pearce et al. 2020) with 10 ensemble members. For our BO experiments we use exact GP with static kernel since the problems are sufficiently low-dimensional, but for our classification ex- periments we need to use approximate GPs. Specifically, we use a grid-interpolated GP with Deep Kernel Learning (Wilson et al., 2016) implemented with the
library (Gardner al., 2018). ensembles, since every member requires a unique set of parameters (and anchor points) the number ensemble members was limited by the memory capacity of our GPUs.
For the machor-regularizee
5.1 SMALL SCALE REGRESSION
Our first experiment a small I-dimensional regression problem where we can easily compute the ground truth. We provide each method with the same 6 observations. For this experiment, we com- pute 100 samples from the approximate posterior for each method (except for the Ensemble where only use 10 samples). For ground truth, use the Metropolis-Hastings MCMC algorithm. Figure 2 shows the results.
5.2 BAYESIAN OPTIMIZATION
The goal of Bayesian Optimization is to find the maximum of function in few samples as possible. This commonly done by first constructing a posterior over possible functions given the samples have collected so far, then, through various methods, choosing sample points that might maximize the true function. By using different methods for predicting the posterior in BO, we can evaluate how well these methods capture epistemic uncertainty.

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Deep GP
Bayesian Dropout
Ensemmle 10
(Ours)
In Distribution
Out of Distribution
Detrribtion
Distibbttin
3
0.0 0.2 2-4
0.6 0.8 1.0 9.0 02 0.6.6 0.6.66 9.8 1.0 0.0 0.4 0.6 0.8.0
MNIST
Deep GP
Beyesian Dropout
Ensemble 10
GPN (Ours)
track
Mgp
603
cet
aatomobile airplanc
0.0 0.4 0.0 08 1.0 0.0.0.2..00 0.4 0.6 0.0
0.10
(b) CIFAR-10
Figure 5: Boxplots of 100 posterior samples from every method for test images, one from class that the model was trained on ("In Distribution") and one from class that the model wasn't 't trained on ("Out Of Distribution").
Table I: Classification performance on both the MNIST and CIFAR-10 datasets for In Distribution and Out of Distribution examples. Cl-Any refers to the proportion data points for which at least sample (out of 100) contains the correct label and CI-All refers the proportion of data points for which every sample predicts the correct label.
Dataset
Deep GP
Dropout
Ensemble 10
GPN (Ours)
Accuracy
99.7%
99.2%
99.0%
99.5%
In Dist.
CI-All
98.9%
89.8%
0.0%
97.9%
MNIST
Out of Dist.
CI-Any
0.0%
9.4%
0.0%
79.1%
Out of Dist. ROC AUC
0.94
0.97
0.88
0.99
Accuracy
88.7%
82.3%
87.3%
88.5%
In Dist.
CI-AIl
87.0%
68.5%
0.0%
87.2%
CIFAR-10 10
Out of Dist.
Cl-Any
0.0%
0.0%
0.0%
67.6%
Out of Dist. ROC AUC
0.71
0.64
0.60
0.84
For all methods we use Thompson Sampling and implemented the experiments using the BOTorch library by Balandat et al. (2020). We tested each method five standard benchmark BO functions: the Branin, Hartmann-6D. Shekel, Michalewiez- and Powell functions. We normalize the out- put of each function between minimum value (found by random sampling) and the optimal value. To make sure that each method had a large enough sample variance, we chose perreeppraeeeers such that 100 prior samples from each method have range of roughly [-1, 1]. To choose the best regularization syyper-paramete we ran each method with 5 different regularization parameters, then chose the hyper-parameters with the highest average performance across all BO problems.
Figure 3 shows the best normalized function value sampled so far during optimization averaged across the BO problems each ran for 5 trials. Our method significantly out-performs Random sampling and even slightly outperforms GPs, one of the best BO techniques for low-dimensional problems. Even though our method trails slightly behind Bayesian Dropout, these results still illus- trate the uncertainty estimation power of our method, even on cow-dimensional problems.
5.3 CLASSIFICATION
We also look at the problem of image classification. Along with our method having high accuracy, also interested in how well our method (and the baseline methods) is able to capture epistemic uncertainty. To measure this we split the dataset into two sets based on class labels: the distri- bution dataset and the of distribution dataset. During training, only provide the labels for
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the in-distribution dataset. During testing, along with recording the accuracy on the in distribution dataset, we also sample our models on the out of distribution dataset. For such out of distribution examples, we expect the models to show high epistemic uncertainty and, thus, the samples for these out of distribution examples should have high variance.
Note that we still provide the full set of unlabeled training data to agent. This obviously favors our method over the baseline methods as our method is the only method that is able to take advantage of this unlabeled data. However, from a practical perspective this is a reasonable setup many real- world datasets have far more unlabeled data than labeled data.
One desired property of an epistemic uncertainty estimator is in constructing confidence sets. For the classification setting, the confidence interval (CI) we consider is the set of possible class predictions. We would expect that this confidence interval contains the true class with high probability. We also expect that, on in-distribution examples, the confidence interval only contains the true label with high probability. To construct these confidence intervals, we take 100 samples from each model on every data point in the test data. We define the metries Cl-Any as the proportion of data points for which at least one sample contains the correct label and CI-All as the proportion of data points for which every sample predicts the correct label.
Another desired property of an epistemic uncertainty estimator predicting when a data point outside the training distribution and, thus, any classification prediction made such data point will likely be incorrect. For out of distribution data points, we expect high sample variance from the posterior model and low sample variance from in distribution data points. Thus, variance be used as a metric to predict which test data points are outside the training distribution. To quantify each method's ability perform out of distribution detection using sample variance, we construct ROC curves for each method and measure the area under the curve (AUC).
We tested all methods on both the MNIST dataset and the CIFAR-10 dataset. For MNIST, the in distribtuion classes were [0, 1, 2, 3, 4, 5] and the out of distribution classes were [6, 7, 8,9.9. For CIFAR- 10, the in distribtuion classes were ["airplane", "automobile", "bird", "deer", "dog". "ship"] and the out-of-distribution classes were ["cat", "frog", "horse", "truck"]. To make sure that each method had a large enough sample variance, we chose hyper- appem-parmmetess such that prior distributions had near 100% CI-AII. To choose the best regularization hyper-paree syper-parameter, as in the BO experiments, we ran each method with different regularization parameters, then chose the model with both a high validation accuracy and CI-AII.
For all other methods, we take 100 samples for measuring CI-Any, CI-AIl, and sample variance.
Table D summarizes the performance of each model using the metrics of test accuracy, CI-Any, CI-AII, and the ROC AUC for out of distribution detection. Full ROC curves for each method are plotted in Figure 4 While all methods are able to achieve high prediction accuracy and high CI-All on in distribution examples, our method is the only one to achieve high Cl-Any on out of distribution examples, illustrating our method's ' ability to construct valid confidence intervals. Additionally, our method significantly out performs all other methods on out of distribution detection.
Figure 5 shows two informative test images, one from the in distribution dataset and one from the out of distribution dataset, for both MNIST and CIFAR- 10, along with box plots of sampled PMFs from each of the learned models. While all methods predict the correct label with high precision the in distribution example, ours has a uniquely wide prediction distribution on the out of distribution example, illustrating high predicted epistemic uncertainty.
CONCLUSION
In this paper we introduce Generative Posterior Networks (GPNs), method to learn generative model of the Bayesian posterior distribution by regularizing the outputs of the network towards samples of the prior. We prove that under mild assumptions, GPNs approximate samples from the true posterior. We then show empirically that our method is not only competitive with the best uncertainty predictions techniques on small Bayesian Optimization problems, significantly outperforms these competing methods on high-dimensional classification tasks.

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Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International Conference on Machine Learning, pp. 1613-1622. PMLR, 2015.
Sebastian Curi*, Felix Berkenkamp", and Andreas Krause. Efficient model- basee reinforcement learning through optimistic policy search and planning. In Neural Information Processing Systems (NeurIPS), 2020. URL https: //a aaaia.iii....... www./org/aaooo abs 2006. 08684
Armen Der Kiureghian and Ove Ditlevsen. Aleatory epistemic? does it matter? Structural safety, 31 2009.
Yarin Gal. Uncertainty in deep learning. PhD thesis, University of Cambridge, 2016.
Yarin Gal and Zoubin Ghahramani. Dropout as bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pp. 1050-1059.
PMLR, 2016.
Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian Q Weinberger, and Andrew Gordon Wilson Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration. In Advances in Neural Information Processing Systems, 2018.
James Hensman, Nicolo Fusi, and Neil D Lawrence. Gaussian processes for big data, arXiv preprint arXiv: 1309.6835, 2013.
Tim Pearce, Felix Leibfried, and Alexandra Brintrup. Uncertainty in neural networks: Approxi- mately bayesian ensembling. In International conference on artificial intelligence and statisties, pp. 234-244 PMLR, 2020.
Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. Advances in neural information processing systems, 25, 2012.
Jost Tobias Springenberg, Aaron Klein, Stefan Falkner, and Frank Hutter. Bayesian optimization with robust bayesian neural networks. Advances in neural information processing systems, 29:
4134-4142, 2016.
David Stutz, Matthias Hein, and Bernt Schiele. Confidence-calibratee adversarial training: Gen- eralizing to unseen attacks. In International Conference on Machine Learning, pp. 9155-9166. PMLR, 2020.
Martin A Tanner and Wing Hung Wong. The calculation of posterior distributions by data augmen- tation. Journal of the American statistical Association, 82(398):528-540 1987.
Michalis Titsias. Variational learning of inducing variables in sparse gaussian processes. In Artificial intelligence and statistics, pp. 567-574. PMLR, 2009.
Andrew Wilson and Hannes Nickisch. Kernel interpolation for scalable structured gaussian pro- cesses (kiss-gp). In International Conference Machine Learning, pp. 1775-1784. PMLR, 2015.
Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric Xing. Deep kernel learning. In Artificial intelligence and statistics, pp. 370-378. PMLR, 2016.
10

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Under review as a conference paper at ICLR 2022
A PROOFS
A.1 PROOF OF LEMMA I
Proof. is standard result that linear feed-forward neural networks have bounded Lipschitz con- stants, We'll denote L(O) as the Lipschitz constant of f for a given set of parameters 0.
Recall that this likelihood defined as follows:
oby Y S.. 1y anm=f 03vi
II Nu'in (a 0), oc) Prrir (0)
Let Xsumple' C Xsample be the subset of points in our discretization, Xsample> that are closest to the observed points, Xotsi in other words, for each element 2'ts Xobs, there exists an element E Xsmplee where:
x =
min || - x|12
x EEXinok
We denote the elements of Xsamplo as [ai...... wl Let Y с Y be the corresponding elements in Y: in other words, Y/ fa²; 0) for E [1.
By definition of our discretiztaion, know that for all E [L...., NN. || =2/1 L M. Using the Lipschitz-continiity f. for any setting of parameters 0, Il 0) - Y S M L(0).
If we take the limit as M - oo, f(rtbs 0) Y; and
Plyohs 8) x / 1 ((x 1a4l
II [N(rsiol Pprior (0)d0
/o 1 0yvi - prior (0rior())r d0
IN 18 (.))
II [ 18, (.)
p())
II N
p())
x Pyyos
Now we know that Ply Y) x P(yots 0). But since both P(yota Y) and P(yob. (0) are both probability distributions of Yous, then P(yoos oobs (8) Pyyots ().
A.2 PROOF OF THEOREM
Proof. We will start by showing that P(Y (ytt)) is normally distributed. By Lemma that lom P(ooss Y P(yobs Y). Thus, in the limit A - co,
1 know
P( yoess) P(yoos P)P(Y)
P(yom P()
II N (x 0). p(P)
[IN((((r' Viiee oe) P(Y)
-N Make, Einee D Ex)
11

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Under review as a conference paper at ICLR 2022
for some mean and covariance Make Like Gaussian, we know that:
Since the product of two multivariate Gaussians is
P( Yotns) = (H Epos)
(17)
where
Σ
1 like
+ 24'
Mpodt = Poss D Mike + pass D N8.
(18)
post
Now we consider the distribution P(fMAP( (Y ano)) where Y ( Dnne). We will show that, when we set
Hane Mvr
Juc Ex E
the distribution P(YMAP( Y )n becomes equal to the posterior distribution N (Hppot, Epoa). There are three steps needed to show equality between these distributions:
1. Show that P(MMAP(Y ann) normally distributed for some mean and variance, RMS
/ post post
RMS
2. Show that RMS
Hgpos--
3. Show that SIRMS
Epsst
post
Using the same reasoning as above, in the limit as M + 0c.
P(ÝMAP V anc) = arg max Poy Pancc Y
&
arg max.axxA N
Miiker Enire N Mane Emne)
Since the max of Gaussian is the mean, then
FMAP Y anc) AY anc +
(19)
where we define:
A Epost I &
(20)
E
post 2 Mlkke
(21)
We will now show that E|YMAP Y an=)]
postt Because set Manc Me:
E EE (Y nnc)]
IE[AY γ anc +
= AEEY ane +
=ApLy + b
-Epont Exe u&g + Epora Dii Mike
Mposa
Finally, we will show that War War[YMAP] anne)] = Epor.
Var(S amm)) = War[AÝ anc + bj
AWar Y anc 1T
= (Eport Exe
EEx + Exe Din Exx) (Epost 2
=(Epose Epost D. like Dg ) E E Eposs)
Epont D. Epunt Epert
D -1
like post
--11 Epost Epor( (Ezl
= Epost
12

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Under review as a conference paper at ICLR 2022
RMS where post Thus, in the limit as M + oo, P(MMAP(Yas)) = P(Y (oba))
So, (MMA((Ya))) (uposs post
RMS Epestt
Mpost and D
Σ post-
A.3 DERIVATION OF LOSS FUNCTION
We start with
= arg max Egg Pamn log Plike (Yobs (Xaamplle aunc; x)) + log Panc (gc Xsumple a x))
If we choose Pane anchor distributions as follows:
My and Eane Egg. we can simplify the logarithm of the likelihood and
log Pike (y obs (Xammle O i +))
Σ log, N
(Vvina ( G Φ) o.) (Whtall(a"
Σ ll 9( amee 1)
20?
log Panc (Xsamplee Oanci xb) log N(g (Xsample O Φ)) k&p Exg
5
where 6, = g( sample Ounc ()) - f (axmpppe Ounc).
Putting these together, we get:
Φ arg max Eg ~ (0m) 20
E Iluses g(a Ounci 2)11
-
1
8 Σ
2
arg min Eg ~ (0m:)
Σ 1ovs g( Oace; Φ)12 +
N
13


Reviews:
[{"review_id": "JiczInBpnh", "reviewer": "Reviewer_vyt5", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n- The paper deals with an important area of research that provides an efficient way to estimate epistemic uncertainty. \n- The method provides somewhat improved results compared to previous methods. \n\nCons:\n- Even though this work is built upon Pearce et al 2020, it largely hinges on the previous work in the manuscript. Specifically, it is hard to understand without knowing several definitions and results in Pearce. So, I’d recommend authors to provide self-contained backgrounds and results in ch.3. Also, the assumed relationship that parameter-likelihood is equal to data-likelihood (Eq 3) seems to be a special case, but there is no comments on that. \n- It is unclear why the authors exclude the input variables X in the data/parameter-likelihoods. \n- Theoretical results and mathematical states in the paper seem not rigorous. Specifically, in the statements and proof of lemma 1 and theorem 1, the authors simply assume equality between two probability distributions in the limit. This can have many potential problems, such as limit/integral interchangability. It would be better to write and develop their theoretical results in terms of convergence behavior. \n- I have also concerns regarding experiments. First of all, the experiment section lacks important experimental details that make reproducing the results in the paper hard. Also, while the primary motivation for this research is to remove the MAP optimization procedure, there is no comparison to baselines in terms of computational complexity. The last paragraph in page 7 is not reasonable since having many unlabeled samples in practice cannot be the reason why the authors have access to test samples during training. In addition, the CI-ALL measure (that requires the same answers to all members of ensemble) seems unnatural since the power of ensemble comes from diversified predictions from individual members and mostly their majority voting results are of interest. Finally, I cannot understand how the authors could compute the classification accuracy for OOD, unknown class samples given fixed architecture. \n", "summary_of_the_paper": "This paper built upon Pearce et al 2020 that shows regularizing parameters to prior distributions in ensemble of NNs results in the approximate Bayesian inference, aiming to improve computational efficiency of the previous method. Specifically, in the method in Pearce et al, sampling one parameter from posterior requires to find a MAP solution for some anchor point, which can be computationally demanding. Therefore, the authors propose to learn MAP function from an anchor point to MAP solution, which removes the required MAP optimization procedure. \n", "summary_of_the_review": "The paper is not self-contained. Also, even though their motivation for the improvement is to improve the computational efficiency, the experiments did not provide any results on that. There are some issues in the mathematical statements and proof thereof. \n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "9pSVYi7xIPH", "reviewer": "Reviewer_15Ce", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n \n(1)\tThe proposed method proposes a novel idea to build a mapping function from prior to posterior. It avoids ensemble training and achieves reasonable uncertainty quantification results. \n\n(2)\tThe proposed method addresses the high-dimensional parameter space issue by constructing the mapping function in low-dimensional output space.\n\nWeakness:\n\nFor the methods: the theory part is not well-written. Some equations are confusing and may have possible mistakes.\n\n(1) The derivation of Eq. 13 and Eq. 14 is confusing. Why the second term $\\log P_{anc}(g(x_{sample,\\phi,\\theta_{anc}}))$ in Eq. 13 includes the posterior sample $g(x_{sample,\\phi,\\theta_{anc}})$ in the prior distribution? When looking into the details in Appendix.A.3, there might be some mistakes. For example, why $\\log P_{anc}(g(x_{sample},\\phi,\\theta_{anc}))$ and $\\log P_{like}(y_{obs}|g(x_{sample},\\phi,\\theta_{anc}))$ both equals to $\\sum_{i} \\log N(y_{obs}^i | g(x_{sample}^i,\\phi,\\theta_{anc}),\\sigma_{\\epsilon})$\n\n(2) The proposed method does not clearly explain how to obtain the parameters of the prior distribution $P(\\hat{Y})$, i.e.,  $\\mu_{\\hat{Y}},\\Sigma_{\\hat{Y}}$.\n\n(3) When we use low-dimensional $z$ for sampling in Eq. 15, there is no explicit equation for $L_{reg}(z_1,z_2,...,z_k)$. The details of such regularization should be provided and there is no theoretical justification of why we need to add $L_{reg}(z_1,z_2,...,z_k)$ to keep $z_1,z_2,...,z_k$  roughly normally distributed.\n\n(4) No training details are provided.\n\nFor the experiments:\n\n(1) Could you explain why the ensemble 10 methods are significantly worse than the proposed method as the proposed method is supposed to only improve efficiency of the Bayesian ensemble method?  It also does not make sense that the MC dropout method outperforms the ensemble method.  Are there any theoretical justifications for such dramatical improvement in accuracy compared to anchor-regularized neural networks ensembles (Pearce et al., 2020)?\n\n(2) For the image classification tasks, the paper evaluates the proposed method based on the out-of-distribution prediction accuracy and the confidence interval. The confidence interval for classification tasks is confusing. The authors should provide more analyses of why those metrics can be used for evaluating uncertainties.\n\n(3) The author should provide some analyses for the convergence of the model. The model may be hard to train especially when the mapping function $g$ is not powerful enough. Additional ablation studies should also be provided such as the sensitivity of hyperparameters.\n\n(4) No experiments for comparing the efficiency and complexity of different methods. From the introduction section, it seems that a major contribution of the paper is to improve efficiency. \n\n\n\n", "summary_of_the_paper": "The paper develops an uncertainty quantification method as an extension of the Bayesian ensemble method. Instead of training multiple ensemble models through MAP optimization, the proposed method tries to learn a mapping function between the prior distribution and the posterior distribution of model parameters, which avoids the complex training of ensemble models and achieves better efficiency. Due to the high-dimensional parameter space, the authors propose to learn the mapping function in the low-dimensional output space with theoretical justifications. The experiments include image classification and Bayesian optimization, where authors compare the proposed method with Bayesian ensemble and MC-dropout in terms of uncertainty estimation.", "summary_of_the_review": "Overall, this paper proposes interesting ideas by building a mapping function from the prior to the posterior. However, the authors did not explain their ideas well and there might be some mistakes in their formulation. The author may also need to provide more detailed empirical results and corresponding analyses.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "9Yb6_Ug6GWf", "reviewer": "Reviewer_v6ij", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "# Strengths\n\nThis paper provides a nifty alternative to ensembling, which significantly reduces the computational complexity as only one network needs to be trained now. Moreover, as opposed to trying to get the posterior over parameter space the authors compute the posterior over function space; this drastically reduces the number of dimensions the GPN needs to predict, making the problem much easier to learn.\n\n# Weaknesses\n\nWhile I am a fan of the approach conceptually, I think there are a number of weaknesses. \n\nFirstly, I think the related works section is lacking. I think most of the discussion on MCMC and VI could have been cut out. More time should have been focusing on other prior works on ensembling neural networks.\nAs the approach can be seen as a form of hypernetwork[https://arxiv.org/abs/1609.09106], I think this should have definitely been discussed by the authors. There is also other work--similar to Pearce et al--showing how to approximately sample from the posterior of neural networks using the NTK which is also pertinent to discuss[https://arxiv.org/abs/2007.05864].\n\nNext, I am confused by section 4.1 as a whole. To be specific and ensure that I am on the same page as the authors, I am going to rewrite the problem formulation (i apologize in advance for the lack of boldface. for some reason it isn't working properly for me). \n\nWe have training data, \n$$ (x_{obs}, y_{obs}) = \\{ (x_{obs}^i, y_{obs}^i) \\}_{i=1}^N $$\n\nwhere the following relationship is assumed\n$$ y_{obs}^i = f(x_{obs}^i; \\theta) + \\varepsilon_i, \\quad \\epsilon_i \\sim \\mathcal{N}(0, \\sigma_\\epsilon). $$\n\nIn supervised Bayesian learning, the goal is to compute the following posterior where I will first write the posterior over the parameters\n\n$$ p(\\theta \\vert x_{obs}, y_{obs} )  \\propto P_{prior}(\\theta) \\prod_{i=1}^N P_{like} (y_{obs}^i \\vert x_{obs}^i, \\theta)  $$\n\nOne can also compute a posterior over functions as well. \nLet \n$$ \\tilde{y}^i_{obs} = f(x_{obs}^i; \\theta) $$\nthus we can express the relationship between the inputs and outputs as \n$$ y_{obs}^i = \\tilde{y}^i_{obs} + \\epsilon_i. $$\n\nLet $ \\tilde{Y} = (\\tilde{y}^1_{obs}, ..., \\tilde{y}^N_{obs}) $, we now want to compute the following posterior\n$$ p(\\tilde{Y} \\vert x_{obs}, y_{obs}) \\propto P_{prior}(\\tilde{Y} \\vert x_{obs}) \\prod_{i=1}^N P_{like}(y^i_{obs} \\vert \\tilde{y}^i_{obs}) $$\nIn Lemma 1, the authors prove that in function space, the likelihood is normally distributed with respect to $\\tilde{Y}$. This is trivially true by construction  i.e., $P_{like}(y_{obs}^i \\vert \\tilde{y}^i) = \\mathcal{N}(y_{obs}^i \\vert \\tilde{y}^i, \\sigma_\\epsilon)  $. When operating in parameter space, the likelihood is complicated but the prior is easier; when moving to function space, this switches.\n\nBesides that, Lemma 1 also makes no sense to me. In the lemma, they take a discretization of the input space but I see no reason to do that as the likelihood is only defined on the inputs provided in the training data. This is also true for theorem 1 as well. In my opinion, section 4.1 could have been entirely removed, which would have left more space for more experiments.\n\nConcerning the experiments section, it was disappointing to not see an experiment similar to Pearce et al, comparing the true posterior to the posterior obtained by GPN on a simple toy dataset. As many assumptions were made, I think it is very important that the approach was empirically verified in a setting where the ground truth can be obtained. Next, I don't understand experiment 2. Regardless of the learning method used, the posterior studied in this paper revolves around using supervised data. I have no idea how the authors were able to use the unlabeled data nor does it seem fair to me.", "summary_of_the_paper": "In this work, the authors introduce generative posterior networks (GPN). GPN is a single network that approximately produces samples from posterior over neural network parameters; this is in contrast to the standard approach of training N individual neural networks.", "summary_of_the_review": "While I like the idea, I think more work needs to be done on the structure of the paper and the experiments. I think section 4.2 could be cut, allowing for more space to do experiments and a more fleshed-out discussion section.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "5oMFmZ-aWH6", "reviewer": "Reviewer_ZgaQ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper introduces a new proposal to make use of ensemble NN methods, GPNs. With the approximations imposed, GPNs perform well in the experiments shown. Although some points about the text could be improved and there are certainly some potentially important drawbacks, it seems an interesting contribution to the field. There seems to be a couple of tests that need to be done, but the results point in the right direction. \n\n# Pros:\n\n* Allow to obtain samples in a more scalable fashion from NN-ensemble-like models. \n* The results obtained support the proposed method in a solid manner when compared to previous methods. \n\n# Cons:\n\n* The conditions imposed on the method makes its predictions behave in a Gaussian-like manner. This could be fine in cases where data does not exhibit complex behaviour, but could be a problem for cases with multimodality, heteroscedasticity, etc.  \n* Notation can be improved to make the description of the method clearer. \n\n\n# Other questions / comments:\n\n* It could be argued that, contrary to what is pointed out in the introduction, VI or GP-based methods can be easy to work with. In fact, if some assumptions are feasible or the data behaves in a certain manner, they can be pretty illustrative and informative models. I would suggest rewriting this part of the introduction, pointing out more clearly why NN ensembles may be more interesting in some cases. \n\n* GPs are a model class, not an inference method. \n\n* I would suggest conducting an experiment on the scalability of the method, measuring convergence times with big datasets. This would help us gauge correctly how well does the method perform in this matter as well.  \n\n* While it is true that basic BNNs degrade their performance in higher dimensionality settings, there have been important advances in the last years to prevent this from happening, especially when using deeper or wider networks. This can be said for the function-space-based optimization systems, such as those of [1,2].\n\n* As a mere suggestion, I find the notation of the data-likelihood and parameter-likelihood PDF unnecessarily complex. I would either simplify this to improve the clarity of the text or explain the notation further (not just relying on referenced sources). In general, through the paper, the notation could be clearer. \n\n* Figure 3 can be improved. I suggest trying alternative visualizations s.a. combining different boxplots into one using colours, or at the very least increasing the font (in the CIFAR-10 case) to better show the results.  \n\n* Have the authors conducted any tests in regression datasets? It could be very interesting to see the resulting behaviour in those cases. \n  \n ## References\n \n [1] Ma, C., Li, Y., and Hernández-Lobato, J. M. (2019). “Variational implicit processes”. In: International Conference on Machine Learning, pp. 4222–4233.\n\n [2] Sun, S., Zhang, G., Shi, J., and Grosse, R. (2019). “Functional variational Bayesian neural networks”. In: International Conference on Learning Representations.\n ", "summary_of_the_paper": "NN ensembles are useful models to estimate epistemic uncertainty in the predictions. Since their computational cost is generally high, they introduce Generative Posterior Networks (GPNs) as a generative NN model to approximate the posterior distribution of the inference process. Under not very restrictive assumptions, GPNs work as well as previous methods, if not better. This is shown through extensive experiments, especially in classification problems.", "summary_of_the_review": "The contribution is novel and significative, although there are a couple of things that should be addressed in the revised version of the paper. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This article proposes a novel uncertainty quantification method, formulating the problem as a Bayesian inference problem. Instead of training multiple ensemble models through MAP optimisation, as in ensemble methods, the proposed approach tries to learn a mapping function between the prior distribution and the posterior distribution of model parameters. This avoids the complex training of ensemble models and achieves better efficiency. 

The approach is novel, and the problem of importance. The paper however suffers from a number of weaknesses:
* Some theoretical results would need to be made mathematically more rigorous
* The presentation is unclear and confusing in some places
* Empirical results are not reproducible due to the lack of details
Although the authors clarified some of the points raised by reviewers in their response, the paper in its current form is not ready for publication, and I recommend rejection.
Decision: Reject