Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks.

The review comment to be evaluated is:

This paper presents Bayesian Hypernetworks; variational Bayesian neural networks where the variational posterior over the weights is governed by a hyper network that implements a normalizing flow (NF) such as RealNVP and IAF. As directly outputting the weight matrix with a hyper network is computationally expensive the authors instead propose to utilize weight normalisation on the weights and then use the hyper network to output scalar scaling variables for each hidden unit, similarly to what was done at [1]. The main difference with this prior work is that [1] consider these NF scaling variables as auxiliary random variables to a mean field Gaussian distribution over the weights whereas this paper attempts to posit a distribution directly on the weights via the NF. This avoids the nested variational approximation and auxiliary models of [1], which can potentially yield a tighter bound. The proposed method is evaluated on extensive experiments.

This paper seems like a plausible idea with extensive experiments but the similarity with [1] make it an incremental contribution and, furthermore, it seems that it has a technical issue with what is explained at Section 3.3. More specifically, if you generate the parameters \theta according to Eq. 7 and posit a prior over \theta then you will have a problematic variational bound as there will be a KL divergence, KL(q(\theta) || p(\theta)), with distributions of different support (since q(\theta) is defined only along the directions spanned by u), which is infinite. For the KL to be valid you will need to posit a prior distribution over `g`, p(g), and then consider KL(q(g) || p(g)), with q(g) being given by the NF. From the experiment paragraph at page 5 though I deduct that you instead employ “an isotropic standard normal prior over the weights”, i.e. \theta, thus I believe that you indeed have a problematic bound. How do you actually compute logq(\theta) when you employ the parametrisation discussed at 3.3? Did you use that parametrisation in every experiment?

Other than that, I believe that it would be interesting to experiment with a `full` hyper network, i.e. generating directly the entire parameter vector \theta, e.g. at the toy regression experiment where the dimensionality is small. This would then better illustrate the tradeoffs you make when you reduce the flexibility of the hyper-network to just outputting the row scaling variables and the effect this has at the posterior approximation.

Typos:
(1) Page 3, 3.1.1 log(\theta) -> logp(\theta).
(2) Eq. 6, it needs to be |det \frac{\partial h(\epsilon)}{\partial \epsilon}|^{-1} or |det \frac{\partial h^{-1}(\theta)}{\partial \theta}| for a valid change of variables formula.

[1] Louizos & Welling, Multiplicative Normalizing Flows for Variational Bayesian Neural Networks.

The evidence to be evaluated is:Bayesian DNNs have been studied since the 1990s (Neal, 1996; MacKay, 1994).

Broadly speaking, existing methods either 1) use Markov chain Monte Carlo (Welling & Teh, 2011; Neal, 1996), or 2) directly learn an approximate posterior distribution using (stochastic) variational inference (Graves, 2011; Gal & Ghahramani, 2016; Salimans et al., 2015; Blundell et al., 2015), expectation propagation (Hernandez-Lobato & Adams, 2015; Soudry et al., 2014), or \(\alpha\)-divergences (Li & Gal, 2017). We focus here on the most popular approach: variational inference.

The other important points of reference for our work are Bayes by Backprop (BbB) (Blundell et al., 2015), and multiplicative normalizing flows (Louizos & Welling, 2017). Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution). More similar is the work of Louizos & Welling (2017), who propose and dismiss BHNs due to the issues of scaling BHNs to large primary networks, which we address in Section 3.3.2 Instead, in their work, they use a hypernet to generate scaling factors, \(\mathbf{z}\) on the means \(\mathbf{\mu}\) of a factorial Gaussian distribution. Because \(\mathbf{z}\) follows a complicated distribution, this forms a highly flexible approximate posterior: \(q(\mathbf{\theta})=\int q(\mathbf{\theta}|\mathbf{z})q(\mathbf{z})\mathrm{d}\mathbf{z}\). However, this approach also requires them to introduce an auxiliary inference network to approximate \(q(\mathbf{z}|\mathbf{\theta})\) in order to estimate the entropy term of the variational lower bound, resulting in lower bound _on the variational lower bound_.

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017). We take advantage of techniques for _invertible_ DDGNs developed in several recent works on generative modeling (Dinh et al., 2014; Dinh et al., 2016) and variational inference of latent variables (Rezende and Mohamed, 2015; Kingma et al., 2016). Training these models uses the change of variables formula, which involves computing the log-determinant of the inverse Jacobian of the generator network. This computation involves a potentially costly matrix determinant, and these works propose innovative architectures which reduce the cost of this operation but can still express complicated deformations, which are referred to as "normalizing flows". We now describe how variational inference is applied to Bayesian deep nets (Section 3.1), and how we compose the methods described in Sections 2.2 and 2.3 to produce Bayesian hypernets (Section 3.2). In variational inference, the goal is to maximize a lower bound on the marginal log-likelihood of the data, \(\log p(\mathcal{D})\) under some statistical model. This involves both estimating parameters of the model, and approximating the posterior distribution over unobserved random variables (which may themselves also be parameters, e.g., as in the case of Bayesian DNNs). Let \(\mathbf{\theta}\in\mathbb{R}^{D}\) be parameters given the Bayesian treatment as random variables, \(\mathcal{D}\) a training set of observed data, and \(q(\mathbf{\theta})\) a learned approximation to the true posterior \(p(\mathbf{\theta}|\mathcal{D})\). Since the KL divergence is always non-negative, we have, for _any_\(q(\mathbf{\theta})\): \[\log p(\mathcal{D}) =\text{KL}(q(\mathbf{\theta})\|p(\mathbf{\theta}|\mathcal{D}))+\mathbb{E} _{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{\theta})-\log q(\mathbf{\theta})] \tag{1}\] \[\geq\mathbb{E}_{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{ \theta})-\log q(\mathbf{\theta})]\,. \tag{2}\] The right hand side of (2) is the evidence lower bound, or "ELBO".

Computing the expectation in (2) is generally intractable for deep nets, but can be estimated by Monte Carlo sampling. For a given value of \(\theta\), \(\log p(\mathcal{D}|\mathbf{\theta})\) and \(\log p(\mathbf{\theta})\) can be computed and differentiated exactly as in a non-Bayesian DNN, allowing training by backpropagation.

Here we approximate the posterior distribution \(p(\mathbf{\theta}|\mathcal{D})\) using the approximate posterior \(q(\mathbf{\theta})\).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\). Since BHNs are both trained and evaluated via samples of \(q(\mathbf{\theta})\), expressing \(q(\mathbf{\theta})\) as a generative model is a natural strategy.

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\). As discussed in Section 2.3, a number of techniques have been developed for efficiently training such invertible DDGNs. In this work, we employ both RealNVP (RNVP) (Dinh et al., 2016) and Inverse Autoregressive Flows (IAF) (Kingma et al., 2016). Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below. In order to scale BHNs to large primary networks, we use the weight normalization reparametrization (Salimans & Kingma, 2016)6:

This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters. We also employ weight normalization within the hypernet, and found this stabilizes training dramatically. Initialization plays an important role as well; we recommend initializing the hypernet weights to small values to limit the impact of noise at the beginning of training. We also find clipping the outputs of the softmax to be within \((0.001,0.999)\) critical for numerical stability.

Parameter uncertainty also has regularization benefits: integrating over the posterior creates an implicit ensemble. Intuitively, when the most likely hypothesis predicts "A", but the posterior places more total mass on hypotheses predicting "B", we prefer predicting "B". By improving our estimate of the posterior, we more accurately weigh the evidence for different hypotheses.

We use an isotropic standard normal prior on the scaling factors (\(g\)) of the weights of the network.

We first demonstrate the behavior of the network on the toy 1D-regression problem from Blundell et al. (2015) in Figure 1.

We also use this experiment to evaluate the effects of our proposal for scaling BHNs via the weight norm parametrization (Section 3.3) by comparing with a model which generates the full set of parameters, and find that the two models produce very similar results, suggesting that our proposed method strikes a good balance between scalability and expressiveness. Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors. For this experiment, we trained an over-parametrized linear (primary) network: \(\hat{y}=a\cdot b\cdot x\) on a dataset generated as \(y=x+\epsilon\), and the BHN learns capture both the modes of \(a=b=1\) and \(a=b=-1\). We now show that BHNs act as a regularizer, outperforming dropout and traditional mean field (BbB).

In our experiments, we find that BHNs perform on par with dropout on full datasets of MNIST and CIFAR10; furthermore, increasing the flexibility of the posterior by adding more coupling layers improves performance, especially compared with models with 0 coupling layers, which cannot model dependencies between the parameters.

In these MNIST experiments, we use MLPs with 2 hidden layers of 800 or 1200 hidden units each. For CIFAR10, we train a convolutional neural net (CNN) with 4 hidden layers of \([64,64,128,128]\) channels, \(2\times 2\) max pooling after the second and the fourth layers, filter size of \(3\), and a single fully connected layer of \(512\) units.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input: Figure 3: Box plot of performance across 10 trials. Bayesian hypernets (BHNs) with inverse autoregressive flows (IAF) consistently outperform the other methods.

Input: Figure 5: Adversary detection: Horizontal axis is the step size of the FGS algorithm. While accuracy drops when more perturbation is added to the data (left), uncertainty measures also increase (first row). In particular, the BALD and Mean STD scores, which measure epistemic uncertainty, are strongly increasing for BHNs, but not for dropout. The second row and third row plots show results for adversary detection and error detection (respectively) in terms of the AUC of ROC (y-axis) with increasing perturbation along the x-axis. Gradient direction is estimated with one Monte Carlo sample of the weights/dropout mask.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on my review of the evidence, the claim "We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks" is well-stated and accurately reflects what is presented in the paper. The evidence thoroughly demonstrates that the authors:

1. Develop and detail a novel framework using hypernetworks for Bayesian inference
2. Provide extensive theoretical foundations and mathematical formulations
3. Show experimental results validating the approach
4. Acknowledge limitations and compare to existing methods
5. Support claims with empirical evidence across multiple experiments and datasets

The reviewer points out some technical concerns but these don't undermine the fundamental claim of proposing this framework. The claim is measured and doesn't overstate the contribution.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks.

The review comment to be evaluated is:

This paper proposes Bayesian hypernetworks to carry out Bayesian learning of deep networks. The idea is to construct a generative model capable of approximating the posterior distribution over the parameters of deep networks. I think that the paper is well written and easy to follow.

I like the idea of constructing general approximation strategies for complex posterior distribution and the proposed approach inherits all the scalability properties of modern deep learning techniques. In this respect, I think that the paper tackles a timely topic and is interesting to read.

It is not entirely clear to me why the Authors name their proposal Bayesian hypernetworks. This seems to suggest that also the hypernetwork is infered using Bayesian inference, but if I understand correctly this is not the case.

I have some comments on novelty and realization of the experiments. In the positioning of the work in the literature, the Authors point out that hypernetworks have been proposed before, so it is not clear what is the actual novelty in the proposal. Is it the use of Real NVPs and IAFs as hypernetworks? These methods have been already proposed and extensively studied in the literature, and even if they have been adapted to be hypernetworks here, I believe that the novelty is fairly limited.

The experimental part is interesting as it explores a number of learning scenarios. However, I think that it would have been useful to add comparisons with standard variational inference (e.g., Graves, 2011) for deep networks to substantiate the claims that this approach underestimates uncertainty. I believe that this would strengthen the comparative evaluation.

I think the paper would have made a stronger case by including other approaches to approximate posteriors using generative models. For example, the variational Gaussian process paper sounds like an ideal method to include here.

[1] D. Tran, R. Ranganath, and D. M. Blei. Variational Gaussian process. arXiv preprint arXiv:1511.06499, 2015.

The evidence to be evaluated is:Bayesian DNNs have been studied since the 1990s (Neal, 1996; MacKay, 1994).

Broadly speaking, existing methods either 1) use Markov chain Monte Carlo (Welling & Teh, 2011; Neal, 1996), or 2) directly learn an approximate posterior distribution using (stochastic) variational inference (Graves, 2011; Gal & Ghahramani, 2016; Salimans et al., 2015; Blundell et al., 2015), expectation propagation (Hernandez-Lobato & Adams, 2015; Soudry et al., 2014), or \(\alpha\)-divergences (Li & Gal, 2017). We focus here on the most popular approach: variational inference.

The other important points of reference for our work are Bayes by Backprop (BbB) (Blundell et al., 2015), and multiplicative normalizing flows (Louizos & Welling, 2017). Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution). More similar is the work of Louizos & Welling (2017), who propose and dismiss BHNs due to the issues of scaling BHNs to large primary networks, which we address in Section 3.3.2 Instead, in their work, they use a hypernet to generate scaling factors, \(\mathbf{z}\) on the means \(\mathbf{\mu}\) of a factorial Gaussian distribution. Because \(\mathbf{z}\) follows a complicated distribution, this forms a highly flexible approximate posterior: \(q(\mathbf{\theta})=\int q(\mathbf{\theta}|\mathbf{z})q(\mathbf{z})\mathrm{d}\mathbf{z}\). However, this approach also requires them to introduce an auxiliary inference network to approximate \(q(\mathbf{z}|\mathbf{\theta})\) in order to estimate the entropy term of the variational lower bound, resulting in lower bound _on the variational lower bound_.

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017). We take advantage of techniques for _invertible_ DDGNs developed in several recent works on generative modeling (Dinh et al., 2014; Dinh et al., 2016) and variational inference of latent variables (Rezende and Mohamed, 2015; Kingma et al., 2016). Training these models uses the change of variables formula, which involves computing the log-determinant of the inverse Jacobian of the generator network. This computation involves a potentially costly matrix determinant, and these works propose innovative architectures which reduce the cost of this operation but can still express complicated deformations, which are referred to as "normalizing flows". We now describe how variational inference is applied to Bayesian deep nets (Section 3.1), and how we compose the methods described in Sections 2.2 and 2.3 to produce Bayesian hypernets (Section 3.2). In variational inference, the goal is to maximize a lower bound on the marginal log-likelihood of the data, \(\log p(\mathcal{D})\) under some statistical model. This involves both estimating parameters of the model, and approximating the posterior distribution over unobserved random variables (which may themselves also be parameters, e.g., as in the case of Bayesian DNNs). Let \(\mathbf{\theta}\in\mathbb{R}^{D}\) be parameters given the Bayesian treatment as random variables, \(\mathcal{D}\) a training set of observed data, and \(q(\mathbf{\theta})\) a learned approximation to the true posterior \(p(\mathbf{\theta}|\mathcal{D})\). Since the KL divergence is always non-negative, we have, for _any_\(q(\mathbf{\theta})\): \[\log p(\mathcal{D}) =\text{KL}(q(\mathbf{\theta})\|p(\mathbf{\theta}|\mathcal{D}))+\mathbb{E} _{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{\theta})-\log q(\mathbf{\theta})] \tag{1}\] \[\geq\mathbb{E}_{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{ \theta})-\log q(\mathbf{\theta})]\,. \tag{2}\] The right hand side of (2) is the evidence lower bound, or "ELBO".

Computing the expectation in (2) is generally intractable for deep nets, but can be estimated by Monte Carlo sampling. For a given value of \(\theta\), \(\log p(\mathcal{D}|\mathbf{\theta})\) and \(\log p(\mathbf{\theta})\) can be computed and differentiated exactly as in a non-Bayesian DNN, allowing training by backpropagation.

Here we approximate the posterior distribution \(p(\mathbf{\theta}|\mathcal{D})\) using the approximate posterior \(q(\mathbf{\theta})\).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\). Since BHNs are both trained and evaluated via samples of \(q(\mathbf{\theta})\), expressing \(q(\mathbf{\theta})\) as a generative model is a natural strategy.

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\). As discussed in Section 2.3, a number of techniques have been developed for efficiently training such invertible DDGNs. In this work, we employ both RealNVP (RNVP) (Dinh et al., 2016) and Inverse Autoregressive Flows (IAF) (Kingma et al., 2016). Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below. In order to scale BHNs to large primary networks, we use the weight normalization reparametrization (Salimans & Kingma, 2016)6:

This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters. We also employ weight normalization within the hypernet, and found this stabilizes training dramatically. Initialization plays an important role as well; we recommend initializing the hypernet weights to small values to limit the impact of noise at the beginning of training. We also find clipping the outputs of the softmax to be within \((0.001,0.999)\) critical for numerical stability.

Parameter uncertainty also has regularization benefits: integrating over the posterior creates an implicit ensemble. Intuitively, when the most likely hypothesis predicts "A", but the posterior places more total mass on hypotheses predicting "B", we prefer predicting "B". By improving our estimate of the posterior, we more accurately weigh the evidence for different hypotheses.

We use an isotropic standard normal prior on the scaling factors (\(g\)) of the weights of the network.

We first demonstrate the behavior of the network on the toy 1D-regression problem from Blundell et al. (2015) in Figure 1.

We also use this experiment to evaluate the effects of our proposal for scaling BHNs via the weight norm parametrization (Section 3.3) by comparing with a model which generates the full set of parameters, and find that the two models produce very similar results, suggesting that our proposed method strikes a good balance between scalability and expressiveness. Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors. For this experiment, we trained an over-parametrized linear (primary) network: \(\hat{y}=a\cdot b\cdot x\) on a dataset generated as \(y=x+\epsilon\), and the BHN learns capture both the modes of \(a=b=1\) and \(a=b=-1\). We now show that BHNs act as a regularizer, outperforming dropout and traditional mean field (BbB).

In our experiments, we find that BHNs perform on par with dropout on full datasets of MNIST and CIFAR10; furthermore, increasing the flexibility of the posterior by adding more coupling layers improves performance, especially compared with models with 0 coupling layers, which cannot model dependencies between the parameters.

In these MNIST experiments, we use MLPs with 2 hidden layers of 800 or 1200 hidden units each. For CIFAR10, we train a convolutional neural net (CNN) with 4 hidden layers of \([64,64,128,128]\) channels, \(2\times 2\) max pooling after the second and the fourth layers, filter size of \(3\), and a single fully connected layer of \(512\) units.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input: Figure 3: Box plot of performance across 10 trials. Bayesian hypernets (BHNs) with inverse autoregressive flows (IAF) consistently outperform the other methods.

Input: Figure 5: Adversary detection: Horizontal axis is the step size of the FGS algorithm. While accuracy drops when more perturbation is added to the data (left), uncertainty measures also increase (first row). In particular, the BALD and Mean STD scores, which measure epistemic uncertainty, are strongly increasing for BHNs, but not for dropout. The second row and third row plots show results for adversary detection and error detection (respectively) in terms of the AUC of ROC (y-axis) with increasing perturbation along the x-axis. Gradient direction is estimated with one Monte Carlo sample of the weights/dropout mask.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim "We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks" appears well-stated. The paper thoroughly presents a novel framework combining hypernetworks with Bayesian inference, providing extensive mathematical formulation, experimental validation, and comparisons with existing methods. While the reviewer notes some concerns about novelty regarding hypernetworks and Real NVPs/IAFs, the core contribution of using hypernetworks for Bayesian inference is clearly demonstrated. The evidence shows detailed implementation, scalability solutions, and empirical results supporting the framework's effectiveness.

<score>1</score>

Justification:
1. The evidence presents comprehensive mathematical foundations (equations 1-10)
2. The paper demonstrates practical implementation details and solutions for scaling issues
3. Multiple experiments validate the approach (toy problems, MNIST, CIFAR10)
4. Results show competitive or superior performance compared to existing methods
5. Limitations and technical challenges are openly addressed
6. The claim modestly states "a framework" rather than making grandiose claims about superiority or revolutionary advances


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks.

The review comment to be evaluated is:

* Edit: I increased my rating to 6. The authors fixed the first error I pointed out below. Regarding the second point: I still think it is possible to take a limit of sigma -> 0 in MNF, which makes the methods very similar.

The authors propose a new method of defining approximate posteriors for use in Bayesian neural networks. The idea of using hypernetworks for Bayesian inference is compelling, and the authors show some promising first results. I see two issues, and would be willing to increase my rating if these were sufficiently addressed.

- The paper says it uses an "isotropic standard normal prior on the weights of the network". However, the stochastic part of the generated weights (i.e. the scales) is of a lower dimension than the weights. It seems to me this means that the KL divergence between prior and posterior is undefined, or infinite, as the posterior is only defined on a sub-manifold. What exactly is the loss term that is added to the training objective? And how is this justified?

- The instantiation of Bayesian hypernetworks that is used in experiments seems to be a special case of the method of multiplicative normalizing flows as proposed by Louizos and Welling and discussed in this paper. If the variances / sigmas are zero in the latter method, their approximation seems functionally equivalent to Bayesian hypernetworks (though with different parameterization). Is my understanding correct? If so, the novelty of the proposed method is limited.

The evidence to be evaluated is:Bayesian DNNs have been studied since the 1990s (Neal, 1996; MacKay, 1994).

Broadly speaking, existing methods either 1) use Markov chain Monte Carlo (Welling & Teh, 2011; Neal, 1996), or 2) directly learn an approximate posterior distribution using (stochastic) variational inference (Graves, 2011; Gal & Ghahramani, 2016; Salimans et al., 2015; Blundell et al., 2015), expectation propagation (Hernandez-Lobato & Adams, 2015; Soudry et al., 2014), or \(\alpha\)-divergences (Li & Gal, 2017). We focus here on the most popular approach: variational inference.

The other important points of reference for our work are Bayes by Backprop (BbB) (Blundell et al., 2015), and multiplicative normalizing flows (Louizos & Welling, 2017). Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution). More similar is the work of Louizos & Welling (2017), who propose and dismiss BHNs due to the issues of scaling BHNs to large primary networks, which we address in Section 3.3.2 Instead, in their work, they use a hypernet to generate scaling factors, \(\mathbf{z}\) on the means \(\mathbf{\mu}\) of a factorial Gaussian distribution. Because \(\mathbf{z}\) follows a complicated distribution, this forms a highly flexible approximate posterior: \(q(\mathbf{\theta})=\int q(\mathbf{\theta}|\mathbf{z})q(\mathbf{z})\mathrm{d}\mathbf{z}\). However, this approach also requires them to introduce an auxiliary inference network to approximate \(q(\mathbf{z}|\mathbf{\theta})\) in order to estimate the entropy term of the variational lower bound, resulting in lower bound _on the variational lower bound_.

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017). We take advantage of techniques for _invertible_ DDGNs developed in several recent works on generative modeling (Dinh et al., 2014; Dinh et al., 2016) and variational inference of latent variables (Rezende and Mohamed, 2015; Kingma et al., 2016). Training these models uses the change of variables formula, which involves computing the log-determinant of the inverse Jacobian of the generator network. This computation involves a potentially costly matrix determinant, and these works propose innovative architectures which reduce the cost of this operation but can still express complicated deformations, which are referred to as "normalizing flows". We now describe how variational inference is applied to Bayesian deep nets (Section 3.1), and how we compose the methods described in Sections 2.2 and 2.3 to produce Bayesian hypernets (Section 3.2). In variational inference, the goal is to maximize a lower bound on the marginal log-likelihood of the data, \(\log p(\mathcal{D})\) under some statistical model. This involves both estimating parameters of the model, and approximating the posterior distribution over unobserved random variables (which may themselves also be parameters, e.g., as in the case of Bayesian DNNs). Let \(\mathbf{\theta}\in\mathbb{R}^{D}\) be parameters given the Bayesian treatment as random variables, \(\mathcal{D}\) a training set of observed data, and \(q(\mathbf{\theta})\) a learned approximation to the true posterior \(p(\mathbf{\theta}|\mathcal{D})\). Since the KL divergence is always non-negative, we have, for _any_\(q(\mathbf{\theta})\): \[\log p(\mathcal{D}) =\text{KL}(q(\mathbf{\theta})\|p(\mathbf{\theta}|\mathcal{D}))+\mathbb{E} _{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{\theta})-\log q(\mathbf{\theta})] \tag{1}\] \[\geq\mathbb{E}_{q}[\log p(\mathcal{D}|\mathbf{\theta})+\log p(\mathbf{ \theta})-\log q(\mathbf{\theta})]\,. \tag{2}\] The right hand side of (2) is the evidence lower bound, or "ELBO".

Computing the expectation in (2) is generally intractable for deep nets, but can be estimated by Monte Carlo sampling. For a given value of \(\theta\), \(\log p(\mathcal{D}|\mathbf{\theta})\) and \(\log p(\mathbf{\theta})\) can be computed and differentiated exactly as in a non-Bayesian DNN, allowing training by backpropagation.

Here we approximate the posterior distribution \(p(\mathbf{\theta}|\mathcal{D})\) using the approximate posterior \(q(\mathbf{\theta})\).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\). Since BHNs are both trained and evaluated via samples of \(q(\mathbf{\theta})\), expressing \(q(\mathbf{\theta})\) as a generative model is a natural strategy.

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\). As discussed in Section 2.3, a number of techniques have been developed for efficiently training such invertible DDGNs. In this work, we employ both RealNVP (RNVP) (Dinh et al., 2016) and Inverse Autoregressive Flows (IAF) (Kingma et al., 2016). Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below. In order to scale BHNs to large primary networks, we use the weight normalization reparametrization (Salimans & Kingma, 2016)6:

This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters. We also employ weight normalization within the hypernet, and found this stabilizes training dramatically. Initialization plays an important role as well; we recommend initializing the hypernet weights to small values to limit the impact of noise at the beginning of training. We also find clipping the outputs of the softmax to be within \((0.001,0.999)\) critical for numerical stability.

Parameter uncertainty also has regularization benefits: integrating over the posterior creates an implicit ensemble. Intuitively, when the most likely hypothesis predicts "A", but the posterior places more total mass on hypotheses predicting "B", we prefer predicting "B". By improving our estimate of the posterior, we more accurately weigh the evidence for different hypotheses.

We use an isotropic standard normal prior on the scaling factors (\(g\)) of the weights of the network.

We first demonstrate the behavior of the network on the toy 1D-regression problem from Blundell et al. (2015) in Figure 1.

We also use this experiment to evaluate the effects of our proposal for scaling BHNs via the weight norm parametrization (Section 3.3) by comparing with a model which generates the full set of parameters, and find that the two models produce very similar results, suggesting that our proposed method strikes a good balance between scalability and expressiveness. Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors. For this experiment, we trained an over-parametrized linear (primary) network: \(\hat{y}=a\cdot b\cdot x\) on a dataset generated as \(y=x+\epsilon\), and the BHN learns capture both the modes of \(a=b=1\) and \(a=b=-1\). We now show that BHNs act as a regularizer, outperforming dropout and traditional mean field (BbB).

In our experiments, we find that BHNs perform on par with dropout on full datasets of MNIST and CIFAR10; furthermore, increasing the flexibility of the posterior by adding more coupling layers improves performance, especially compared with models with 0 coupling layers, which cannot model dependencies between the parameters.

In these MNIST experiments, we use MLPs with 2 hidden layers of 800 or 1200 hidden units each. For CIFAR10, we train a convolutional neural net (CNN) with 4 hidden layers of \([64,64,128,128]\) channels, \(2\times 2\) max pooling after the second and the fourth layers, filter size of \(3\), and a single fully connected layer of \(512\) units.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input: Figure 3: Box plot of performance across 10 trials. Bayesian hypernets (BHNs) with inverse autoregressive flows (IAF) consistently outperform the other methods.

Input: Figure 5: Adversary detection: Horizontal axis is the step size of the FGS algorithm. While accuracy drops when more perturbation is added to the data (left), uncertainty measures also increase (first row). In particular, the BALD and Mean STD scores, which measure epistemic uncertainty, are strongly increasing for BHNs, but not for dropout. The second row and third row plots show results for adversary detection and error detection (respectively) in terms of the AUC of ROC (y-axis) with increasing perturbation along the x-axis. Gradient direction is estimated with one Monte Carlo sample of the weights/dropout mask.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll evaluate the claim "We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks."

This claim appears to be well-stated based on the evidence provided. The paper thoroughly describes the theoretical framework, implementation details, and experimental validation of Bayesian hypernetworks. The evidence shows:

1. Clear mathematical formulation of the approach
2. Comprehensive experimental results demonstrating effectiveness
3. Comparisons with existing methods
4. Acknowledgment of limitations and trade-offs
5. Practical solutions for scaling issues

The reviewer's comments, while raising important technical points, don't dispute the fundamental proposal of the framework. The issues raised are about specific technical details rather than the core claim itself.

The claim is straightforward and accurately reflects what the paper delivers without overstating capabilities or novelty.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

A Bayesian hypernetwork \(h\) is a neural network which learns to transform a simple noise distribution, \(p(\mathbf{\epsilon})=\mathcal{N}(\mathbf{0},\mathbf{I})\), to a distribution \(q(\mathbf{\theta}):=q(h(\mathbf{\epsilon}))\) over the parameters \(\mathbf{\theta}\) of another neural network (the "primary network").

The evidence to be evaluated is:Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution).

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\).

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\).

Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below.

We only output the scaling factors \(g\) from the hypernet, and learn a maximum likelihood estimate of \(\mathbf{v}\).7 This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters.

Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After careful review, the claim appears to be well-stated and accurately describes what a Bayesian hypernetwork is based on the provided evidence. The evidence thoroughly supports the key elements of the claim:

1. The hypernetwork (h) is indeed a neural network that outputs parameters of another network
2. It transforms simple noise distribution (ε ~ N(0,I)) to a distribution q(θ) over parameters
3. These parameters θ are for the "primary network"

The mathematical formulation and implementation details in the evidence, including equations 5-10, along with Figures 1, 2, and 6, demonstrate this functionality. The claim doesn't overstate capabilities or make unsupported assertions - it simply defines the core concept accurately.

<score>1</score>

Justification:
- Direct evidence confirms the definition: "Bayesian hypernets (BHNs) express a flexible q(θ) by using a DDGN... to transform random noise ε ~ N(0,I) into independent samples from q(θ)"
- Figures demonstrate the practical implementation, showing parameter distributions and transformations
- The claim sticks to fundamental definition without making exaggerated claims about performance or capabilities
- No significant gaps between claim and supporting evidence
- Mathematical formalism in evidence matches claim's description


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

A Bayesian hypernetwork \(h\) is a neural network which learns to transform a simple noise distribution, \(p(\mathbf{\epsilon})=\mathcal{N}(\mathbf{0},\mathbf{I})\), to a distribution \(q(\mathbf{\theta}):=q(h(\mathbf{\epsilon}))\) over the parameters \(\mathbf{\theta}\) of another neural network (the "primary network").

The review comment to be evaluated is:

This paper presents Bayesian Hypernetworks; variational Bayesian neural networks where the variational posterior over the weights is governed by a hyper network that implements a normalizing flow (NF) such as RealNVP and IAF. As directly outputting the weight matrix with a hyper network is computationally expensive the authors instead propose to utilize weight normalisation on the weights and then use the hyper network to output scalar scaling variables for each hidden unit, similarly to what was done at [1]. The main difference with this prior work is that [1] consider these NF scaling variables as auxiliary random variables to a mean field Gaussian distribution over the weights whereas this paper attempts to posit a distribution directly on the weights via the NF. This avoids the nested variational approximation and auxiliary models of [1], which can potentially yield a tighter bound. The proposed method is evaluated on extensive experiments.

This paper seems like a plausible idea with extensive experiments but the similarity with [1] make it an incremental contribution and, furthermore, it seems that it has a technical issue with what is explained at Section 3.3. More specifically, if you generate the parameters \theta according to Eq. 7 and posit a prior over \theta then you will have a problematic variational bound as there will be a KL divergence, KL(q(\theta) || p(\theta)), with distributions of different support (since q(\theta) is defined only along the directions spanned by u), which is infinite. For the KL to be valid you will need to posit a prior distribution over `g`, p(g), and then consider KL(q(g) || p(g)), with q(g) being given by the NF. From the experiment paragraph at page 5 though I deduct that you instead employ “an isotropic standard normal prior over the weights”, i.e. \theta, thus I believe that you indeed have a problematic bound. How do you actually compute logq(\theta) when you employ the parametrisation discussed at 3.3? Did you use that parametrisation in every experiment?

Other than that, I believe that it would be interesting to experiment with a `full` hyper network, i.e. generating directly the entire parameter vector \theta, e.g. at the toy regression experiment where the dimensionality is small. This would then better illustrate the tradeoffs you make when you reduce the flexibility of the hyper-network to just outputting the row scaling variables and the effect this has at the posterior approximation.

Typos:
(1) Page 3, 3.1.1 log(\theta) -> logp(\theta).
(2) Eq. 6, it needs to be |det \frac{\partial h(\epsilon)}{\partial \epsilon}|^{-1} or |det \frac{\partial h^{-1}(\theta)}{\partial \theta}| for a valid change of variables formula.

[1] Louizos & Welling, Multiplicative Normalizing Flows for Variational Bayesian Neural Networks.

The evidence to be evaluated is:Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution).

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\).

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\).

Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below.

We only output the scaling factors \(g\) from the hypernet, and learn a maximum likelihood estimate of \(\mathbf{v}\).7 This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters.

Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim appears to be well-stated as it accurately describes the fundamental concept of Bayesian hypernetworks without exaggeration. The evidence directly supports this through mathematical formulations and descriptions, particularly in equations 5 and 6 which show how the hypernetwork h transforms the noise distribution. The figures demonstrate the practical implementation, showing how BHNs can learn multimodal distributions (Figure 2) and correlations between parameters (Figure 6). The reviewer's technical concerns about KL divergence, while valid, don't directly impact the accuracy of this definitional claim.

<score>1</score>

Justification:
1. The claim sticks to mathematical definitions without overstating capabilities
2. Evidence directly shows the transformation from noise distribution via equations 5-6
3. Figures validate that the described transformation can capture complex parameter distributions
4. The claim avoids making broader statements about performance or capabilities
5. While the reviewer raises implementation concerns, these don't affect the fundamental definition being claimed


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

A Bayesian hypernetwork \(h\) is a neural network which learns to transform a simple noise distribution, \(p(\mathbf{\epsilon})=\mathcal{N}(\mathbf{0},\mathbf{I})\), to a distribution \(q(\mathbf{\theta}):=q(h(\mathbf{\epsilon}))\) over the parameters \(\mathbf{\theta}\) of another neural network (the "primary network").

The review comment to be evaluated is:

This paper proposes Bayesian hypernetworks to carry out Bayesian learning of deep networks. The idea is to construct a generative model capable of approximating the posterior distribution over the parameters of deep networks. I think that the paper is well written and easy to follow.

I like the idea of constructing general approximation strategies for complex posterior distribution and the proposed approach inherits all the scalability properties of modern deep learning techniques. In this respect, I think that the paper tackles a timely topic and is interesting to read.

It is not entirely clear to me why the Authors name their proposal Bayesian hypernetworks. This seems to suggest that also the hypernetwork is infered using Bayesian inference, but if I understand correctly this is not the case.

I have some comments on novelty and realization of the experiments. In the positioning of the work in the literature, the Authors point out that hypernetworks have been proposed before, so it is not clear what is the actual novelty in the proposal. Is it the use of Real NVPs and IAFs as hypernetworks? These methods have been already proposed and extensively studied in the literature, and even if they have been adapted to be hypernetworks here, I believe that the novelty is fairly limited.

The experimental part is interesting as it explores a number of learning scenarios. However, I think that it would have been useful to add comparisons with standard variational inference (e.g., Graves, 2011) for deep networks to substantiate the claims that this approach underestimates uncertainty. I believe that this would strengthen the comparative evaluation.

I think the paper would have made a stronger case by including other approaches to approximate posteriors using generative models. For example, the variational Gaussian process paper sounds like an ideal method to include here.

[1] D. Tran, R. Ranganath, and D. M. Blei. Variational Gaussian process. arXiv preprint arXiv:1511.06499, 2015.

The evidence to be evaluated is:Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution).

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\).

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\).

Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below.

We only output the scaling factors \(g\) from the hypernet, and learn a maximum likelihood estimate of \(\mathbf{v}\).7 This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters.

Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me assess the claim:

The claim defines a Bayesian hypernetwork as a neural network that transforms a simple noise distribution to a distribution over parameters of another neural network. Based on the evidence:

1. The mathematical formulation and implementation details are thoroughly described in the paper, including equations (5), (6), and (10) that specify exactly how the transformation works.

2. The figures demonstrate that the BHN can indeed learn complex parameter distributions:
- Figure 2 shows multimodal posterior distributions
- Figure 6 confirms parameter dependencies
- Figure 1 illustrates the practical application

3. The reviewer's main criticism is about the naming convention rather than the technical definition.

The claim is precise, technically accurate, and supported by both theoretical foundations and empirical evidence. It avoids overstating capabilities while capturing the essential aspects of the method.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

A Bayesian hypernetwork \(h\) is a neural network which learns to transform a simple noise distribution, \(p(\mathbf{\epsilon})=\mathcal{N}(\mathbf{0},\mathbf{I})\), to a distribution \(q(\mathbf{\theta}):=q(h(\mathbf{\epsilon}))\) over the parameters \(\mathbf{\theta}\) of another neural network (the "primary network").

The review comment to be evaluated is:

* Edit: I increased my rating to 6. The authors fixed the first error I pointed out below. Regarding the second point: I still think it is possible to take a limit of sigma -> 0 in MNF, which makes the methods very similar.

The authors propose a new method of defining approximate posteriors for use in Bayesian neural networks. The idea of using hypernetworks for Bayesian inference is compelling, and the authors show some promising first results. I see two issues, and would be willing to increase my rating if these were sufficiently addressed.

- The paper says it uses an "isotropic standard normal prior on the weights of the network". However, the stochastic part of the generated weights (i.e. the scales) is of a lower dimension than the weights. It seems to me this means that the KL divergence between prior and posterior is undefined, or infinite, as the posterior is only defined on a sub-manifold. What exactly is the loss term that is added to the training objective? And how is this justified?

- The instantiation of Bayesian hypernetworks that is used in experiments seems to be a special case of the method of multiplicative normalizing flows as proposed by Louizos and Welling and discussed in this paper. If the variances / sigmas are zero in the latter method, their approximation seems functionally equivalent to Bayesian hypernetworks (though with different parameterization). Is my understanding correct? If so, the novelty of the proposed method is limited.

The evidence to be evaluated is:Bayes by Backprop can be viewed as a special instance of a Bayesian hypernet, where the hypernetwork only performs an element-wise scale and shift of the input noise (yielding a factorial Gaussian distribution).

A hypernetwork (Ha et al., 2017; Brabandere et al., 2016; Bertinetto et al., 2016) is a neural net that outputs parameters of another neural net (the "primary network").3 The hypernet and primary net together form a single model which is trained by backpropagation.

Our proposed Bayesian hypernetworks employ a differentiable directed generator network (DDGN) (Goodfellow et al., 2016) as a generative model of the primary net parameters. DDGNs use a neural net to transform simple noise (most commonly isotropic Gaussian) into samples from a complex distribution, and are a common component of modern deep generative models such as variational autoencoders (VAEs) (Kingma and Welling, 2013; Jimenez Rezende et al., 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014; Goodfellow, 2017).

Bayesian hypernets (BHNs) express a flexible \(q(\mathbf{\theta})\) by using a DDGN (section 2.3), \(h\in\mathbb{R}^{D}\to\mathbb{R}^{D}\), to transform random noise \(\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{D})\) into independent samples from \(q(\mathbf{\theta})\). This makes it cheap to compute Monte Carlo estimations of expectations with respect to \(q\); these include the ELBO, and its derivatives, which can be backpropagated to train the hypernet \(h\).

\[q(\mathbf{\theta})= \int q(\mathbf{\theta};h(\mathbf{\epsilon}))q(\mathbf{\epsilon})d\mathbf{ \epsilon}\,. \tag{5}\] To avoid this issue, we use an invertible \(h\), allowing us to compute \(q(\mathbf{\theta})\) simply by using the change of variables formula: \[q(\mathbf{\theta})=q_{\mathbf{\epsilon}}(\epsilon)\left|\det\frac{\partial h(\epsilon )}{\partial\mathbf{\epsilon}}\right|^{-1}\,, \tag{6}\] where \(q_{\mathbf{\epsilon}}\) is the distribution of \(\mathbf{\epsilon}\) and \(\mathbf{\theta}=h(\epsilon)\).

Note that the latter can be efficiently applied, since we only require the ability to evaluate likelihood of generated samples (not arbitrary points in the range of \(h\), as in generative modeling applications, e.g., Dinh et al. (2016)); and this also means that we can use a lower-dimensional \(\mathbf{\epsilon}\) to generate samples along a submanifold of the entire parameter space, as detailed below.

We only output the scaling factors \(g\) from the hypernet, and learn a maximum likelihood estimate of \(\mathbf{v}\).7 This allows us to overcome the computational limitations of naively-parametrized BHNs noted by Louizos & Welling (2017), since computation now scales linearly, instead of quadratically, in the number of primary net units. Using this parametrization restricts the family of approximate posteriors, but still allows for a high degree of multimodality and dependence between the parameters.

Next, we demonstrate the distinctive ability of Bayesian hypernets to learn multi-modal, dependent distributions. Figure 6 (appendix) shows that BHNs do learn approximate posteriors with dependence between different parameters, as measured by the Pearson correlation coefficient. Meanwhile, Figure 2 shows that BHNs are capable of learning multimodal posteriors.

We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters. BHNs feature efficient training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations. We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.

In this paper, we employ weight normalization in the primary network (7), treating (only) the scaling factors \(\mathbf{g}\) as random variables. We choose an isotropic Gaussian prior for \(\mathbf{g}\): \(p(\mathbf{g})=\mathcal{N}(\mathbf{g};\mathbf{0},\lambda\mathbf{I})\), which results in an \(L_{2}\) weight-decay penalty on \(\mathbf{g}\), or, equivalently, \(\mathbf{w}=\mathbf{g}\frac{\mathbf{v}}{||\mathbf{v}||_{2}}\). Our objective and lower bound are then: \[\log p(\mathcal{D};\mathbf{v},\mathbf{b}) =\log\int_{\mathbf{g}}p(\mathcal{D}|\mathbf{g};\mathbf{v},\mathbf{ b})p(\mathbf{g})d\mathbf{g} \tag{8}\] \[\geq\mathbb{E}_{q(\mathbf{g})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{g})]\] (9) \[\geq\mathbb{E}_{\mathbf{\epsilon}\sim q_{\mathbf{\epsilon}}(\mathbf{ \epsilon}),\mathbf{g}=h_{\phi}(\mathbf{\epsilon})}[\log p(\mathcal{D}|\mathbf{g}; \mathbf{v},\mathbf{b})+\log p(\mathbf{g})-\log q(\mathbf{\epsilon})+\log\left| \det\frac{\partial h_{\phi}(\mathbf{\epsilon})}{\partial\mathbf{\epsilon}}\right|] \tag{10}\] where \(\mathbf{v}\) and \(\mathbf{b}\) are the direction and bias parameters of the primary net, and \(\phi\) is the parameters of the hypernetwork. We optimize this bound with respect to \(\{\mathbf{v},\mathbf{b},\phi\}\) during training.

Input: Figure 2: Learning the identity function with an overparametrized network: ŷ = a · b · x. This parametrization results in symmetries shown by the dashed red lines, and the Bayesian hypernetwork assigns significant probability mass to both modes of the posterior (a = b = 1 and a = b = −1).

Input: Figure 6: Histogram of Pearson correlation coefficient p-values (left) and a scatter matrix (right) of samples from a hypernet approximate posterior. We see that the hypernet posterior includes correlations between different parameters. Many of the p-values of the Pearson correlation test are below .05.

Input: Figure 1: Illustration of BHNs (second and third) and a traditional non-Bayesian DNN (first) on the toy problem from Blundell et al. (2015). In the second subplot, we place a prior on the scaling factor g and infer the posterior distribution using a BHN, while in the third subplot the hypernet is used to generate the whole weight matrices of the primary net. Each shaded region represents half a standard deviation in the posterior on the predictive mean. The red crosses are 50 examples from the training dataset.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim appears to be well-stated and accurately reflects what Bayesian hypernetworks do. The evidence thoroughly demonstrates that:

1. The hypernetwork h indeed transforms noise ε from a normal distribution to parameter distributions θ
2. The mathematical formulation in equations 5-6 directly supports the claim
3. Figures 1 and 2 show concrete examples of the hypernetwork generating distributions over parameters
4. The reviewer's comments don't contradict the basic definition stated in the claim

The claim is precise and avoids overstating capabilities, focusing on the fundamental definition rather than performance claims. It aligns closely with the mathematical and empirical evidence presented.

<score>1</score>

Justification:
- The claim precisely matches the mathematical formulation in equations 5-6 showing how h transforms ε to θ
- Figures demonstrate the hypernetwork successfully generating parameter distributions
- Evidence shows detailed derivations supporting the transformation mechanism
- The claim avoids overstating capabilities or making unsupported generalizations
- Reviewer comments focus on specific implementation details rather than the fundamental definition
- The actual experimental results (Figures 1,2,6) validate that the network functions as described in the claim