Low-Budget Simulation-Based Inference with Bayesian Neural Networks

First, thank you for your review. We are happy to hear that you found the contribution broadly useful to the scientific community, sensible, novel and clever, that the paper mostly well-structured and that the experiments interesting and that the choice of metrics is appropriate.

We adress your concerns bellow.

Lack of method summary: There's no clear summary of the method, e.g. in the contributions section, an algorithm, or the start of section 3. A brief overview of 3-4 sentences would be very beneficial for readers. It's only towards the end of section 3.2, that the overarching method begins to come together on a first read. In my opinion, the abstract also does not include enough information on the method.

Thanks for pointing this out. We have added the following sentence at the beginning of section 3 to guide the reader. "In this section, we propose a novel SBI algorithm based on Bayesian neural networks with tuned prior on weights. The algorithm first optimizes a prior on weight to have desirable conservativeness properties. This tuned prior is then used to compute an approximate posterior on weights that is itself used to make predictions through Bayesian model averaging."

Lack of clear definitions: The concept of an "a priori-calibrated Bayesian model" should be defined more clearly when it is first introduced. I presume that an example of a well calibrated a priori model, would be one for which the Bayesian model average at initialization is equal to the simulator parameters prior $p(\theta)$ for any $x \ in \mathcal{X}$? Presumably this includes a large variety of useful and non-useful models for modelling epistemic uncertainty (as suggested in equation 8).

An a-priori calibrated Bayesian model is defined as a prior over weights that lead to a calibrated Bayesian model average. If this Bayesian model average is the prior $p(\theta)$ for all $x$, this is indeed a calibrated model. We added the following sentence to make things clearer: "This means that the Bayesian model average computed using the prior normal on weights $p(w)$ is not calibrated."

Figures: The credibility figure in Figure 1 is confusing, either the standard deviations are not in order, or there is a typo one of the standard deviations. Figure 3 should have a labeled legend. Many of the font sizes are too small.

Indeed, there are two standard deviations that are not in order, this has been changed. We also added the meaning of the legend of Figure 3 in the caption.

Limited experimental runs: Only 3 runs are performed for each experiment, and only the median is reported. This makes it hard to assess if differences between methods are significant. Repeating with more runs would be beneficial, although I am sympathetic to limitations in computational budget.

We agree that more runs would have been preferable. However, we estimated the computational cost of 3 runs to be approximately 25,000 GPU hours. This is because each point in the reported curves corresponds to 3 full training and testing procedures, and testing with Bayesian neural networks involves the approximation of the Bayesian model average for each test sample. Unfortunately, we do not have the computational power to do many more runs. That being said, we see from Figure 2 that all the methods but BNN-NPE show a coverage AUC below -0.1 for low simulation budgets on at least one benchmark. BNN-NPE shows only positive or very slightly negative coverage AUC. We believe we can then conclude that using BNN improves the conservativeness even if we have only 3 runs.

Why is the convergence to the NPE solution so slow in terms of the number of simulations? Can the prior be altered to avoid this problem, rather than introducing a temperature parameter?

The prior we define is the reason why we obtain calibrated approximations. Altering the prior would then probably negatively affect the coverage results, as seen in Figure 1. Using a temperature that is too low could also affect the coverage results; it should be tuned to have the appropriate trade-off.

Overall, we think we have addressed each concern raised in your review and implemented changes accordingly. While these improvements appear to resolve the identified issues, we'd value your perspective on whether any aspects still fall short of expectations. If the changes adequately address your concerns, we'd appreciate having this reflected in the evaluation.

As a potential user, I would find it very worrisome that BNN-NRE has significantly lower nominal log-posterior than standard methods. Indeed, for some tasks, it seems to require 5 orders of magnitude (for many other tasks 3 orders of magnitude) more simulations than standard methods.

It depends on what kind of application one is working on and what we are trying to achieve. This work follows the philosophy highlighted by Hermans et al., 2022, where they argue that to be useful for scientific application, posterior approximations must be conservative before anything else. If that is not the case, then they cannot be used reliably for downstream scientific tasks. In this work, the main objective is to build SBI methods that are conservative for low simulation budgets. In some settings, people might indeed not care that much about conservativeness and focus on statistical power. Our work does not aim to address this setting, and in those cases, other methods might indeed be preferred. We would also like to highlight that we focus on conservative simulation-based inference in the low simulation budget regime, a regime where other methods fall short. For large datasets, others methods like standard or balanced NPE/NRE might be prefered.

Joeri Hermans, Arnaud Delaunoy, Francois Rozet, Antoine Wehenkel, Volodimir Begy, and Gilles Louppe. A crisis in simulation-based inference? beware, your posterior approximations can be unfaithful. Transactions on Machine Learning Research, 2022

Overall, we think we have addressed each concern raised in your review and implemented changes accordingly. While these improvements appear to resolve the identified issues, we'd value your perspective on whether any aspects still fall short of expectations. If the changes adequately address your concerns, we'd appreciate having this reflected in the evaluation.

First, thank you for your review. We are happy to hear that you found the contribution novel, interesting, potentially impactful, elegant and rigorous.

We adress your concerns bellow.

The paper overstates its claims. My main issue with this paper is that it overstates its claims and does not acknowledge the weakness of empirical results. Looking at Figure 2, and in particular for NRE: BNN-NRE never reaches the log posterior of the other methods (even for 1M simulations). Yet the authors state that the nominal log posterior density is on par with other methods for very high simulation budgets. Where does this claim come from. Similarly, in Figure 2 (NRE), the authors do not acknowledge that the AUC is not particularly much higher for BNN-NRE as compared to NRE, even for 10 simulations. Indeed, for three of the four tasks NRE has a higher coverage AUC than BNN-NRE for 10 simulations. On top of this, for many tasks and simulation budgets, BNN-NRE does have a negative AUC, but the authors simply claim that they show positive coverage AUC. Finally,

We totally agree that our method has limitations. We have tried to be as transparent as possible regarding those limitations by including a limitation paragraph at the end of the conclusion highlighting all the mentioned limitations. The differences between BNN-NPE and BNN-NRE are also highlighted in lines 416-418. We understand that some sentences might sound like overstatements, and we have now modified those. We have modified the sentence about log posterior density as follows: "We also observe that the nominal log posterior density is on par with other methods for very high simulation budgets, on most benchmarks and with an appropriate temperature, but that more samples are required to achieve high values." Regarding the coverage AUC of NRE and BNN-NRE, in our opinion, BNN-NRE has better coverage values. We observe that NRE reaches coverage AUC values of -0.05 on SLCP, -0.1 on Two Moons, almost -0.1 on Lotka-Volterra, and almost -0.05 on Statial SIR. In opposition, BNN-NRE shows coverage AUC that is at worse, around -0.03, and is much more often positive. The coverage curves of BNN-NRE are above NRE at least 90% of the time. We agree that stating that BNN-NRE always shows positive coverage is, strictly speaking, wrong. We have updated the claim as follows: "Figure 2 compares simulation-based inference methods with and without accounting for computational uncertainty. We observe that BNNs equipped with our prior and without temperature show positive, or only slightly negative, coverage AUC even for simulation budgets as low as $O(10)$."

The empirical results are weak.

We agree that our method shows empirical limitations as it requires higher simulation budgets to reach similar log posterior density. However, the empirical results clearly show that using Bayesian neural networks with our prior leads to better coverage AUC. All other methods are shown to have dangerously low coverage AUC in low simulation budget regime. We lose in statistical power to avoid risking false inference, which is the main objective of this work.

Second, to me, the empirical results are difficult to interpret. (A) Many of the curves shown in Figure 2 are very noisy and irregular. It might be beneficial to average across more seeds to observe clear trends.

We agree that more runs would have been preferable. However, we estimated the computational cost of 3 runs to be approximately 25,000 GPU hours. This is because each point in the reported curves corresponds to 3 full training and testing procedures, and testing with Bayesian neural networks involves the approximation of the Bayesian model average for each test sample. Unfortunately, we do not have the computational power to do many more runs. That being said, as mentioned above, even if the curves are a bit noisy, they are sufficient to validate the fact that using Bayesian neural networks with our prior leads to better coverage AUC.

The results in Figure 3 seem to indicate that NPE is indeed outperforming BNN-NPE here. Would the authors argue that BNN-NPE is preferable over NPE for this task?

If we have a close look to what happens for a budget of 256, we observe that BNN-NPE takes into account the epistemic uncertainty and provide a large posterior. However, NPE provides a narrower posterior with its mass centered at the wrong place. NPE is then confident and wrong, while BNN-NPE is not overconfident. This behavior of NPE is what we are trying to avoid with BNN-NPE, and it works for that example.

What do the numbers in the legend in Figure 3 mean?

They are the simulation budgets. We have updated the caption to make it clearer.

Overall, we think we have addressed each concern raised in your review and implemented changes accordingly. While these improvements appear to resolve the identified issues, we'd value your perspective on whether any aspects still fall short of expectations. If the changes adequately address your concerns, we'd appreciate having this reflected in the evaluation.

First, thank you for your review. We are happy to hear that you found the contribution novel, interesting, and very relevant to the community, that the motivation and derivation of the tailored prior is particularly well done and convincing, that the method seems to empirically achieve the motivated goal and that the paper is well-written and easy to follow. We address your concerns below.

The authors evaluate their method with the expected coverage (EC). In Figure 2, it is in my opinion very difficult to draw conclusions which methods works best. This is likely due to the fact that only 3 runs have been evaluated. Given the complexity of SBI and the high variance of the inferential results, I think they should do at least 10 evaluations and report these.

We agree that more runs would have been preferable. However, we estimated the computational cost of 3 runs to be approximately 25,000 GPU hours. This is due to the fact that each point in the reported curves corresponds to 3 full training and testing procedures and that testing with Bayesian neural networks involves the approximation of the Bayesian model average for each test sample. Unfortunately, we do not have the computational power to do many more runs. That being said, we see from Figure 2 that all the methods but BNN-NPE show a coverage AUC below -0.1 for low simulation budgets on at least one benchmark. BNN-NPE shows only positive or very slightly negative coverage AUC. We believe we can then conclude that using BNN improves the conservativeness even if we have only 3 runs.

The authors use as a second evaluation metric the expected posterior log density (EPLD). It is not clear what a good value should look like here or even if a high log density should be desired. is to have a conservative posterior, is a extremely high EPLD a good measure? The authors should, despite possible drawbacks, have used some divergence to the true posterior, such as MMD, in addition to the other two and report this.

To evaluate the conservativeness of a posterior approximation, coverage is the only relevant measure. We reported EPLD to compare the methods as to how close they are to the real posterior. EPLD is actually the training objective of NPE methods and can be derived from the expected KL divergence between the approximate posterior and true posterior.

$w^* = \arg \min_w \mathbb{E}_{p(x)} \left[ KL\left[ p(\theta | x) || \hat{p}_w(\theta | x) \right] \right]$

$w^* = \arg \min_w \mathbb{E}_{p(\theta, x)} \left[ \frac{p(\theta | x) }{\hat{p}_w(\theta | x)}\right]$

$w^* = \arg \max_w \mathbb{E}_{p(\theta, x)} \left[\hat{p}_w(\theta | x)\right]$

We indeed do not know what a good EPLD value is, but we know that a method with high EPLD is closer to the real posterior than one with low EPLD, according to the KL divergence.

It is not clear how crucial hyperparameters such as the temperature T or the covariance functions of the reference GP are chosen in practice.

If you are referring to the temperature $T$ applied on the prior during the posterior approximation, we have given guidelines on lines 408-409. If you are referring to the parameters variance and lengthscale of the kernel, those are indeed hard to choose. The lengthscale is chosen to be a quantile of observed distances when taking two samples from the measurement set. The variance should be guided by domain knowledge as to how close the posterior is expected to be from the prior. In this setting, the priors are uniform and the prior density is then equal on all the parameter space. We have set the standard deviation to half that density value but more refined choices could be made with domain knowledge. This is explained in lines 686-692 in Appendix A.

The authors propose to use cold posteriors, e.g., using a temperature of T=0.01. The motivation of this is not clear, given that the entire idea is to use a prior that enforces calibration. Why would it be desirable to in fact reduce the impact of the prior?

Indeed, this is counterintuitive. To obtain conservative approximations, it is better not to use a temperature. However, we have observed that in some cases, we were not able to reach satisfactory EPLD values and proposed using temperature as a way to set a trade-off between conservativeness and EPLD values. Note that tempering the prior is something that is often done in practice in Bayesian deep learning.

Figures 2 should show the standard errors.

They are shown in Figure 11. We didn't include those in Figure 2 because it makes the figure hard to read.

Figure 3 should show the reference posterior and not only the true parameter value.

The reference posterior is unfortunately unknown, but we can assume that NPE with a simulation budget of 1 million samples should not be far from the reference posterior.

Thank you for addressing my comments.

We agree that more runs would have been preferable. However, we estimated the computational cost of 3 runs to be approximately 25,000 GPU hours.

I understand that and agree that running SBI experiments is very costly. Given that the results of the paper are, however, of empirical nature and the fact that all reviewers seem to agree, I feel like the results are not conclusive enough with 3 replications. As has been pointed out in another review, I believe limiting the simulation budget to $10^4$ in lieu of more experiments, could make the manuscript more convincing. (Further, given that the paper proposes a solution for low-budget SBI, I find the argument that the BNNs need to compute costly model averages a bit antithetic).

After having read the other reviews, I will leave my score as is.

the experimental results are difficult to interpret because they show the media of only three repetitions. More repetitions and error bars would be better here.

We agree that more runs would have been preferable. However, we estimated the computational cost of 3 runs to be approximately 25,000 GPU hours. This is due to the fact that each point in the reported curves corresponds to 3 full training and testing procedures and that testing with Bayesian neural networks involves the approximation of the Bayesian model average for each test sample. Unfortunately, we do not have the computational power to do many more runs. That being said, although a bit noisy, we belive that those 3 runs are sufficient to demonstrate that using BNNs with our prior leads to better coverages in the low-simulation regime, which is our main claim.

the high underconfidence of the BNN approach in how-data regimes is concerning. Additional evaluation of the posterior predictive distributions would be appropriate.

We would argue that it depends on what we are trying to achieve. This work follows the philosophy highlighted by Hermans et al., 2022, where they argue that to be useful for scientific application, posterior approximations must be conservative before anything else. If that is not the case, then they cannot be used reliably for downstream scientific tasks. In this work, the main objective is to build SBI methods that are conservative for low simulation budgets. Of course, we still wish to maintain reasonable statistical power but only if conservativeness is observed in the first place. In that regard, overconfidence is a much bigger issue than underconfidence and we mainly aim to avoid overconfidence. We agree that in some settings, people might not care that much about conservativeness and focus on statistical power. Our work does not aim to address this setting and other methods might be preferred for such applications.

Joeri Hermans, Arnaud Delaunoy, Francois Rozet, Antoine Wehenkel, Volodimir Begy, and Gilles Louppe. A crisis in simulation-based inference? beware, your posterior approximations can be unfaithful. Transactions on Machine Learning Research, 2022

the choice and construction of the prior from simulated should be explained more clearly. A better explanation ideally will resolve the questions on the general procedure and on Figure 1 above.

We hope our clarifications above made everything clearer. We dedicated an entire page (section 3.2) to this. Could you point out specific elements that would be interesting to add or clarify in your opinion?

NPE / NRE ensembles seems to perform similarly well compared to the BNN approach. How does it perform on the cosmology example? How does it compare to the BNN approach in terms of computational budget? Do you think ensembles could be a good alternative when computational budgets are limited? What are the disadvantages of ensembles compared to BNN for SBI?

More generally, what is the computational complexity of the BNN approach compared to NPE (ensembles)?

From Figure 2, we can conclude that they do not perform similarly well, as the NPE ensemble has bad coverage for low simulation budgets, which is the setting of interest in this work, on all the benchmarks. BNN-NPE does not show the same behavior. We believe that the same observations would be made on the cosmological examples as it has been observed on all the benchmarks. In terms of computational budget, ensembles need more computational power to train than Bayesian neural networks with variational inference. For an ensemble of $N$ members, $N$ training procedure have to be performed, while for BNNs, only one training procedure outputing the posterior on weights has to be performed. At inference time, approximating the Bayes model average requires $M$ neural network evaluation if $M$ neural networks are used for the Monte-Carlo approximation. The two methods are then equivalent on that point if $M=N$.

Overall, we think we have addressed each concern raised in your review and implemented changes accordingly. While these improvements appear to resolve the identified issues, we'd value your perspective on whether any aspects still fall short of expectations. If the changes adequately address your concerns, we'd appreciate having this reflected in the evaluation.

First, thank you for your review. We are happy to hear that you found the contributions valuable and that it addresses a timely problem, that you think that the technical contributions are sound and that the experimental results support the claims, and that you found the paper well-written and clearly structured. We address your concerns below.

The only paper with a related approach that seems to be missing in the discussion is Lueckmann et al. 2017, section 2.2, who propose performing SVI on the neural network weights for continual learning across SNPE rounds. Also, it seems a bit odd to me that the benchmark used are not cited (except for the spatial SIR by Hermans et al.). The SLCP benchmark was introduced by Papamakarios et al. (SNLE paper), two moons by Greenberg et al.; and the overall set of SBI benchmarking tasks was introduced in Lueckmann et al. 2021.

Thanks for pointing it out. We were not aware of the work by Lueckmann et al. (2017), and we now discuss it in the introduction. The benchmarks are appropriately cited in Appendix B.

In section 3, especially in section 3.2, it should be explained more clearly why and how the prior has to be tuned to be used for the BNN approach. In the classical SBI setting, the prior is usually set a priori using expert knowledge. This step usually does not involve using simulated data for an optimization procedure. In the BNN approach, it seems that the prior has to be tuned with actual simulations (e.g., lines 254-261). Do these simulations have to be run in addition to the training data? Are they accounted for the in the budgets in the benchmarks?

Do these simulations for prior tuning have to be run in addition to the training data? Are they accounted for the in the budgets in the benchmarks?

First, we would like to clarify what kind of prior we are talking about. In the classical SBI setting, a prior over the simulator's parameters $p(\theta)$ is set a priori using expert knowledge. From that, an approximate posterior model based on a neural network with fixed weights $w$ is computed $p(\theta | x, w)$. In this paper, we are also using a prior over simulator's parameters $p(\theta)$ that is set using expert knowledge and not learned or optimized in any way. However, we use a Bayesian neural network as an approximate model, and the approximate posterior becomes $\hat{p}(\theta | x) = \int p(\theta | x, w) p(w | D) dw$. The term $p(w | D)$ requires the introduction of a prior $p(w)$ that is, this time, over the neural network's weight and not the simulator's parameters. We optimize this prior to match the behavior of a Gaussian process in the functional domain. We mention in lines 258-265 several ways to construct a measurement set for the optimization. In the experiments, we only use simulations to derive the support of the distribution, and the simulations used are the same as the training simulations. No additional simulations are then used, and all the simulations that are used are taken into account in the simulation budget.

Figure 1 needs a couple of clarifications. The caption says it's a visualization of the tuned prior. Then, the next sentence says the left panels show posterior functions sampled from the tuned prior prior over the neural network's weights. How are these samples obtained and what are thet supposed to show? Are they obtained before or after SBI training? Similarly question for the last panel: is the calibration calculated after the full SBI training or only after the prior tuning?

Again, the confusion might come from the fact that there are two types of priors used in this work. This figure shows a visualization of the tuned prior over weight $p(w|\phi)$. To show this prior, we plot the approximate posterior function $p(\theta | x, w)$ with $w \sim p(w)$. This is then before SBI training, as during SBI training, we compute the posterior over weights $p(w | D, \phi)$. Approximate posterior samples after SBI training would then be $p(\theta | x, w), w \sim p(w | D, \phi)$. We added some precisions in the caption to avoid any confusion.

In general, it seems that a better explanation of the steps involved in setting up and training the BNN-NPE (NRE) approach. I suggest adding more details in the next and adding an algorithm scheme in the text or appendix.

We added a few sentences at the beginning of Section 3 indicating the different steps involved. There are three main steps. The first step is to optimize an appropriate prior $p(w | \phi)$. The second step is to use that prior to compute a posterior over weights $p(w | \phi, D)$ and the third step is to use that posterior over weight to compute an approximate posterior over simulator's parameters $\hat{p}(\theta | x) = \int p(\theta | x, w) p(w | D, \phi) dw$.

Thank you for the additional clarifications. I will take your responses into account for the later discussion among the reviewers and AC.

In the meantime, if you really cannot run additional experiments, could you please show the results with SEM error bars (instead of median)? This will provide more transparency regarding the empirical results - thanks!

Here are the plots with the SEM errors plotted along the mean https://imgur.com/a/AZ1aots . We also invite you to see Figures 11 and 12, which contain similar information as they show the 3 runs and, hence, the minimum, median, and maximum values obtained.

Thanks for engaging in the discussion and taking our responses into account. If that improved your opinion about the paper, could you reflect this in your evaluation?