GATSBI: Generative Adversarial Training for Simulation-Based Inference
Authors: Poornima Ramesh
Jan-Matthis Lueckmann
Jan Boelts
Álvaro Tejero-Cantero
David S. Greenberg
Pedro J. Goncalves
Jakob H. Macke
Venue: ICLR 2022 Poster (2022)
Keywords: Machine Learning
simulation-based inference
generative adversarial networks
approximate bayesian computation
data-driven modelling
GANs
SBI
likelihood-free inference
implicit models

Abstract: Simulation-based inference (SBI) refers to statistical inference on stochastic models for which we can generate samples, but not compute likelihoods.
Like SBI algorithms, generative adversarial networks (GANs) do not require explicit likelihoods. We study the relationship between SBI and GANs, and introduce GATSBI, an adversarial approach to SBI. GATSBI reformulates the variational objective in an adversarial setting to learn implicit posterior distributions. Inference with GATSBI is amortised across observations, works in high-dimensional posterior spaces and supports implicit priors. We evaluate GATSBI on two common SBI benchmark problems and on two high-dimensional simulators. On a model for wave propagation on the surface of a shallow water body, we show that GATSBI can return well-calibrated posterior estimates even in high dimensions. 
On a model of camera optics, it infers a high-dimensional posterior given an implicit prior, and performs better than a
state-of-the-art SBI approach. We also show how GATSBI can be extended to perform sequential posterior estimation to focus on individual observations.
Overall, GATSBI opens up opportunities for leveraging advances in GANs to perform Bayesian inference on high-dimensional simulation-based models.

Published as a conference paper at ICLR 2022
GATSBI: GENERATIVE ADVERSARIAL TRAINING FOR SIMULATION-BASE INFERENCE
Poornima Ramesh University of Tübingen
Jan-Matthis Lueckmann
Jan Boelts TU Munich
Álvaro Tejero-Cantero University of Tübingen
University of Tübingen
David S. Greenberg Helmholtz Centre Hereon
Pedro J. Gonçalves University of Tübingen
Jakob H. Macke
University of Tübingen
ABSTRACT
Simulation-base inference (SBI) refers to statistical inference stochastic models for which we can generate samples, but not compute likelihoods. Like SBI algo- rithms, generative adversarial networks (GANs) do not require explicit likelihoods. We study the relationship between SBI and GANs, and introduce GATSBI, an adversarial approach to SBI. GATSBI reformulates the variational objective in an adversaria setting to learn implicit posterior distributions. Inference with GATSBI is amortised across observations, works in high-dimensional posterior spaces and supports implicit priors. We evaluate GATSBI on two SBI benchmark problems and on two high-diman dige-dimensional simulators. On a model for wave propagation on the surface of shallow water body, show that GATSBI can return cell-calibrated posterior estimates even in high dimensions. On a model of camera optics, it infers high-dimensional posterior given implicit prior, and performs better than state-of-the-art SBI approach. We also show how GATSBI can be extended to perform sequential posterior estimation focus on individual observations. Over- all, GATSBI opens up opportunities for leveraging advances in GANs to perform Bayesian inference on high-dimensional simulation based models.
INTRODUCTION
Hypothesis-making in many scientific disciplines relies on stochastic simulators pont-unike ex- pressive statistical models such as neural networks have domain-relevant, interpretable parameters. Finding the parameters 0 of a simulator that reproduce the observed data To constitutes inverse problem. In order to use these simulators to formulate further hypotheses and experiments, one needs to obtain uncertainty estimates for the parameters and allow for candidate parameters accounting for the same observation. These requirements are met by Bayesian inference, which attempts approximate the posterior distribution p(0)x..). While variety of techniques exist to calculate posteriors for scientific simulators which allow explicit likelihood calcu- lations (()) inference on black-box simulators with intractable likelihoods a.k.a. 'simulation- based inference' (Cranmer et I. 2020).. poses substantial challenges. In traditional, so-called Approximate Bayesian Computation (ABC) approaches SBI (Beaumont al. 2002] 2009; (009) MMajjoram et al. 20033] Sisson et al. (207),, one samples parameters 0 from prior (8) and accepts only those parameters for which the simulation output x p(x:) 0) close to the observation d(x,, xo) < c. With increasing data dimensionality N, this approach incurs an exponentially growing simulation expense ('curse of
i.e., different
dimensionalityy) ); also requires suitable choices of distance function d and acceptance threshold
Recent SBI algorithms (e.g., Greenberg et al.) 2019; Gutmann and Corander, 2015 Hermans et al.)
2020 Lueckmann et al. 2017 Meeds and Welling 2014] Papamakarios and Murray 2016) Radey et al. 2020 Thomas et al. (0221) 1 draw on advances in machine learning and are often based on Gaussian Processes (Rasmussen and Williams 2006) or neural networks for density estimation (e.g.
using normalizing flows; Papamakarios et al. 2021). While this has led to substantial improvements over classical approaches, and numerous applications in fields such as cosmology (Cole et al.) (021). neuroscience (Gonçalves al. 2020) robotics (Muratore al. 2021), statistical inference for

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prior
simulations
generator
approximate posterior
.T(O)
x- p(xx0)
folz.x)
golllra)
discriminator
D₂(O.x)
Figure I: GATSBI uses conditional generative adversarial network (GAN) for simulation- based inference (SBI). We sample parameters from a prior (0), and use them to generate synthetic data from black-box simulator. The GAN generator learns implicit approximate posterior qa (0]zc), i.e., it learns to generate posterior samples O' given data x. The discriminator is trained differentiate between l' and conditioned on T.
high-dimensional parameter spaces is an open challenge (Cranmer et al. 202T:: This prevents the application of these methods many real-world scientific simulators.
2020 Lueckmann et al.
In machine learning, generative adversarial networks (GANs, (Fooffllode et al. 2014) have emerged as powerful tools for learning generative models of nigh-dimensional data, for instance, images (Isola ee al. 22166] Radford et al. 2013: Zhu et al. 2017). Using minimax game between a 'generator" and "discriminator"" GANs can generate samples which look uncannily similar the data they have been trained on. Like SBI algorithms, adversarial training is 'likeihhodd-fre" i.e., it does not require explicit evaluation of a density function, and also relies on comparing empirical and simulated data. This raises the question of whether GANs and SBI solve the same problem, and in particular whether and how adversarial training can help scale SBI to high-dimensional regimes.
First, show how one can use adversarial training to learn an implicit posterior density for SBI. We refer to this approach as GATSBI (Generative Adversarial Training for SBI Second, we show that GATSBI can be formulated as an adversarial variational inference algorithm (Huszár) 2017 Makhzani et al. 2015 Mescheder et al. 2017), thus opening up potential area of exploration for SBI approaches. Third, we show how it can be extended to refine posterior estimates for specific observations, i.e., sequential posterior estimation. Fourth, two low-dimensional SBI benchmark problems,
Here, address these questions and make the following contributions:
show that GATSBI's performance is comparable to common SBI algorithms. Fifth, we illustrate GATSBI on high-dimensional inference problems which are of reach for most SBI methods: In fluid dynam- ics problem from earth science, we learn an implicit posterior over 100 parameters. Furthermore, we show that GATSBI (in contrast to most SBI algorithms) can be used on problems in which the prior is only specified implicitly: by recasting over (784-dimensional) images. Finally, we discuss the opportunities and limitations of GATSBI relative to current SBI algorithms, and chart out avenues for future work.
as an SBI problem, we learn posterior
METHODS
2.1 SIMULATION-BASED INFERENCE AND ADVERSARIAL NETWORKS
Simulation- inference (SBI) targets stochastic simulators with parameters & that we can use to generate samples æ. many scientific applications, simulators are 'black-box': we cannot evaluate the likelihood p(z (), take derivatives through the simulator, or access its internal random numbers. The goal of SBI is to infer the parameters 6, given empirically observed data x, i.c., to obtain the posterior distribution p(0)æ.). In most applications of existing SBI algorithms, the simulator is mechanistic forward model depending on low (~ 10) number of parameters
Note that the name is shared by recently proposed, and entirely unrelated, algorithms (Min et al.
2021
Yu
2020
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Generative adversarial networks (GANs) (Goodfellow et al. 2014) consist of two networks trained adversarially. The generator f2(2) produces samples E from latent variables 2; the discriminator De acts as a similarity measure between the distribution of generated and observations 2, from a distribution (r.) The two networks are pitted against each other: the generator attempts to produce samples that would fool the discriminator; the discriminator is trained to tell apart generated and ground-truth samples. This takes the form of a minimax game ming maxo L(ø, eb) between the networks, with entss-yn as the value function:
L(ó, v) = (x.) log (x(r)) + Epts) log(1 - (f
At the end of training, the networks are sufficiently expressive, samples from fg are hard to distin- guish from the ground-truth samples, and the generator approximates the observed data distribution (oo) GANs can also be used to perform (implicit) conditional density estimation, by making the generator and discriminator conditional on external covariates / (Mirza and Osindero 2014). Note that the GAN formulation does not evaluate the intractable distribution Pdata (2..) or Pdata ( (4) at any point. Thus, GANs and SBI appear be solving similar problems: They are both 'likelihood-free' i.c., they do not evaluate the target distribution, but rather exclusively work with samples from it.
Nevertheless, GANS and SBI differ in multiple ways: GANs use neural networks, and thus are always differentiable, whereas the numerical simulators in SBI may not be. Furthermore, the latent variables for the GAN generator are typically accessible, while in SBI, one may not have access to the simulator's s internal random variables. Finally, in GANs, the goal is to match the distribution of the generated samples x to that of the observed data fo: and parameters of the generator are learnt as point estimates. In contrast, the goal of SBI is to learn distribution over parameters 0 of a simulator consistent with both the observed data To and prior knowledge about the parameters. However, conditional GANs can be used to perform conditional density estimation. How can repurpose them for SBI? We propose an approach using adversarial training for SBI, by employing GANs as implicit density estimators rather than generative models of data.
2.2 GENERATIVE ADVERSARIAL TRAINING FOR SBI: GATSBI
Our approach leverages conditional GANs for SBI. We train a deep generative neural network fa with parameters Φ, i.e., takes as inputs observations and noise z from fixed source with distribution p(z), so that f0, 2) deterministically transforms z to generate 0. This generative network f is an implicit approximation (0) x) of the posterior distribution, i.e..
}
p(z) fo(, x)
0 0~ (0)z)
(1)
We train f adversarially: We sample many parameters 0 from the prior, and for each 0, simulate æ p(x[0) i.e., we sample tuples x0,z) ~ T(0)p(z(0) = p(0)=)p(z) These samples from the joint constitute training data. We define discriminator D. parameterised by s, to differentiate between samples 6 drawn from the approximate posterior g (0)x) and the true posterior p(f]x). The discriminator is conditioned on æ. We calculate the cross-entropy of the discriminator outputs, and maximize it with respect to the discriminator parameters vo and minimize it with respect to the generator parameters @ i.e., minş maxş L(ő, 1)), with
L( vb) = E (())(c=1)))) log (0, æ) + log (1 - (f
Ep(x) [[p(ot=) log (0, x)) + Eg(ols) ( [log (1 D (0, ]]]]]
(2)
This conditional GAN value function targets the desired posterior p(0)x:):
Proposition 1. Given fixed generator for the discriminator De- maximizing equation 2 satisfies
p(0|z)
D Dw (8, p(f x) + (0)x))
and the corresponding loss function for the generator parameters is the Jensen- divergence (JSD) beiween the true and approximate posterior,
Ly- 2.1SD((((01=))laa000))))) log 4.
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Using sufficiently flexible networks, the GATSBI generator trained with the cross-entropy loss converges in the limit of infinitely many samples to the true posterior, since the JSD is minimised if and only if (0)x)) (2014) and is given in detail in App. GATSBI thus performs amortised inference, i.e., learns a single inference network which can be used perform inference rapidly and efficiently for wide range of potential observations, without having to re-train the network for each new observation. The training steps for GATSBI are described in App. B
p(0)xz). The proof directly follows Prop. 1 and 2 from Goodfellow et al.
2.3 RELATING GATSBI TO EXISTING SBI APPROACHES
There have been previous attempts at using GANs to fit black-box scientific simulators to empirical data. However, unlike GATSBI, these approaches do not perform Bayesian inference: Kim et al. (2020) and Louppe al. (0117) train discriminator to differentiate between samples from the black- box simulator and empirical observations. They use the discriminator to adjust a proposal distribution for simulator parameters until the resulting simulated data and empirical data match. However, neither approach returns a posterior distribution representing the balance between a prior distribution and fit empirical data. Similarly, Jethava and Dubhashi (2017) use two generator networks, one to approximate the prior and one to learn summary statistics, as well as two discriminators. This scheme yields a generator for parameters leading realistic simulations, but does not return a posterior over parameters. Parikh et al. (2020) train a conditional GAN (c-GAN), as well as two additional GANs (r-GAN and G-GA within an active learning scheme, in order to solve 'population of models' problems (Britton et al. 2013 Lawson et al. (018)). While c-GAN exhibits similarities with GATSBI, the method does not do Bayesian inference, but rather solves a constrained optimization problem. Adler and Öktem (2018) use Wasserstein-GAN (Arjovsky et al. (2017) to solve Bayesian inverse problems in medical imaging. While their set-up is similar to GATSBI, the Wasserstein metric imposes stronger conditions for the generator to converge the true posterior (App. Sec. A.3).
Several SBI approaches train discriminators to circumvent evaluation of the intractable likelihood (e.g... Cranmer
Gr..merr et al. 2015 Gutmann al. 2018 Pham et al. [2014] Thomas et al. 2021). Hermans
et al. (2020) train density-ratio estimators by discriminating samples (6, x) ~ T (()p((z()) from
(6, 2) . (())p(=), which be used to sample from the posterior. Unlike GATSBI, this requires
a potentially expensive step of MCMC sampling. A closely related approach (Durkan et al.) 2020
Greenberg al. 2019) directly targets the posterior, but requires density estimator that can be evaluated- unlike GATSBI, where the only restriction on the generator is that it remains differentiable.
Finally, ABC and synthetic likelihood methods have been developed to tackle problems in high- dimensional parameter spaces (> 100-dimensions) although these require model assumptions re- garding the Gaussianity of the likelihood (Ong al. 2018), or low-dimensional summary statisties (Kousathanas et al. 2015 Rodrigues et al. 2019). in addition to MCMC sampling.
2.4 RELATION TO ADVERSARIAL INFERENCE
GATSBI reveals a direct connection between SBI and adversarial variational inference. Density estimation approaches for SBI usually use the forward Kullback-Leibler divergence (DKL). The reverse DKL can offer complementary advantages (e.g. by promoting mode-seeking rather than mode- covering (Turner and Sahani 201T)), ), but is hard to compute and optimise for conventional density estimation approaches. We here show how GATSBI, by using an approach similar to adversarial variational inference, performs optimization that also finds minimum of the reverse DKL.-
First, we note that density estimation approaches for SBI involve maximising the approximate log posterior over samples simulated from the prior. This is equivalent to minimising the forward DKL:
Lewa (d) = Ep(x)
= - -E(())p(01)) log go (0]z) + Ep(x))(((=) logp(0) x). (4)
rather than reverse DKL.:
Lee (0) = Ep(z)) DEL (ea)) = (p())000)2)) log (0(())))
(5)
p(0)x)
This is because it is feasible to compute Liwd with an intractable likelihood: it only involves evaluating the approximate posterior under samples from the true joint distribution p(0, x)) = p(x=0)=((8 The
problem with the reverse DKI., however, is that it requires evaluating the true posterior p(flzr) under
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samples from the approximate posterior (=(x2) which is impossible with intractable likelihood. This is a problem also shared by variational--utoencoooee (VAEs, Kingma and Welling 2013) but which can be solved using adversarial inference.
VAE typically consists of an encoder network qa (u/z) (u(z;) approximating the posterior over latents u given data æ, and decoder network approximating the likelihood pa (a/lu. The parameters & and of the two networks are optimised by maximising the Evidence Lower Bound (ELBO):
ELBO +) - ppt))) [DRI. (I(x)) + E (ule)) log Po (x/[x)]
Lo (o(u(z)p(()) + const.
- --p(s)ga(ula) log
p( (x/u)
Das (u (x/u)) + const.
Note that the DKL. is minimised when maximising the ELBO. If any of the distributions used to compute the ratio in the DKL. in equation 66 is intractable, one can do adversarial inference, i... the ratio can be approximated using a discriminator (see Prop. I and (0(u)z) is recast as the generator. Various adversarial inference approaches approximate different ratios, depending on which of the distributions intractable (see App. Table T for a non-exhaustive comparison of different discriminators and corresponding losses), and their connection to GANs has been explored extensively (Chnn et al. 2018 Hu et al. 2017 Mohamed and Lakshminarayanan 2016, [Srivastava et al.) 2017).
Recasting SBI into the VAE framework, the intractable distribution is the likelihood Pa (z|0) with a constant, i.e., the simulator only has parameters to inference over. Hence, p(x) is defined as the marginal likelihood. In this setting, the SBI objective becomes the reverse DKL. and we can use an adversarial method, c.g. GATSBI, to minimise it for learning go ( (xr). There is an additional subtlety at play here: GATSBI's generator loss. given an optimal discriminator, is not the reverse DKL. but the JSD (see Prop. D However, following the proof for Prop. 3 in Mescheder et al. (2017). we show that the approximate posterior that minimises the JSD also maximises the ELBO (see App. A.2). Thus, GATSBI is an adversarial inference method that performs SBI using the reverse DKL.
Note that the converse is not true: not all adversarial inference methods can be adapted for SBI. Specifically, some adversarial inference methods require explicit evaluation of the likelihood. Of those methods listed in Table only likelihood-free variational inference (LFVT) (Tran al. 2017) is explicitly used to perform inference for a simulator with intractable likelihood. LFVI is defined as performing inference over latents and global parameters shared across multiple observations æ, which are sampled from an empirical distribution p' (z) * p(z:), i.e., it learns go (u. / J...) However, p1(x)= = p(z), and is constant, i.e., the simulator does not have tunable parameters (but only parameters one does inference over) then LFVI and GATSBI become (almost) equivalent: LFVI differs by single term the loss function, and we empirically found the methods to yield similar results (see App. Fig. 6). We discuss LFVI's relationship to GATSBI in more detail in App. A.3
2.5 SEQUENTIAL GATSBI
The GATSBI formulation in the previous sections allows us to learn an amortised posterior p(0)x) for any observation x. However, one is often interested in good posterior samples for particular experimental observation xo- and not for all possible observations x. This can be achieved by training the density estimator using samples 0 from a proposal prior (0) instead of the usual prior (0). The proposal prior ideally produces samples @ that are localised around the modes of p(0) (())). and can guide the density estimator towards inferring a posterior for T. using less training data. To this end, sequential schemes using proposal distributions to generate training data over several rounds of inference (e.g. by using the approximate posterior from previous rounds as the proposal prior for subsequent rounds) be more sample efficient (Lueckmann et al. 2021).
Sequential GATSBI uses a scheme similar to other sequential SBI methods (c.g., Greenberg et al. 2019 Hermans et al. 2020 Lueckmann et al.) [017]] Papamakarios and Murray [0166] Papamakarios et al. 2019): the generator from the current round of posterior approximation becomes the proposal distribution in the next round. However, using samples from proposal prior (0) to train the GAN would lead to the GATSBI generator learning proposal posterior p(0 (z) rather than the true posterior
Note the difference in notation: the parameters 0 correspond to the latents u of VAF, and the simulator in SBI to the decoder of the VAE.
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p(0]z)). Previous sequential methods solve this problem either by post-hoe correcting the learned density (Papamakarios and Murray] (2066), by adjusting the loss function (Lueckmann et al., (2017). or by reparametrizing the posterior network (Greenberg et al. 2019). While these approaches might work for GATSBI (see App. A an outline), we propose a different approach: at training time, we sample the latents 2 that the generator transforms into parameters f-fromm corrected distribution Pe(z), so that the samples 0 are implicitly generated from an approximate proposal posterior 4s (0)z):
~ p=(2)
= 0 ~~ (0)))
(7)
== fo
where pe(z) (f))(2,))) ((0
x(0) p(z) and f(x) is the marginal likelihood
x(0) p(z)
under samples from the proposal prior. Since #0), p(z) and p(x) are intractable, we approximate
w0,xz) x)) using ratio density estimators for =(0)
_(() and p(ar) This ensures that at inference time, the
p(#)
the true posterior p(0)xr) with highest accuracy at z = To (see App. A.4 BB. Azadi et al. (2019); [Ch et al. (2020) use similar approaches to improve GAN training. Note that this scheme may be used with other inference algorithms evaluating a loss under approximate posterior samples, e.g. LFVI (see Sec. 2.4).
GATSBI generator transforms samples from p(2) into an approximation
3 RESULTS
We first investigate GATSBI on two common benchmark problems in SBI (Lueckmann et al. (021). We then show how GATSBI infers posteriors in a high-dimensional regime where most ffate-ofthhe-sse SBI algorithms are inadequate, as well as in problem with an implicit prior (details in App. D.I.
3.1 BENCHMARK PROBLEMS
We selected two benchmark problems with low-dimensional posteriors on which state-of-the- art SBI
algorithms have been tested extensively. We used the benchmarking- and setup from Lueckmann
et al. (201)). Briefly, we compared GATSBI against five SBI algorithms: classical rejection ABC
(Tvaaeé et al. 1997, REJ-ABC), sequential Monte Carlo ABC (Beaumont et al. [009)] SMC-ABC)
and the single-round variants of flow-based neural likelihood estimation (Papamakarios et al. 2019
NLE), flow-based neural posterior estimation (Greenberg et al. 2019 Papamakarios and Murray
2016 NPE) and neural ratio estimation (Durkan et | 2020; Hermans al. [020]] NRE). GANs have previously been shown to be poor density estimators for low-dimensional distributions (Zaheer et al. 2017). Thus, while we include these examples to provide empirical support that GATSBI works, we do not expect it to perform as well state-of-the samples from the approximate posterior against a reference posterior using a classifier trained on the two sets of samples (classification-bo based two-sample test, C2ST). A C2ST accuracy of 0.5 (chance level) corresponds to perfect posterior estimation.
SBI algorithms on these tasks. We compared
"Simple Likelihood Complex Pusterior" (SLCP) is challenging SBI problem designed to have a simple likelihood and a complex posterior (Papamakarios et al. 2019). The prior x(0) is a 5- dimensional uniform distribution and the likelihood for the 8-dimensional x is Gaussian whose mean and variance are nonlinear functions of 0. This induces a posterior over 0 with four symmetrical modes and vertical cut-offs (see App. Fig. 7) making it challenging inference problem. We trained all algorithms on a budget of 1k, 10k and 100k simulations. The GATSBI C2ST scores were on par with NRE, better than REJ-ABC and SMC-ABC, but below the NPE and NLE scores (Figure 2A)).
The "Two-Moons Model" (Greenberg et al. 2019) has a simple likelihood and a bimodal posterior, where each mode is crescent-shaped (see App. Fig. (8). Both x and 0 are vvo-dinensional. We initially trained GATSBI with the same architecture and settings as for the SLCP problem. While the classification score for GATSBI (see Figure 23) was on par with that of REJ-ABC and SMC-ABC for 1k simulations, performance did not significantly improve when increasing training budget. When we tuned the GAN architecture and training hy hyperparameters specifically for this model, we found a performance improvement. Lueckmann et al. (2021) showed that performance can be significantly improved with sequential inference schemes. We found that although using sequential schemes with GATSBI leads to small performance improvements, it still did not ensure performance on on-par with the flow-baser methods (see App. Fig. 99. Moreover, since it had double the number of hyperparameters as amortised GATSBI, sequential GATSBI also required more hyperparameter tuning.

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SLCP
B
1.00
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0UC-ADC
SMC ABC
lacuroo)
NPE
GAT381
GATSEUS
1000
000
100 000
1000
10:000 Number simulations
100 000
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Figure 2: GATSBI performance benchmark problems. Mean C2ST score (+ standard error of the mean) for 10 observations each. A. The classification score for GATSBI (red) on SCLP decreases with increasing simulation budget, and is comparable NRE. It outperforms rejection ABC and SMC-ABC, but has worse performance than NPE and NLE. B. The classification score for GATSBI (red) on the two-moons model decreases with increasing simulation budget, comparable to REJ-ABC and SMC-ABC for simulation budgets of 1k and 10k, and is outperformed by all other methods for 100k simulation budget. However, GATSBI's classification score improves when its architecture and optimization parameters are tuned to the two-moons model (red, dashed).
A
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Figure 3: Shallow water model inference with GATSBI, NPE and NLE. A. Ground truth, obser- vation and prior samples. Left: ground-truth depth profile and prior samples. Middle: surface wave simulated from ground-truie -tutth profile as function of position and time. Right: wave amplitudes at three different fixed times for ground-truth depth profile (black), and waves simulated from multiple prior samples (gray). B. GATSBI inference. Left: posterior samples (red) versus ground-truth depth profile (black). Middle: surface wave simulated from single GATSBI posterior sample. Right: wave amplitudes for multiple GATSBI posterior samples, at three different times (red). Ground-truth waves in black. C. NPE inference. Panels as in B. D. NLE inference.
3.2 HIGH-DIMENSIONAL INFERENCE PROBLEMS
We showed above that, on low-dimensional examples, GATSBI does not perform well as the best SBI methods. However, we expected that it would excel on problems on which GANs excel: high-dimensiooal and structured parameter spaces, in which suitably designed neural networks can be used as powerful classifiers. We apply GATSBI to two problems with -00--iimessional parameter spaces, one of which also has implieit prior.
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Shallow Water Model The ID shallow water equations describe the propagation of an initial disturbance across the surface of a shallow basin with an irregular depth profile and bottom friction (Figure 3/). They are relevant to occanographers studying the dynamics on the continental sea shelf 1973). We discretised the partial differential equations on a 100-point spatial (Backhaus 1983 Holton
grid, obtaining simulator parameterised by the depth profile of the basin, 0 E 1R100 (substantially higher than most previous SBI-applications, which generally are Re 10 dim). The resulting amplitude of the waves evolving over a total of 100 time steps constitutes the 10K-dimensional raw observation from which we aim to infer the depth profile. The observation is taken into the Fourier domain, where both the real and imaginary parts receive additive noise and are concatenated, entering the inference Our task was to estimate p(0[zz), i.e., to infer the basin depth profile from a noisy Fourier transform of the surfaces waves. We trained GATSBI with 100k depth profiles sampled from the prior and the resulting surface wave simulations. For comparison, we trained NPE and NLE on the same data, using the same embedding net as the GATSBI discriminator to reduce the high dimensionality of the observations z E 220k to R 100
pipeline as an array E R20r 220k
Samples from the GATSBI posterior captured the structure of the underly- ing ground-trutt depth profile (Fig ure 33, left). In comparison, only NPE, but not NLE, captured the true shape of the depth profile (Figure 3C, D. left). In addition, the inferred pos- terior means for GATSBI and NPE, but not NLE, were correlated with the ground- truth parameters (0.88 = 0.10 for GATSBI and 0.91 + 0.09 for NPE, mean + standard deviation across 1000 different observations). Further- more, while GATSBI posterior predic-
1.0
empirical CDF
Uniform CDF
Uniform CDF NPE
GATSBI
0.5
0.0
500
1000
500
1000
Rank
Rank
Figure 4: SBC results on shallow water model. Empirical edf of SBC ranks, one line per posterior dimension (red, GATSBI; blue, NPE). In gray, region showing 99% of devia- tion expected under the ideal uniform edf (black).
tive samples did not follow the ground-truth observations as closely as NPE's, they captured the structure and were not as phase-shifted as NLE's (Figure 3 B-D). This is expected since wave- propagation speed depends strongly depth profile, and thus incorrect inference of the depth parameters leads to temporal shifts between observed and inferred waves. Finally, simulation-based calibration (SBC) (Talts al., 2020) shows that both NPE and GATSBI provide well-calibrated posteriors on this challenging, 100 dimensional inference problem (clearly, neither is perfectly cali- brated, Figure 4). This realistic simulator example illustrates that GATSBI can learn posteriors with a dimensionality far from feasible for most SBI algorithms. It performs similar to NPE, previously reported to be a powerful approach on high-dimensional SBI problems (Lueckmann al. 202T).
Noisy Camera Model Finally, we demonstrate a further advantage of GATSBI over many SBI algorithms: GATSBI (like LFVI) can be applied in cases in which the prior distribution is only specified through an implicit model. We illustrate this by considering inference over images from a noisy and blurred camera model. We interpret the model images 0 are samples from an implicit prior over images π (0)) and the simulated sensor activations are observations x. We can then recast the task of denoising the images as an inference problem (we note that our goal is not to advance the state-of-the-art for super-resolution algorithms, for which multiple powerful approaches are available, e.g. Kim et al. 2016 Zhang et al. 2018).
black-box simulator. Thus, the clean
Since images are typically high- dishnsiieniin objects, learning posterior over images is already a difficult problem for many SBI algorithms either due to the implicit prior or the curse of dimensionality making MCMC sampling impracticable. However, the implicit prior poses no challenge for GATSBI since is trained only with samples.
We chose the EMNIST dataset (Cohen et al.) 2017) with 800k 28x 28 2i-diiennio images as the implicit prior. This results in an inference problem in a 784-dimensional parameter spacc-much higher dimensionality than in typical SBI applications. The GATSBI generator was trained to produce clean images, given only the blurred image as input, while the discriminator was trained with clean images from the prior and the generator, and the corresponding blurred images. GATSBI indeed recovered crisp sources from blurred observations: samples from the GATSBI posterior given blurred observations accurately matched the underlying clean ground-truth images (see Figure 5). In contrast,
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8
3
C
n
3
Q
7
2
b
7
Neu
3
A
3
4
7
G41 50
SD
Figure 5: Camera model inference. Top: ground-truth parameter samples from the implicit prior and corresponding blurred camera model observations. Bottom: mean and standard deviation (SD) of inferred GATSBI and NPE posterior samples for matching observation from top.
NPE did not produce coherent posterior samples, and took substantially longer to train than GATSBI (NLE and NRE are not applicable due the implicit prior).
4 DISCUSSION
We proposed new method for simulation-based inference in stochastic models with intractable likelihoods. We used conditional GANs to fit neural network functioning an implicit density, allowing efficient, scalable sampling. We clarified the relationship between GANs and SBI, and showed how method is related to adversarial VAEs that learn implicit posteriors over latent variables. We showed how GATSBI can be extended for sequential estimation, fine-tuning inference a particular observation the expense of amortisation. In contrast conventional density- estimation based SBI-methods, GATSBI targets the reverse DRL-by thus extending both amortised and sequential SBI, we made possible to explore the advantages of the forward and reverse DKL-
We found GATSBI to work adequately on two low-dimensional benchmark problems, though it did outperform neural network-bass based methods. However, its real potential comes from the ability of GANs to learn generative models in nigh-dimensional problems. We demonstrated this by using GATSBI infer well-calibrated posterior over a 100-dimensional parameter space for the shallow water model, and 784-dimensional parameter space for the camera model. This is far beyond what is typically attempted with SBI: indeed we found that GATSBI outperformed NPE on the camera model, and performed similarly to NPE and substantially outperformed NLEE on the shallow water model (both SBI approaches were previously shown to be more powerful than others in high dimensions; Lueckmann et al., 2021). Moreover, we showed that GATSBI can learn posteriors in model in which the prior is only defined implicitly (i.e., through database of samples, not a parametric model).
Despite the advantages that GATSBI confers over other SBI algorithms in terms of flexibility and scalability, comes with computational and algorithmic costs. GANs are notoriously difficult to train-they highly sensitive to hyperparameters (as demonstrated in the two-moons example) and problems such as mode collapse are common. Moreover, training time is significantly higher than in other SBI algorithms. In addition, since GATSBI encodes an implicit posterior, i.e., density from which we can sample but cannot evaluate, it yields posterior samples but not a parametric model. Thus, GATSBI unlikely be the method of choice for low-dimensional problems. Similarly, sequential GATSBI requires extensive hyperparameter tuning in order to produce improvements over amortised GATSBI, with the additional cost of training a classifier approximate the correction factor. Hence, sequential GATSBI might be useful in applications where the computational cost of training offset by having an expensive simulator. We note that GATSBI might benefit from recent improvements on GAN training speed and sample efficiency (Sauer al. 2021).
Overall, we expect that GATSBI will be particularly impactful on problems in which the parameter space is high-dimensional and has GAN-friendly structure, e.g. images. Spatially structured models and data abound in the sciences, from climate modelling to ecology and economics. High- nigh-dimenssonal parameter regimes common, but inaccessible for current SBI algorithms. Building the strengths of GANs, we expect that GATSBI will help close this gap and open up new application-dome for SBI, and new directions for building SBI methods employing adversarial training.

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ACKNOWLEDGMENTS
We thank Kai Logemann for providing code for the shallow water model. We thank Michael Deistler, Marcel Nonnenmacher and Giacomo Bassetto for in-depth discussions; Auguste Schulz, Richard Gao, Artur Speiser and Janne Lappalainen for feedback on the manuscript and the reviewers at ICML 2021, NeurIPS 2021 and ICLR 2022 for insightful feedback. This work was funded by the German Research Foundation (DFG; Germany's 's Excellence Strategy MLCoE EXC number 2064/1 PN 390727645; SPP 2041; SFB 1089 'Synaptic Microcircuits' ), the German Federal Ministry of Education and Research (BMBF; project ADIMEM, FKZ 011S A-D; Tübingen AI Center, FKZ: 0110 (8039) and the Helmholtz Al initiative.
ETHICS STATEMENT
Since this is a purely exploratory and theoretical work based on simulated data, we do not foresee major ethical concerns. However, also note that highly realistic GANs have a potential of being exploited for applications with negative societal impacts e.g. deep fakes (Tolosana et al. 2020).
REPRODUCIBILITY STATEMENT
To facilitate reproducibility, we provide detailed information about simulated data, model set up, training details and experimental results in the Appendix. Code implementing the method and experiments described in the manuscript is available at httpps://g thub. com ma httc/maakk ab/ gatsbi
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APPENDIX
A PROOFS
A.1 PROOF OF CONVERGENCE FOR GATSBI
Proposition 1. Given fixed generator for the discriminator D. (0,z) maximising equation
satisfies
p(0 x
(0, p(6 + (0)x))
and the corresponding loss function for the generator parameters is the Jensen-Shannon divergence (JSD) between the true and approximate posterior:
Lg (Ø) 2JSDDp((0a) II (0) - log
Proof. We start with equation 2
The proof proceeds as for Proposition and 2 in Goodfellow et al. (2014). For convenience, elide the arguments of the various quantities, so that qø denotes gs(0 (0(z). ())
denotes p(0 and Du denotes (0, (0,
L(o, v) Ep(x)
log Do D& +Bgg log (1 D
/ D Do d0 + / go log(11 Dv) d0
Ep(z)
/ (plog Do t4o log(1 1og(1-Da)) d0}
= Ep(x)
For any function g(x) = a log x + + blog(l 1-11)))) r), where a, b E R R 0} and E (0, 1), the maximum of the function is 9/o + Hence, L(d, 1b) for fixed achieves it's maximum
D
p+ q
Note that p(z) drops out of the expression for Da- since it is common to both terms in L(0, v) Plugging this into equation 2 and dropping the expectation over z without loss of generality:
p
Leg- (0) Ep log
+ Egg log
9s
p 9a
2
g0
Ep log
+ Egg log
log
+
p + 96
JSD( (&) log 4.
The JSD always non-negative, and zero only when p(0 x) (0)x). Thus, the global minimum
Lg (Φ*) log 4 is achieved only when the GAN generator converges to the ground-truth posterior p())z).
A.2 PROOF FOR GATSBI MAXIMIZING ELBO LOSS
Proposition 2. 8- and u/² denote the Nash equilibrium of min-max game defined by equation then o' also maximises the evidence lower bound of a VAE with fixed decoder i.e.,
arg max Lg(ő)
(8)
x(0)
arg Bp(n) E
log
log (ogg(1101)
(0(x))))
Proof. The proposition is trivially true if 90- (f]z) = p(f|z), i.., p(0 (z) is the true minimum of both the ISD and DKL.- However, here show that the equivalence holds even when go- (0]::) is not

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the true posterior, e.g. if (0) belongs to family of distributions that does not include the true posterior.
Given that o' and 3 denote the Nash equilibrium in equation 2 the optimal discriminator is given by
know from Proposition | that
p(0
D (0,z) p(0 a:) 46- (0
To lighten notation, we clide the parameters of the networks and their arguments, denoting De- (0)z)
D pir. 9s- (0|zz) as q², and If we plug D into Eq.
2 we have
e= arg min (p(())) log D +Bus
log (1 - (1-Dx.)
arg min Egg log (1 (D-Dw-)
(10)
Note that this is true only at the Nash equilibrium, where D is function of dø- and not ds- This allows us to drop the first term from the equation. In other words, if we switch out 4s- with any other ds in the expectation, equation 10 is no longer minimum w.r.t Φ, even though Dii is optimal.
Let us define D. : (Be)), where o(-) == 1/1 + exp((-)). Then from equation
p(0 x)
o(Rw))=
p(0|x) 9o-
e + 96" /p(0(z)
Comparing the l...s. and r.h.s. of the equation above, we get
Rig = log p(0)x))
(11)
Since both log and are monotonically increasing functions, also have from equation T0 Ø² argg min Eg log (1-o((tt))
arg min Egg (1 o ((Rp)))
arg max Ego (RR)))
arg max Egg (Re.))
(12)
In other words, ls- maximises the function Eyg (Rw-), ). Now, to prove equation we need to show that (Ø(Ø) LE(Ø") ò 6- (Ø( = Ep(z) Egg (log (6) - log 90 + +lgp(xz10)
log p(0[z) + log p(x(0)=(0)
= EEpc) Ens
p(0lar)
log p(0|z)
= Eno
+ log p(x)
96
log p(f[z)
= Bg
+ log p(x)
- Ey(ay) DKL(9s)) (9oll0)))))
46-
log p(0[z)
Eppc
+ log p(x)
ts-
= Ep(c) (Ra. + -llog(z)))
from equation 11
< Eag (Rw-) + E (x(() Eyg log p(x)
from equation 12
= Eno (Ry- + +logp(z)) + Ep(x) ) Egg log - Eppe) log p(z)
Eng (log p(0|z)
+ logp(z)
Z
= LB(Ø")
LE( < Lx(Ø")
Hence, the approximate posterior obtained by optimising the GAN objective also maximises the evidence lower bound of the corresponding VAE.
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A.3 CONNECTION BETWEEN LFVI, DEEP POSTERIOR SAMPLING AND GATSBI
Adversarial inference approaches maximise the Evidence Lower Bound (ELBO) equation to train VAE, and use discriminator approximate intractable ratios of densities in the loss (see Table Likelihood- free variational inference (Tran et al. 2017 LFVT) is one such method.
Table 1: Comparison of adversarial inference algorithms: BiGAN (Donahue et al. 2019). ALI DDumoulin et al. 2016), AAE (Makhzani et al. 2015), AVB (Mescheder et al. 2017) and LFVI
Tran et al. 2017)
ALGORITHM
DISCRIMINATOR RATIO
GENERATOR LOSS FUNCTION
BIGAN, ALI
Pa(elu)p(u)/e (ull)))) (aa)) p(u)p(x)/ (a(x))(()
JSD (u, x) Io (o(ux)))
AAE
JSD(p(u) ( (u))
AVB
DKL (qa(u x (p(u))
LFVI
/a plula)p(a) (o (v(=))((
DKL (go (u (p((r))
GATSBI
JSD (a)())
In the most general formulation, LFVI learns posterior over both latents u and global parameters B which are latents shared across multiple observations i.e., go(z, B|x) and maximises the ELBO given by
((3) (3)
+ (us(u())) log p(zlu)p(u) + const.
LoF(O) = (g) log
(13)
4o (x(x)))/((x)
where (xx) an empirical distribution over observations, and not necessarily p(x), the marginal likelihood of the simulator. A discriminator, (2, u) is trained with the cross-entropy loss to approximate the intractable ratio p(=lu)p(u)) [o(a]2)p'(a) in the second term. Using the nomenclature from Huszár (2017), note that Da, is joint-contrastive: it simultaneously discriminates between tuples [z, u) p( pp()) and (ě, ú) ~ go (alá))' GATSBI, by contrast, is prior-contrastive: its discriminator only discriminates between parameters 0, given a fixed x.
However, when p' (x) p(x) and 3 is constant, the LFVI discriminator becomes prior-contrastive, equation 13 is function of both the discriminator parameters th and generator parameters Φ, and differs from GATSBI only by a single term i.c., from equation and equation and ignoring the constant:
LGAISEI (ò, )b) Lo +b) (x)o))))) log (1,
(14)
The second term corresponds the sot-satung GAN loss (Goodfellow et al. 2014). In this
A
Two Moons
B 1.00
SLCP
1.00
C2ST (acauracy)
C2ST (acouracy)
0.75
0.75
DPS
LFVI
GATSBI
0.50
0.50
1000
000
100 000
1000
10 000
100 000
Number of simulations
Number of simulations
Figure GATSBI, LFVI and Deep Poseterior Sampling (DPS) on benchmark problems. Mean C2ST score (# standard error of the mean) for 10 observations each. A. On Two Moons, the C2ST scores for GATSBI (red), LFVI (navy) and Deep Posterior Sampling (DPS, yellow) are qualitatively similar across all simulation budgets B. On SLCP, DPS is slightly worse than LFVI and GATSBI.
'In Tran et al. 2017 p' (z) ) is denoted ((x)
denoted (3, u) Tran al. (017
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setting, with optimal discriminator, GATSBI minimises JSD, whereas LFVI minimises the reverse DKL-
Adler and Öktem 2018 introduce Deep Posterior Sampling which also implements an adversarial algorithm for posterior estimation. The set-up is similar to GATSBI, but GANs are trained using a Wasserstein loss as in Arjovsky et al. (2017) The Wasserstein loss imposes stronger conditions on the GAN networks in order for the generator to recover the target distribution, i.e., the discriminator has
be I-Lipschitz and the generator K-Lipschitz (Arjovsky al. 2017 Qi 2018). However, the JSD loss function allowed us to outline GATSBI's connection to adversarial VAÉs and subsequently its advantages for SBI (see Sec. 2.4 Suppl. Sec. Sec.AA.2|an and the discussion above). Whether this would also be possible with the Wasserstein loss remains a subject for future work. Nevertheless, the sequential extension of GATSBI using the energy-based correction (see Sec. 2.5 and Supp. Sec. A.4) could in principle also be used with the Wasserstein metric.
Mescheder al. (2118) state that the WGAN converges only when the discriminator minimizes the Wasserstein metric at every step, which does not happen in practice. [Fduu et al. (2017) argue that GAN generator does not necessarily minimise a JSD at every update step since the discriminator optimal only in the limit of infinite data. Hence, neither asymptotic property can be used to reason about GAN behaviour in practice. As a consequence, it is difficult to predict the conditions under which LFVI, Deep Posterior Sampling would outperform GATSBI, or vice-versa. Nevertheless, given the same networks and hyperperparameters (with slight modifications to the discriminator for Deep Posterior Sampling, see App. D.I details), we found empirically that LFVI, Deep Posterior Sampling and GATSBI are qualitatively similar on Two Moons, and that Deep Posterior Sampling slightly worse that the other two algorithms on SLCP: i.e., there is no advantage using one algorithm over the other on the problems investigated (see Fig. 6
A.4 SEQUENTIAL GATSBI
For many SBI applications, the density estimator only required to generate good posterior samples for a particular experimentally observed data Tg. This can be achieved by training the density estimator using samples from a proposal prior o (0) instead of the prior т (0). The proposal prior ideally produces parameters & that are localised around the modes of p(0]2.) and can guide the density estimator towards inferring a posterior that is accurate for ze To. If we replace the true prior (0) with proposal prior (0) (0lor)) i.e., the posterior estimated by GATSBI, and sample from the respective distributions
0,2 (0)p(()))
the corresponding GAN loss (from equation equation 2 is:
i(o }/ = [((0))((()))(=) log (0,, x) + log(1 D (f((2,,)) [)(() log (0, x) + gg =))(ar) log(1 - D. (0,z))
(15)
(16)
This loss would allow us to obtain generator that produces samples 0 that are likely to generate outputs x close to Fa when plugged into the simulator. However, Proposition [ in Appendix [A.] shows that this loss would result in yo (0]=c) converging to the proposal posterior p(0[=r) rather than the ground-truth posterior ()) In order to learn a conditional density that is accurate for x As %o but nevertheless converges to the correct posterior, we need to correct the approximate posterior for the bias due to the proposal prior. We outline three different approaches to this correction step:
Using ensrg GANs Although it possible to use correction factors directly in the GATSBI loss function, as we outline in the next section, these corrections can lead to unstable training (Papamakarios et al., 2019). Here, outline approach in which we train on samples from the proposal prior without explicitly introducing correction factors into the loss function. Instead, we change the setup of the generator to produce 'corrected' samples, which are then used to compute the usual cross-entropy loss, and finally we train the discriminator and generator.
Let us start by introducing the correction factor w(0
=(0) p(z) such that
=(0) p(x)
p(0|z)) - 5(0|z)) w (0,
(17)
In the original formulation of GATSBI sampling from qu (f|=) entails sampling latents & ~ p(2). and
transforming them by deterministic function fa 2) get parameters (see equation equation
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Following the energy-based GAN formulation (Azadi et al. 2019 Che al. 2020), define an intermediate latent distribution pr(z):
Pe (=(=) (f)
(18)
Pe(2) is the distribution of latent variables that, when passed through the function fa( 2). are most likely to produce samples from the upproximate proposal posterior qa (0]zz) gs (w( x)) For GANs, p(=) ) is typically a tractable distribution whose likelihood be computed, and from which one can sample, and thus, we can use MCMC or rejection sampling to also sample from pr (2) (see Appendix for details). The resulting loss function is:
[(o, +) = [p(()=)p(c) log (0, + (p)()))(s) log(1 - D=((o(1))))0)) = Ep(s) [1 + L2
(19)
Note that there are no explicit correction factors in the loss.
We now show that optimising the loss function equation 19 leads to the generator converging the correct posterior distribution. Let us first focus on the second term Lz:
L2 = (x) log(1 (x(f
/ (x) Do (fo(a,
/ p(z) (w(f log(1 - (f,(,
from equation 18
/ (01x)) (w(0. x)) 1 log(1 2))
reparam. trick
(01x)) log(1 - (0,
(0 lo (0,, x)).
Thus, from Proposition we can conclude that by optimising the loss function equation 19 9o (0) + p(0 r). This implies that the generator network, which represents (0)xr) converges to p00=z)) since both the approximate and target proposal posteriors are related respectively to the approximate and true posteriors by the same multiplicative factor w(f.x). Note that the generator is more accurate in estimating the posterior given To (or x Rs F.), i.e., p(0 than given x far from T.. since it is trained samples from the proposal prior.
In practice, this scheme does produce improvements in the learned posterior. However, is compu- tationally expensive, because every update to the generator and discriminator requires round of MCMC or rejection sampling to obtain the 'corrected' samples. Moreover, if we use the generator from the previous round the proposal prior in the next round, need to train classifier to approximate w(O, x) at every round [5] Finally, this approach has additional hyperparameters that need to be tuned during GAN training, which could make it prohibitively difficult to use for most applications.
Below, outline theoretical arguments for two additional approaches, although we only provide empirical results for the second approach.
Using importance weights Lueckmann et al. (017) ) solve the problem of bias from using a proposal prior by introducing importance weights their loss function. One can the same trick for GATSBI,y introducing the importance weights w0, x)
*(0) ((x) into the loss defined in
#(0) p(a)
equation 16
L( v) (=(0)p((0))())) log + log(1 (f L1 L2. (20)
NNote that the correction factor could be computed in closed form if we had generator with an evaluable density: would have train classifier approximate it.
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Optimising this loss allows (0(0)z) to converge to the true posterior p(0 (z). Let us focus the first term L1:
L1 = B(0 (()) (0, log (0, x)
(())((0))) (0) p(x))
log (0,zz)
(00) p(x)
p(010)=(0) T(0) p(x)
log D (0,
#(0) p(x)
p(x10))(() p(xx)
log (0,
p(x)
p(x)
(())p(0)z)
log (0,
p(x)
J. 1. p(x)p(()))) log (0,
(a))((()))) log (H.
Thus, from Proposition 1
conclude that by optimising the loss function equation 20 the generator qo (0(z)z) (z) converges to the true posterior. However, the importance- weight correction could lead to high-variance gradients (Papamakarios et al. 2019). This would be particularly problematic for GANs, where the loss landscape for each network is modified with updates to its advensary, and there no well-defined optimum. High-variance gradients could cause training take longer prevent it from converging altogether.
Using inverse importance weights Since using importance weights in the loss can lead to high-variance gradients, could instead consider using the inverse of the importance weights (()) p(a) in the second term equation 16
((00,x))= x()) p(a)
L(ø, (b) [p(()))((a)) (0, (0 (w(0,z))) log(1 (0,zz)) L1 + L2.
(21)
Optimising L(ó, ) from equation 21 will result in the generator approximating the true posterior at convergence. Focusing on the second term the loss function L2:
L2 ( (w(6,z))) log(1 - ())),z)
/f (0)))s(0)z)) #(0) p(x)
log(1 (2x(0.z))
(()) p(x))
|ʃ p(z) log( - D (0,x))
(p(o))s(01)) log(1 (+
Thus, from Proposition] I
we can conclude that by optimising the loss function equation 21 7o (0|=)
converges p(0]z)) Since d (f]zz) differs from (0(xr) by the same factor as p(0)z)) from p(0]z).
i.e., (0) (see equation equation 17). this implies that qo (0)xr) p(0)x)).

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B TRAINING ALGORITHMS
Algorithm GATSBI
Input: prior (0), simulator p(r 0), generator fo, discriminator De, learning rate À Output: Trained GAN networks for and Dø-
θ 02, 0x) Lid (0)
X { 22. In (,)
while not converged do for discriminator iterations do Sample mini-batch Xd, Oa from X, θ
(2), 6₂ fo Xx) - Ex, Ee e log a (0a, Xa) + Es, log(1 D& Db X.))))
1b < 1/2
end for for generator iterations do Sample mini-batch x., θ from X, θ
~ p(z). 6, = fo Xg) L - - Ex, Es, log(1 - ( x)))
- 6 + AV
end for end while
Algorithm 2 Sequential GATSBI with energy-based correction
Input: (8), simulator p(z classifier w 1, observation Fu. fo. D., learning rate Output: Trained GAN networks fa- and Da--
for i 1... number of rounds
+ {01, 02.00.................. mn) Lid
~ (0)
X; {II,
Fn ((x10)
if i > then:
We max log σ (w)
(On) (b)) + log(1 (wo(())))
we
W2 max log a (w,, (Xn)) + log(1 - a (X:)))
We
w
we
end
while not converged do
for discriminator iterations do
Sample mini-batch X O₂ from X, O X,), X,))-1
Z ~ (Z)) =
= fo
L L= Ex (Σ e log D (0 X + Es, log(1 ( Xx)))
+ A
end for
for generator iterations do
Sample mini batih X g. θ, from X; O;
Z (Z) = p(Z) (w fo Z, x,), x,)))
6, fa
L Ex, Es, log(1 Dr X,))
Φ - o + L
end for end while (0) (0;
end for
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Algorithm 3 Sequential GATSBI with inverse importance weights
Input: (0), simulator p(=(0), classifier w = I,, observation Zo. fo. Di, learning rate Output: Trained GAN networks fa- and Dw-
for .... number of rounds do θ; {01, ,t, 0,) x(0) X; = {I1, T2, tan ~
p( 0
if i then:
Wg < max log σ (wg + log(1 (w)(())))
we
Wy max log の (wz (X)) + log(1 o (x:))
end if
while not converged do
for discriminator iterations do
Sample mini-batch Xa, O₂ from X;, O;
Z p(Z)
θ, (Z,
= Ex, (E o log (Oa, + (w(Ba, (w)) log(1 ( x,))) 3 t & A
end for for generator iterations do
Sample mini-batch Xg. O from X,, O;
Z ~ pZ))
θ, f,,,
L -= Ex, Es, (w(O,, -x)))=log(( (r x,))
Φ - à + A
end for end while (0)
(0:
end for
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C ADDITIONAL RESULTS
C.I POSTERIORS FOR BENCHMARK PROBLEMS
We show posterior plots for the benchmark problems: SLCP (Fig. 7 and Two Moons (Fig. 8). In both figures, panels the diagonal display the histograms for each parameter, while the off- off-diagonal panels show pairwise posterior marginals, i.e., 2D histograms for pairs of parameters, marginalised over the remaining parameter dimensions.
GATSDI
8,
0,
Figure 7: Inference for one test observation of the SLCP problem. Posterior samples for GATSBI trained on 100k simulations (red), and reference posterior samples (black). The GATSBI posterior samples cover well the disjoint modes of the posterior, although GATSBI sometimes produces samples in regions of low density in the reference posterior.
C.2 SEQUENTIAL GATSBI
We found that sequential GATSBI with the energy-based correction produced a modest improvement amortised GATSBI for the Two Moons model with 1k and 10k simulation budgets. and no improvement at with 100k (see Fig. (9) The inverse importance weights correction did not produce an improvement for any simulation budget. Sequential GATSBI performance also sensitive to hyperparameter settings and network initialisation. We hypothesise that further improvement is possible with better hyperparameter network architecture tuning.

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Rel. Posterior
GATSBI
-1.0
1.0
-1.0
1.0
Figure 8: Inference for one test observation of the Two Moons problem. Posterior samples for GATSBI trained on 100k simulations (red), and reference posterior samples (black). GATSBI captures the global bi-modal structure in the reference posterior, but not the local crescent shape. also generates some samples in regions of low density in the reference posterior.
Two Moons
(accuracy)
Amort. GATSBI
Seq, GATSBI
Seq, Imp.
1000
10 000 Number simulations
100 000
Figure 9: Sequential GATSBI performance for the Two Moons Model. The energy-based correction (EBM) results in slight improvement over amortised GATSBI for 1k and 10k simulations, but the inverse importance weights correction does not.
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D IMPLEMENTATION DETAILS
All networks and training algorithms were implemented in PyTorch (Paszke et al. 2019). We used Weights and Biases (Biewald) 2020) to log experiments with different hyperparameter sets and applications. We ran the high-dimensional experiments (camera model and shallow water model) on Tesla- 100 GPUs: the shallow water model required training to be parallelised across 2 GPUs at a time, and took about 4 days converge and about 1.5 days for the camera model on one Tesla V V100. We used RTX-2080Tis for the benchmark problems: the amortised GATSBI runs lasted a maximum of 1.5 days for the 100k budget; the sequential GATSBI runs took longer with the maximum being days for the energy-based correction with a budget of 100k. On similar resources, NPE took about 1 day to train on the shallow water model, and 3 weeks and days to train on the camera model. NPE took about 10 min for 100k simulations on both benchmark problems. SMC-ABC and rBjection-ABC both took about 6s on the benchmark problems with a budget of 100k.
D.1 SIMPLE- LIVELI-LKKE COMPLEX POSTERIOR (SLCP) AND TWO-MOONS
Prior and simulator For details of the prior and simulator, we refer Lueckmann et al. (001)
GAN architecture The generator was a 5-layer MLP with 128 hidden units in the first four layers. The final layer had output features equal to the parameter dimension of the problem. A leaky ReLU nonlinearity with slope 0.1 followed each layer. A noise vector sampled from a standard normal distribution -ivendimensional for Two Moons, and 5-dimensional for SLCP) was injected at the fourth layer of the generator. The generator received the observations as input to the first layer, used the first three layers to embed the observation, and multiplied the embedding with the injected noise at the fourth layer, which was then passed through the subsequent layers to produce an output of the same dimensions as the parameters. The discriminator was a 6-layer MLP with 2048 hidden units in the first five layers and the final layer returned a scalar output. A leaky ReL.U nonlinearity with slope 0.1 followed each layer, except for the last, which was followed by a sigmoid nonlinearity. The discriminator received both the observations and the parameters sampled alternatively from the generator and the prior as input. Observations and parameters were concatenated and passed through the six layers of the network. For Deep Posterior Sampling, the discriminator did not have the final sigmoid layer.
Training details The generator and discriminator were trained in parallel for 1k, 10k and 100k simulations, with a batch size = min(10% of the simulation budget, 1000). For each simulation budget, 100 samples were held out for validation. We used 10 discriminator updates for Ik and 10k simulation budgets, and 100 discriminator updates for the 100k simulation budget, per generator update. Note that the increase in discriminator updates for 100k simulations is intended compensate for the reduced relative batch size i.e., 1000 batches = 0.01 %. The networks were optimised with the cross-entropy loss. We used the Adam optimiser (Kingma and Ba 2015) with learning rate=0.0001, 3,=0.9 and 33=0099 for both networks. We trained the networks for 10k, 20k and 20k epochs for the three simulation budgets respectively. To ensure stable training, we used spectral normalisation (Miyato et al. 2018) for the discriminator network weights. For the comparison with LFVI, we kept the architecture and hyperparameters the same as for GATSBI, but trained the generator to minimise L(Ø)= = (0)) [log 10,00_)) Similarly, for Deep Posterior Sampling, we kept the same architecture (minus the final sigmoid layer for the discriminator) and hyperparameters, but trained the generator and discriminator on the Wasserstein loss: L(o, +) = Ep(0lx) (+) (0)(()=)) (8).
Optimised hyperparameters for Two Moons model The generator was 2-layer MLP with 128 hidden units in the first layer and output features in the second layer (same as the parameter dimension). Each layer was followed by a leaky ReLU nonlinearity with slope 0.1. Two-dimensional white noise was injected into the second layer, after it was multiplied with the output of the first layer. The discriminator was a 4-layer MLP with 2048 hidden units in the first 3 layers each followed by a leaky ReLU nonlinearity of slope 0.1, and a single output feature in the last layer followed by a sigmoid nonlinearity. We trained the networks in tandem for approximately 10k, 50k and 25k epochs for the 1k, 10k and 100k simulation budgets respectively. There were 10 discriminator updates and 10 generator updates per epoch, with the batch size set to 10% of the simulation budget (also for
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100k simulations). All other hyperparameters (learning rate, $1, B2, etc.) were the same for the non-optimised architecture.
Hyperparameters for sequential GATSBI The architecture of the generator and discriminator were the same for amortised GATSBL. We trained the networks for 2 rounds. In the first round, the networks were trained with the same hyperparameters as for amortised GATSBI, on samples from the prior: the only exceptions were the number of samples we held out for the 1 budget (10 instead of 100) and the number of discriminator updates per epoch for the 100k budget (10 instead of 100). Both exceptions were to ensure that there were always 10 discriminator updates per epoch, and speeding up training as much possible. In the second round, the networks were trained using the energy-based correction with samples from the proposal prior, as well as samples from the prior used in the first round, All other hyperparameters were the same as for round one. The simulation budget was split equally across the two rounds, and the networks were trained anew for each 10 different observations. The number of epochs was the same for the first and second round: 5k, 10k and 20k for the 1k, 10, and 100k budget respectively. We trained 2 classifiers at the beginning of round two: one x(e) and the other to approximate p(x)
p(=)) Both classifiers were 4-layer MLPs with 128 hidden units in each layer, and a ReLU nonlinearity following each layer. The classifiers were trained on samples from the proposal prior and prior, and the proposal marginal likelihood and likelihood respectively, using the MLPClassifeer class with default hyperparameters (except for "max_ Imax_iter"see jiterr =5000) from scikit-learn (Pedregosa et al. (011). For the energy- rejection sampling to sample from the corrected distribution pz(:): for particular observation sampled p(2 2) (0, 1), evaluated the probability of acceptance p (w (L(((zz,)) z)) *M and accepted z if < p where is uniform random variable. To compute the scale factor M, simply took the maximum value of
to approximate the ratio *(n)
correction, we used
within each batch. For the inverse importance weights correction, we computed (0, x))-² -1 used it to calculate the loss as Equation equation 21 for each discriminator and generator update in the second round.
D.2 SHALLOW WATER MODEL
Prior 0 N E), 8 E 100
10, Σg =
15, 100.
The values for 1a and Σ were chosen to ensure that the different depth profile samples produced discernible differences in the corresponding simulated surface waves, particularly in Fourier space. For example, combinations of μ values > 25 (deeper basins), a values < 10 and t values > 100 (smoother depth profiles) resulted in visually indistinguishable surface wave simulations.
Simulator
x
[(0) + 0.25e
61,1
11,100
Gj ~(0,
E200.1
100
(() is obtained by solving the 1D Saint- Venant equations on 100-element grid, performing a 2D Fourier transform and stacking the the real and imaginary part to form 2 100x 100-dimensional array.
The equations were solved using semi-implicit solver (Backhaus) 1983) with weight of 0.5 for each time level, implemented in Fortran (F90). The time-step size was set to 300s and the simulation was run for total of 3600s. The grid spacing dz was 0.01, with dry cells at both boundaries using depth of -00. We used a bottom drag coefficient of 0.001 and gravity=9.88 10.vity9..11//..... An initial surface disturbance of amplitude 0.2 was injected at x 2, to push the system out of equilibrium.
We chose to perform inference with observations in the Fourier domain for the following reason. Since waves naturally periodic delocalised phenomenon, it makes sense run inference their tranfforrdd amplitudes, so that convolutional filters can pick up on localised features. We used the scipy fft2 package (Virtanen et al. 2020
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GAN architecture The generator network was similar to the DCGAN generator (Radford et al. 2013). There were five sequentially stacked blocks of the following form: a 2D convolutional layer, followed by batch-norm layer and ReLU nonlinearity, except for the last layer, which had only convolutional layer followed by a tanh nonlinearity. The observations input to the generator were of size batch size 200 100. The input channels, output channels, kernel size, stride and padding for each of the six convolutional layers were as follows: - (2, 512, 4, 1, 0), 2- (512, 256, 4, 2, 1), 3 (256, 128, 4, 2. 1), 4 (128, 128, 4. 2, 1), (128. I, 4, 1). The final block was followed by a fully-connected readout layer that returned 1 100-dimensional vector. Additionally, we sampled 25- d--dimensional noise, where each element of the array was drawn independently from a standard Gaussian. and added it to the output of the lanh layer, just before the readout layer.
The discriminator network was similar to the DGCAN discriminator: it contained embedding layers that mirrored the generator network minus the injected noise. The input channels, output channels, kernel size, stride and padding for each of the six convolutional layers in the embedding network were follows: 1 (2, 256, 4, 1, 0). 2 (256, 128, 4, 2. 1), 3 (128, 64, 4, 2. 1), 4 (64, 64, 4, 2. 1), 5 (64, I, 4, 2, 1). This is followed by 4 fully-connected layers with 256 units each and a leaky ReLU nonlinearity of slope 0.2 after each fully- culleccomeected layer. The final fully-connected layer, however, was followed by a sigmoid nonlinearity. The discriminator received both the Fourier-transformee waves and depth profile alternatively from the generator and the prior as input. The Fourier- transformed waves were passed through the embedding layers, the embedding was concatenated with the input depth profile and then passed through the fully-connected layers.
Training details The two networks were trained in parallel for 440k epochs, with 100k training samples, of which 100 were held out for testing. We used batch size of 125, the cross-entropy loss and the Adam optimiser with learning rate= 0.0001, B1 = 0.9 and 83 0.99 for both networks. In each epoch, there was discriminator update for every generator update. To ensure stability of training, we used spectral normalisation for the discriminator weights, and clipped the gradient norms for both networks to 0.01, unrolled the Discriminator(Met et al. 2017) with 10 updates i.e., in each epoch, we updated the discriminator 10 times, but reset it to the state after the first update following the generator update.
NPE and NLE We trained NPE and NLE as implemented in the sbi package [Tejero Caroo-CCnnrooooooo et al. 2020) on the shallow water model. We set training hyperparameters as described in Lueckmann et al.) (, except for the training batch size which we set to 100 for NPE and NLE. The number of hidden units in the density and ratio estimators which we set to 100 (default is 50). For NPE, we included embedding net embed the 20k-dimensional observations to the number of hidden units. This embedding net was identical to the one used for the GATSBI discriminators, and it was trained jointly with the corresponding density or ratio estimators. We trained with exactly the same 100k training samples used for GATSBL. MCMC sampling parameters for NLE were set as in Lueckmann al. (2021).
To calculate correlation coefficients for the GATSBI and NPE posteriors, we sampled 1000 depth profiles from the trained networks for each of 1000 different observations from test set. We then calculated the mean of the 1000 depth profile samples per observation, and computed the correlation coeffiecient of the mean with the corresponding groundtruth depth profile. Thus, we had 1000 different correlation coefficients; we report the mean and the standard deviation for these correlation coefficients.
iouulatiio-based calibration (Talts et al. 2020) offers a way to evaluate simulation- camulation-based inference in the absence of ground posteriors. SBC checks whether the approximate posterior qo (0)z), when marginalised over multiple observations ær converges to the prior x =0). A posterior that satisfies this condition well-calibrated, although it is not a sufficient test of the quality of the learned posterior, since posterior distribution that is equal to the prior would also be well-calibrated.
However, when complemented with posterior predictive checks it provides a good test for intractable inference problems.
We performed SBC on the shallow water model. To obtain a test data set (6,, x; N we sampled N = 1000 parameters 0, from the prior and generated corresponding observations x; from the simulator. For each 2;, we then obtained a set of L = 1000 posterior samples using the GATSBI generator and calculated the rank of the test parameter 0; under the L. GATSBI posterior samples
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described in algorithm in Talts et al. (2020), separately for each posterior dimension. For the ranking we used a Gaussian random variable with zero mean and variance 10. We then used bins of 20 to compute and plot the histogram of the rank statistic. According to SBC, if the marginalised approximate posterior truly matched the prior, the rank statistic for each dimension should be uniformly distributed. Performing SBC can be computationally expensive because the inference has to be repeated for every test data point. In our scenario it was feasible only because GATSBI and NPE perform amortised inference and do not require retraining or MCMC sampling (as in the case of NLE) for every new z. We followed the same procedure to do SBC on NPE as for GATSBI.
D.3 NOISY CAMERA MODEL
Prior The parameters / were 28x 28-dimmeniio images sampled randomly from the 800k images in the EMNIST dataset.
Simulator
The simulator takes a clean image as input, and corrupts it by first adding Poisson noise,
followed by convolution with a Gaussian point-spread function: m ~ Poisson(O) (0)
<(th- = f * m; f(() exp(- _ )) where denotes a convolution operation with a series of 1D
Gaussian filters given by f. We set the width of the Gaussian function o = 3.
GAN architecture The generator network was similar the Pix2Pix generator (Isola et al. 2016):
there were 9 blocks stacked sequentially; the first 4 blocks consisted of 2D convolutional layer, followed by a leaky ReLU nonlinearity with slope 0.2 and a batchnorm layer; the next 4 blocks consisted of transpose a convolutional layer, followed by Icaky ReLU layer of slope 0.2 and a batchnorm layer; the final block had a transposed convolutional layer followed by a sigmoid nonlinearity. The input channels, output channels, kernel size, stride and padding for each of the convolutional or transposed convolutional layers in the 9 blocks were as follows: 1 (1, 8, 2, 2, 2, 1), 2 (8, 16, 2, 2, 1), 3 (16, 32, 2, 2, 1), 4 (32, 64, 3, 0), 5 (128, 32, 3, 2, 1), 6 (64, 16, 2, 2, 1). (32, 8, 3, 2. 1), 8 (16, 4, 2, 2, 1), 9 (4, 1, 1, 1, 0). There were skip-connections from block 1 to block 8, block to block 7. block 3, block and from block 4 block 5. 200-dimensional white noise was injected into the fifth block, after convolving it with ID convolutional filter and multiplying it with the output of the fourth block.
The discriminator network was again similar to the Pix2Pix discriminator: we concatenated the image from the generator or prior with the noisy image from the simulator, and passed this through blocks consisting of a 2D convolutional layer, leaky ReLU nonlinearity of slope 0.2 and batch-norm layer, and finally through a 2D convolutional layer and a sigmoid nonlinearity. The input channels, output channels, kernel size, stride and padding for each of the convolutional were follows: 1 (2, 8, 2, 1), 2 (8, 16, 2, 2, 1), 3 (16, 32, 2, 2, 1), 4 (32, 64, 2, 2, 1), 5 - (64, 1, 3, 0).
Training details The generator and discriminator were trained in tandem for 10k epochs, with 800k training samples, of which 100 were held out for testing. We used a batch size of 800, the cross-entropy loss and the Adam optimiser with learning rate= 0.0002, B1 0.5 and 33 = 0.99 for both networks. In each epoch, there was a single discriminator update for every second generator update. To ensure that training was stable, we used spectral normalisation for the discriminator weights, and clipped the gradient norms for both networks to 0.01.
We trained NPE using the implementation the sbi package Tejero-Cantero et al., 2020). We set training hyperparameters batch size which we set to (in order to ensure that we did not run out of memory while training), and the number of hidden units in the density estimators which set to 100. 28x 28 dimensional
NPE
described in Lueckmann et al. 12211) except for the training
observations were passed directly to the flow, without an embedding net or computing any summary statistics, was also the case for GATSBI. We trained with exactly the same 800k training samples used for GATSBI.
14

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Published as a conference paper at ICLR 2022
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16


Reviews:
[{"review_id": "bNGtu9Sl5Q6", "reviewer": "Reviewer_DidY", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I think the novelties of this work are highly related to the importance of the considered simulation-based inference (SBI). Accordingly, it is worth discussing the wide applications of the SBI.\n\nIn Section 2.4, it's stated that ``GATSBI, ..., performs an optimization that also finds a minimum of the reverse $D_{KL}$\". I think this is an obvious fact, because GATSBI ideally minimizes the JSD whose minimum is identical to that of the reverse $D_{KL}$. I don't understand why using a whole subsection to state that fact? ", "summary_of_the_paper": "This work proposes to exploit GANs to do simulation-based inference (SBI) for black-box simulators. Generally speaking, the proposed GATSBI is identical to do the original GAN in a joint space, with the true distribution $p(x,\\theta)=p(\\theta)p(x|\\theta)$ ($p(x|\\theta)$ denotes the simulator) and the fake distribution $q(x,\\theta)=p(x)q_{\\phi}(\\theta|x)$, where $p(x)$ is the marginal of $p(x,\\theta)$. The key is that for a black-box simulator, one can query a lot of data pairs $(x,\\theta)$. \n", "summary_of_the_review": "Technically, the novelties are limited, as both the original GAN and the reverse-KL GAN are well-known in GAN research fields.\nHowever, if the simulation-based inference (SBI) for black-box simulators is quite important in some research fields, then there are certainly contributions from this work.\n\nEmpirically, the performance of the proposed GATSBI is only comparable to the baselines. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "SoeOUvjlvK", "reviewer": "Reviewer_eKE2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n- This work contributes a new algorithm for SBI for the setup of high-dimensional parameter spaces or implicit priors. This is a significant contribution in itself since this setup is currently out-of-reach for most (if not all) simulation-based inference algorithms.\n- The paper is generally very well documented. The position of GATSBI within the context of previous methods for SBI is discussed in detail, much beyond what is usually found in other SBI papers.\n- The connection to LFVI is made explicit and discussed in detail in the supplementary materials. Experimental results indicate that both methods perform similarly on the low-dimensional problems.\n- The experiments demonstrate the usefulness of GATSBI on two high-dimensional problems. The Noisy Camera model experiment is the most convincing, as it shows its real potential in comparison to other methods.\n- The quality of the approximate posteriors is diagnosed with SBC. It reveals that GATBSI provides well-calibrated posteriors.\n\nWeaknesses:\n- GATSBI is technically very close to LFVI, proposed initially for likelihood-free in hierarchical implicit models. For this reason, the novelty is limited.\n- The experimental validation is limited to two common benchmark problems and two high-dimensional problems. GATSBI does not perform as well as NPE and NLE on the low-dimensional benchmarks.\n- GATSBI comes with all the challenges and issues of training GANs.", "summary_of_the_paper": "This paper introduces GATSBI, an algorithm for simulation-based inference based on adversarial training. This algorithm paves a promising avenue for applying SBI in high-dimensional parameter spaces or with implicit priors -- a setup currently out-of-reach for simulation-based inference algorithms. Experiments deliver a convincing proof-of-concept on a 784-dimensional parameter space, which none of the current SBI algorithms can solve. Experiments on lower-dimensional parameter spaces are less convincing.", "summary_of_the_review": "In this work, the authors try to address a major obstacle in simulation-based inference. Very few SBI approaches can operate in high-dimensional parameter spaces or with implicit priors, but GATSBI constitutes a convincing proof-of-concept that ought to be investigated further. Experiments are carried out properly, although the experimental validation could have been more thorough. The technical novelty is limited.\n\nDespite some acknowledged weaknesses, I am overall positive about GATSBI. I am willing to recommend it for acceptance.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Kb1S4R-1cr", "reviewer": "Reviewer_cWGX", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I've enjoyed reading the paper: it is written with great attention to detail. The problem is introduced well, the method is clearly motivated and presented in great detail. The highlighted connections to existing methods are nice.\n\nThe biggest question mark for me is in novelty. I commend the authors for discussing the connection to LFVI, but this does highlight the fact that the differences between the proposed method and LFVI are arguably minor. Apart from the additional term in the loss (the role of which is unclear), the difference in the used divergence is the main one. As also pointed out by authors in the appendix, the best choice for the divergence is not clear, and also likely problem-specific. It would be interesting to better understand the role of the divergence in such methods, at least empirically.\n\nIn addition, work by Adler & Öctem (2018) seems to cover similar ground: a comparable method (but, again, with a difference divergence measure) is proposed, while Adler & Öctem also highlight that their method works with an implicit prior, and evaluate the method on a high-dimensional inverse problem.\n\nFinally, the experimental results are somewhat underwhelming. It's not surprising that we don't win much on the low-dimensional problems, but I wish authors explored more high-dimensional inverse problems, perhaps also more realistic ones. \n\nReferences:\n- Adler & Öctem, Deep Bayesian Inversion, 2018. https://arxiv.org/abs/1811.05910", "summary_of_the_paper": "Authors propose to use a GAN as an implicit posterior model for performing likelihood-free Bayesian inference, where the generator is trained to sample from the posterior, while the discriminator is trained to distinguish between parameter samples from the prior and the posterior *conditioned on the data sample*. Authors connect the method to prior work in adversarial (variational) inference, and also consider a *sequential* variant of the method, where the posterior approximation is refined iteratively for a particular datapoint. Authors show that the method matches the results of baselines on low-dimensional inference tasks, while also scaling to high-dimensional problems where classic likelihood-free inference methods don't work well.", "summary_of_the_review": "Overall, while the paper is written excellently, my doubts about novelty and the extent of evaluation are significant enough to put the paper slightly below the bar for acceptance. More extensive evaluation and/or discussion in the context of the prior work would help.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "BPKyo7IuNC3", "reviewer": "Reviewer_MYcd", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper has the potential for broad impact. The number of applications for simulation based inference in science and engineering are far broader than recognized in the ML community.\n\nMuch of the prior art, as referenced in the paper, uses ABC, which is really a method of last resort and should be avoided. By presenting a GAN-style alternative to ABC it allows simulation based inference to be applied in much higher dimensional settings than prior methods. The authors show in the paper that the GAN equilibrium distribution is the one that performs exact Bayesian inference through the simulator. It is a clever approach.\n\nWeaknesses:\n- The VAE comparison doesn't seem as useful to the reader as more details on how to do actually get training to work. Perhaps there should be a swap of what goes in the appendix.\n- The sequential GATSBI setup in eq (7) doesn't make much sense. The reweighting factor requires the computation of the marginal p(x), which is not tractable in general. This setup needs to be clarified.\n\nQuestions:\n- The paper assumes we do not have access to the likelihood or gradients of the simulator output. In general, the likelihood will be hard to compute for complex simulations. However, with modern autodiff, it is not unreasonable to assume that the gradients may be available even in a complex simulation. Have the authors considered an extension where gradients are available?\n\nNitpicks:\n- Subfigure captions typically go below the figure\n- The figures are not BW friendly\n- It is typically i.e., not i.e.", "summary_of_the_paper": "This paper presents a new method to do inference in simulation based systems. It performs \"compiled inference\" in a sense using a neural network to infer the latent inputs to the simulation. It accomplishes this using a clever GAN-based training objective.", "summary_of_the_review": "The paper has big potential impact. However, some of the extensions in the paper don't make much sense as presented.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper explores the application of generative adversarial networks as posterior models in simulation-based inference. A new method is proposed, and its connections with related work are studied. The proposed method is empirically evaluated on joint inference of up to 784 parameters.

The reviews are borderline, with one weak reject, two weak accepts, and one strong accept. Overall, the paper is well-written and well-executed. Its main strength is the promising performance of the proposed method in high-dimensional parameter spaces, which are out-of-reach for many existing approaches. The main weakness of the paper is its lack of novelty: the proposed method is only marginally different from already existing ones, while the paper could have explored the differences to a greater extent.

On balance, I'm leaning towards recommending the paper for acceptance. Despite the lack of novelty, the paper is well executed with potential impact in high-dimensional simulation-based inference.
Decision: Accept (Poster)