Maximum Likelihood Learning of Energy-Based Models for Simulation-Based Inference
Authors: Pierre Glaser
Michael Arbel
Arnaud Doucet
Arthur Gretton
Venue: Submitted to ICLR 2023 (2023)
Keywords: Simulation Based Inference
Energy Based Models
Maximum Likelihood

Abstract: We introduce two Synthetic Likelihood methods for Simulation-Based Inference (SBI), to conduct either amortized or targeted inference from experimental observations when a high-fidelity simulator is available. Both methods learn a Conditional Energy-Based Model (EBM) of the likelihood using synthetic data generated by the simulator, conditioned on parameters drawn from a proposal distribution. The learned likelihood can then be combined with any prior to obtain a posterior estimate, from which samples can be drawn using MCMC. 
Our methods uniquely combine a flexible Energy-Based Model and the minimization of a KL loss: this is in contrast to other synthetic likelihood methods, which either rely on normalizing flows, or minimize score-based objectives; choices that come with known pitfalls. Our first method, Amortized Unnormalized Neural Likelihood Estimation (AUNLE), introduces a tilting trick during training that allows to perform inference using efficient MCMC techniques. Our second method, Sequential UNLE (SUNLE), employs a doubly intractable approach in order to re-use simulation data and improve posterior accuracy for a specific observation. 
We demonstrate the properties of both methods on a range of synthetic datasets, and apply it to a neuroscience model of the pyloric network in the crab, matching the performance of other synthetic likelihood methods at a fraction of the simulation budget.

Under review as a conference paper at ICLR 2023
MAXIMUM LIKELIHOOD LEARNING OF ENERGY- BASED MODELS FOR SIMULATION-BASED INFERENCE
A Anonymous authors Paper under double-blind review
A ABSTRACT
We introduce two synthetic likelihood methods for Simulation-Based In- ference (SBI), to conduct either amortized or targeted inference from ex- perimental observations when high-fidelity simulator is available. Both methods learn a conditional energy-based model (EBM) of the likelihood using synthetic data generated by the simulator, conditioned on parameters drawn from a proposal distribution. The learned likelihood can then be combined with any prior to obtain a postcrior estimate, from which sam- ples can be drawn using MCMC. Our methods uniquely combine a flexible Energy-Baser Model and the minimization of a KL loss: this is in contrast to other synthetic likelihood methods, which either rely on normalixing flows, or minimize score-based objectives; choices that come with known pitfalls. Our first method, Amortized Unnormalized Neural Likelihood Estimation (AUNLE), introduces a tilting trick during training that allows to significantly lower the computational cost of inference by enabling the use of efficient MCMC techniqucs. Our second method, Sequential UNLE (SUNLE), utilizes new conditional EBM learning technique in order to re-use simulation data and improve posterior accuracy on a specific dataset. We rate the properties of both methods on range of synthetic datasets, and apply them to a neuroscience model of the pyloric network in the crab, matching the performance of other synthetic likelihood methods at fraction of the simulation budget.
INTRODUCTION
Simulation-baset modeling expresses system as a probabilistic program (Ghahramani, 2015), which describes, in a mochanistic manner, how samples from the system are drawn given the parameters of the said system. This probabilistic program can be concretely implemented in a computer- - as a simvulator from which synthetic parameter-samples pairs can be drawn. This setting is common in many scientific and engineering disciplines such stellar events in cosmology (Alsing et al., 2018; Schafer & Freeman, 2012), particle collisions in a particle accelerator for high energy physics (Fberl, 2003; Sjöstrand et al., 2008), and biological neural networks in neuroscience (Markram et al., 2015; Pospischil et al., 2008). Describing such systems using a probabilistic program often turns out be easier than specifying the underlying probabilistic model with a tractable probability distribution. We consider the task of inference for such systems, which consists in computing the posterior distribution of the parameters given observed (pon-synthetic) data. When likelihood function of the simulator available alongside with prior belief on the parameters, inferring the posterior distribuution of the parameters given data is possible using Bayes' rule. Traditional inference methods such as variational techniques (Wainwright & Jordan, 2008) or Markov Chain Monte Carlo (Andrieu et al., 2003) can then be used to produce approximate posterior samples of the parameters that are likely to have generated the observed data. Unfortunately, the likelihood function of simulator is computationally intractable in general, thus making the direct application of traditional inforence techniques unusable for simulation-base modelling.
Simulation-Bon Based Inference (SBI) methods (Cranmer et al., 2020) are methods specifically designed to perform inference in the presence of a simulator with an intractable likelihood. These methods repeatedly generate synthetic data using the simulator to build estimate of

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the posterior, that either can be used for any observed data (resulting in so-called amortized inference procedure) or that is taryeted for a specific observation. While the accuracy of inference increases more simulations are rum, so does computational cost, especially when the simulator is expensive, which is common in many physics applications (Cranmer et al., 2020). In high-dimensional settings, early simulation-based inference techniques such as Approximate Bayesian Computation (ABC) (Marin et al., 2012) struggle to generate high quality posterior samples a reasonable cost, since ABC repeatedly rejects simulations that fail to reproduce the observed data (Beaumont et al., 2002). More recently, model-based inference methods (Wood, 2010; Papamakarios et al., 2019; Hermans et al., 2020; Greenberg el al., 2019), which encode information about the simulaatr via a parametric densily (-ratio) estimator of the data, have been shown to drastically reduce the number of simulations needed to reach a given inference precision (Tueckmann et al., 2021). The computational gains are particularly important when comparing ABC to targeted SBI methods, implemented in a multi-round procedure that refines the estimation of the model around the observed data by sequentially simulating data points that are closer to the observed ones (Greenberg et al., 2019; Papamakarios ct al., 2019; Hermans ct al., 2020).
Previous model-based SBI methods have used their parametric estimator to learn the likelihood (e.g. the conditional density specifying the probability of an observation being simulated given a specific parameter set, Wood 2010; Papamakarios et. al. 2019; Pacchiardi & Dutta 2022), the setiihhood----arrgien
ratio (Hermans et al., 2020), or the posterior function directly (Greenberg et al., 2019). We focus in this paper on likelihood-basee Syuthetic Likelihood; SL, in short) methods, of which two main instances exist: (Sequential) Neural Likelihood (Papamakarios et al., 2019), which learns a likelihood estimate using a normalizing flow trained by optimizing a Maximum Likelihood (ML) loss; and Score Matched Neural Likelihood (Pacchiardi & Dutla, 2022), which learns unnormalized (or Eneryy-Basser LeCun et al. 2006) likelihood model trained using conditional score matching. Recently, SNL was applied successfully to challenging neural data (Deistler et al., 2021). However, limitations still remain in the approaches taken by both SNL and SMNL. One the one hand, flow-based models may need to use very complex architectures to properly approximate distributions with rich structure such as multi-modality (Kong & Chaudhuri, 2020; Cornish et al., 2020). On the other hand, score matching, the objective of SMNLE, minimizes the Fisher Divergence between the data and the model, a divergence that fails to capture important features of probability distributions such as mode proportions (Wenliang & Kanagawa, 2020; Zhang et al., 2022). This is unlike whose maximizers satisfy attractive theoretical properties (Bickel & Doksum, 2015).
(also called
watniiimmm-iikiihho based-objectives,
Contributions. In this work, we introduce Amortized Unnormalized Likelähood Neural Estimation (AUNLE), and Sequential UNLE, a pair of SBI Synthotic Likelihood methods performing respectively sequential and targeted inference. Both methods Icarn Conditional Energy Based Model of the simulator's 's likelihood using a Maximum Likelihood (ML) objective, and perform MCMC on the posterior estimate obtained after invoking Bayes' Rule. While posteriors arising from conditional EBMs exhibit a particular form of intractability called double intractability, which requires the use of tailored MCMC techniques for inference, we train AUNLE using a new approach which we call tilting. This approach automatically removes this intractability in the final posterior estimate, making AUNLE compatible with standard MCMC methods, and significantly reducing the computational burden of inference. Our second method, SUNLE, departs from AUNLE by using a new training technique for conditional EBMs which is suited when the proposal distribution is not analytically available. While SUNLE returns a doubly intractable posterior, we show that inference can be carried out accurately through robust implementations of doubly-intractale demonstrate the properties of AUNLE and SUNLE on an array of synthetic benchmark models (Lueckmann el. al., 2021), and apply SUNLE to a neuroscience model of the crab Cancer borealis, increasing posterior accuracy over prior art while needing only a fraction of the simulations required by the most efficient prior method (Glöckler et al., 2021).
MCMC methods. We
2 BACKGROUND
Simulation Based Inference (SBI) refers to the set of methods aimed at estimating the posterior p(f]x,,) of some unobserved parameters @ given some observed variable F, recorded from physical system, and a prior p(0). In SBI, one assumes access to simulator
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SMNLE
NLE
AUNLE (Ours)
Ground Truth
Likelihood
Posterior
Figure Performance of SMNLE, NLE and AUNLE traing using a simulator with bimodal likelihood p(z(0), and gaussian prior p(0) using 1000 samples. Top: Simulator likelihood p(x(0)) for some fixed dg- Bottom: posterior estimate.
G: (0,u) + y G(O, u), from which samples y/0 can be drawn, and whose associated likelihood p() accurately matches the likelihood p(x(0) of the physical system of interest. Here, u represents draws of all random variables involved in performing draws of 2/0. By slight abuse of notation, representing data from the physical syslem of interest, and the simulated random variable y draw from the simulator:
will not distinguish between the physical random variable
will use r for both. The complexity of the simulator (Cranmer et al., 2020) prevents access to a simple form for the likelihood p(r(0), (0), making standard Bayesian inference impossible. Instead, SBI methods perform inference by drawing parameters from a proposal distribution л (0), and use these parameters as inputs to the simulator G to obtain a set of simulated pairs (Ir, 0) which they use to compute posterior estimate of p(0)z). Specific SBI submethods have been designed to handle separately the case of amortized inference, where the practitioner seeks to obtain a posterior estimate valid for any zo (which might not be known a priori), and taryeted inference, where the posterior estimate should maximize accuracy for a specific observed variable Zg. While amortized inference methods set their proposal distribution methods iteratively refine their proposal to focus their simulated observations around the targeted To through sequence of simulation-training rounds (Papamakarios et al., 2019).
to be the prior p. targeted inference
2.1 (CONDITIONAL) ENERGY- BASED MODELS.
Energy-Baser Models (LeCun et al., 2006) are unnormalized probabilistic models the form
(x)
(x)
Z(b)= - - (x),())
Z(
where ((b) the intractable normalizing constant of the model, and Eg is called the eneryy function, usually set to be neural network with weights v. By directly modelling the density p(z) of the data through a flexible energy function, simple EBMs can capture rich geometries and multi-modality, whereas other model classes such normalizing flows may require more complex architecture (Cornish et al., 2020). The ficxibility of EBMs comes at the cost of having an intractable density qo (x) due to the presence of the normalizer Z(v), increasing the challenge of both training and sampling. In particular, an EBM's log-likelihood log and associated gradient V log 9t both contain terms involving the (intractable) normalizer (w²):
intractable
intractable
(1)
(x) - Ess (z) log Z(b),
- + ExNgo (z).
making exact training of EBMs via Maximum Likelihood impossible. Approximate likelihood optimization can be performed using Gradient-Based algorithm where at each iteration
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k the intractable expectation (under the EBM qex) present in Vp log ge. to, one under a particle approrimation q= * D wyyy of Iv- The particles y(u forming q are traditionally set to be samples from MCMC chain with invariant distribution Tox with uniform weights w; N, while recent work EBM for high-dimensional image data uses an adaplation of Langevin Dynamics (Raginsky et al., 2017; Du & Mordatch, 2019; Nijkamp et al., 2019; Kelly & Grathwohl, 2021). We outline the traditional ML learning procedure for EBM in Algorithm 2, where make_ laaeee_articee__e_ee apppoooapppppp.pppp q0) is a generic routine producing a particle approximation of target unnormalized density q and initial particle approximation qu-
is replaced by
Energy- Based Models naturally extended to both joint EBMs qo(( -.(000))) (Kelly
Z
& Grathwohl, 2021; Grathwohl et al., 2020) and conditional EBMs (CEBMs Khemakhem et al. 2020; Pacchiardi & Dutta 2022) of the form:
(x<(0 e Ex,(00
7(0:b) f e-
7(0, ())
Unlike joint and standard EBMs, conditional EBMs define family of conditional densities qu (x](),, each of which is endowed with an intractable normalizer 2(0, v)).
2.2 SYNTHETIC LIKELIHOOD METHODS FOR SBI
Synthetic Likelihood (SL) methods (Wood, 2010; Papamakarios al., 2019; Pacchiardi & Dutta, 2022) form a class of SBI methods that learn a conditional density model go (x/0) of the unknown likelihood p(x[0) for every possible pair of observations and parameters (I, 0). The set (1(0), & E v is model class parameterised by some vector & E V, which recent methods set be neural network with weights v.. We describe the existing Neural SL variants to date.
Neural Likelihood Estimation (NLE, Papamakarios et al. 2019) sets qu to (normalizod) flow-based model, and is optimized by maximizing the average conditional log- likelihood Ex(0)p(10)) log qu (x)0). NLE performs inference by invoking Bayes' rule to obtain an unnor- (=())0)p()) malized posterior estimate (0|x)) J au (=)0)) x (x0) from which samples can be drawn either using MCMC, or Variational Inference (Glöckler et al., 2021).
Score Matched Neural Likelihood Estimation (SMNLE, Pacchiardi & Dutta 2022) models the unknown likelihood using conditional Energy-Based Model (x(6)) of the form of Equation (2), trained using a score matching objective adapted for conditional density estimation. The use of an unnormalized likelihood model makes the posterior estimate obtained via Bayes' Rule known up to a O-dependent term:
go (0)z) x (z x e (2,0
Z(0)= = / (10)=/e-Bwte.0) (-Eo(e,0)
2(0)
(3)
Intractable
Posteriors of this form are called doubly intractable posteriors (Moller et al., 2006). In the case where the likelihood go (z(()) can be sampled from, Møller et al. (2006); Murray et al. (2006) have proposed tractable MCMC methods that draw an auxiliary variable y go (x]() at every iteration to compute the acceptance probability of the proposed sample. Importantly, these MCMC methods still admit qu (flz)) as their invariant distribution, making inference as exact as in standard MCMC methods. In the case of SMNLE however, (=(xx(0) cannot be tractably sampled from; SMNLE instead uses an approzimate doubly intractable method, which replaces the exact sample by the result of an MCMC chain with invariant distribution qu (z)0). Even though this variant introduces an additional approximation not present in standard ( ("singly" intractable) MCMC algorithms, the distance between the true posterior and the distributions of the MCMC samples can be bounded under specific assumptions (Alquier et al., 2016).
Both the likelihood objective NLE and the score-based objective of SMNLE do not involve the analytic expression of the proposal n,, making it easy to adapt these methods for either
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amortized or targeted inference. To address the limitations of both methods mentioned in the introduction, we next propose a method that combines the use of flexible Energy-Basee Models as in SMNLE, while being optimized using likelihood loss as in NLE.
3 UNNORMALIZED NEURAL LIKELIHOOD ESTIMATION
In this section, we present our two methods, Amortized-UNT.E and Sequential-UNL.. Both AUNLE and SUNLE approximate the unknown likelihood p(z for any possible pair of (æ, 0) using conditional Energy-Based -Baed Model (x(())) () in Equation (2), where Ey is some neural network. Additionally, AUNLE and SUNLE are both trained using a likelihood based loss; however, the training objectives and inference phases differ to account for the specificities of amortized and targeted inference, as detailed below.
3.1 AMORTIZED UNLE
Given a likelihood model (2 0), a natural learning procedure would involve fitting a model (x(()))=(0) of the true "joint synthetic" distribution =(0)p(=]0), NLE does. However, show that using an alternative tilted version of this model allows to compute posterior that more tractable than the ones computed by other SL methods relying on conditional EBMs such as SMNLE (Pacchiardi & Dutta, 2022). Our method, AUNLE, fits joint probabilistic model oyir of the form:
20 0) m(0)e- (1,00)
(x) π (00e (o..0))dddd.
(4)
7x(1)
x (x, 0)] using an instance of by maximizing its log-likelihood L. = (x(0)1(210) log Jo.
Algorithm 2. The gain in tractability offered by AUNLE is direct consequence of the following proposition, its joint model.
Proposition 1. Let Pa == aa (-10), vb E }}, and qu E Pa. Then we have:
(likelihood modelling) a (x = (x(10))
(joint model tilting) a (I, 0) = (()) for f(0) ())x(()))) ((Z, ()mufffrrmiittion) If p(-10) E Pe. then the vp- minémizing (a( satisfies:
(x(2)) p(z|0), and (60, 1²) = Zz (v)).
Proof. The first point follows by holding 0 fixed in Lo.x (x. (x,0). To prove the second point,
notice that I (z. 0) Z(0,ab) =(0)e Bla.9) 7(0,0)
2.(v) (0) 200.) For the last point, note that
B(n.0)
2(0.+)) (+(w)
at the optimum, we have that a (x,, 0) = 7 (z))) Integrating out x on both sides of
the equality yields (0))=(0) (0),, proving the result.
Proposition shows that AUNLE indeed learns likelihood model go (a) () through joint model g0.= tilting the prior æ with f(0). Importantly, this tilting guarantees that the optimal likelihood model will have normalizing function [(; v) constant (or uniform) in 0,
standard unnormalized posterior qu- (0|z) p(0)= E (0.r). from which samples can be drawn using classical MCMC techniques, as for NLE. AUNLE's posterior contrasis with the posterior of SMNLE (Pacchiardi & Dutla, 2022), an amorlized SBI method which also computes posterior using a conditional EBM of the likelihood, but that remains doubly intractable, as discussed in Section 2. The gain in tractability of AUNLE's posterior is beneficial from an inference accuracy standpoint as it removes the need use an otherwise approzimate doubly-intractable technique when performing inference. Importantly, such a property is also beneficial from a computational cost standpoint, since approximate coubly-intractable methods require running (inner) MCMC chain with target qu- (xl0) for every iteration of the (outer) MCMC chain with target qu- (f]x), roughly squaring the computational cost of standard MCMC methods. This computational advantage is all the more important since AUNLE returns an amortized posterior, valid for any observed data To and which may be thus sampled from more than once. We confirm in Appendix B.3 that
reducing AUNLE's posterior
(0+)
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the (Z, mniifoormnninoonn of AUNLE's posterior, which is only guaranteed in well-specified setting, at the true optimum v², holds well in practice.
Algorithm 1 Amortized-UNL.
Algorithm 2 maximize_ebm__og_u. v/b)
Input: prior p(0), simulator G, budget N Output: Posterior estimate (f(x))
Input: Training Data D = (x', 03 Ji=1:
0') Initial
EBM parameters v/n
Initialize vo. " x xe- C (o.(())) ((1) P
Output: Density estimator qo (z, 0)
Initialize 40h (2) x e Es (1, @0 x Σi 0(01,9))
for i NN do
for k = 0,.... K - do
Draw 0 ~ n, x G(D,,)
i == make particllteelaaa www.clle_pppooxpppe Ayn 7 G = N Σ V. Faye ('),0'))))))))) +Bgg V Eap (z,, 0)
Add (0,z) to D end for
Get v/*t == max.aiizzeeeo.. ebm lareeeeemmm____e log_1 vo)
vk+1 = ADAM( G)
Set ge- (0)z) == Bu- (z.00 p(0))
end for Return 9ox
Infer using MCMC on 2v- (0)z))
3.2 TARGETED INFERENCE USING SEQUENTIAL- UNLE
In this section, we introduce our second method, Sequential-UNLE (or SUNLE in short), which performs largeted inference for a specific observation ø SUNLE follows the traditions methodology of largeled inference by splitting the simulator budget N over R rounds (often equally), where in each round r, a likelihood estimate (x(()) in the form of a conditional EBM is trained using all the currently available simulated data D. This allows to construct new posterior estimate qu: (0|z)=e Eup (z.0) ((0))))))))))))))))))))))))))( p(0)/Z(v)t 8) which is used to sample parameters
(0()) y that are then provided to the simulator for generating new data x² G(O()). The new data are added to the set D and are expected to be more similar to the observation of interest xo- This procedure allows to focus the simulator budget on regions relevant to the single observed data of interest Io, and, as such, is expected to be more efficient in terms of the simulator use than amortized methods (Lueckmann et al., 2021). Next, we discuss the learning procedure for the likelihood model and the posterior sampling.
Learning the likelihood. At each round r, the effective proposal of the training data. available can be understood (provided the number of data points drawn at reach rounds is randomized) as a mixture probability: т == () (0]x,)++... +ww;-, (0]r.) (0 which is used to update the likelihood model. In this case, the analytical form of v is unavailable as it requires computing the normalizing constants of the posterior estimates at each round, thus making the tilting approach introduced for AUNLE impractical in the sequential setting. Since currently available likelihood objectives (Kelly & Grathwohl, 2021; Du & Mordatch, 2019) for EBMs take as input unconditional (or joint) EBMs, likelihood learning approach building on such objectives would require modeling and learning the entire joint distribution
o (0))(z:()) including the proposal r0)). This latter point. is problematic since is not needed for inference, and can be highly complex (as is set to be the current posterior estimate), increasing the difficulty of training. Instead, SUNLE learns a likelihood model maximizing the average conditional likeliiod,,,,,,
intractable
N Σ Eo 0¹) + E Ee (5)
((v) Σ log (z'10)) VoL(E)
1
where (') are the current samples. Unlike standard EBM objectives, this loss directly targets the likelihood (x)0), thus bypassing the need for modelling the proposal æ. We propose in Algorithm 4 a method that optimizes this objective (previously used for normalizing flows in Papamakarios et al., 2019) when the density estimator is a conditional EBM. The intractable term of Equation (5) is an average over the EBM probabilities conditioned on all parameters from the training set, and thus differs from the intractable term of (1), composed of a single integral. Algorithm 4 approximates this term during training by keeping track of
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òg, per conditional density qu (-16') comprised of single particle. The algorithm proceeds by updating only a batch of size B of such particles using an MCMC update with target probability chain 9s, (-10"), where 1 is the EBM iterate at iteration & of round r. Learning the likelihood using Algorithm 4 allows to use all the existing simulated data during training without re-learning the proposal, maximizing sample efficiency while minimizing learning complexity. The multi-round procedure of SUNLE is summarized in Algorithm 3.
one particle approximation
Posterior sampling. Unlike AUNLE, SUNLE's likelihood estimate du does not inherit the (Z. mnnfforrmmni property guaranteed by Proposition 1. As a consequence, its posterior quz (0|z) is doubly intractable as it involves the intractable normalizing constant ( Nevertheless, we propose to sample from qu (0|z) using tochniques. We consider a custom robust doubly intractable implementation that allows for accurate inference even on challenging posteriors with no paramcter tuning other than compute-related parameters like the number of warmup steps.
MCMC
Algorithm 3 Sequential-UNLE
Algorithm 4 maximizee_cebm 1oo__ogg__11 (D, vo)
Input: prior p(0), simulator G, budget N, no. In-utt: Training data D = {0()) 10 Ini- tial EBM parameters 4/ Output: Cond. Density estimator (x())
rounds R
Output: Posterior estimate (0(z)
Initialize z(0) p. vu, q X 6 Bvg (())) Initialize quu X e (0,2), { = 6z J.v
Get D (0()) π (0), x())) G(O,-))
for k = 0, ... K - 1 do
for r= 1...., R do
for i 0,.... N do
Get ,;, == maximize cebm log_ 1D, 1
== make laake_particle.e appprtiiceeappro((a... ),
Set x
e Bup 9) p(0) ()
end for
Get {6) N/R / mu.. via Doubly- Intr. MCMC
G= N Σv Ev (x' (x', +Ba E. (x' 0*)
Get D = DU 10(0, (()) ~ -)))
W ADAM the G)
end for
end for
Infer using Doubly-Intr. MCMC on Iop (0|z) Return 40x
4 EXPERIMENTS
In this section, study the performance and properties of AUNLE and SUNLE in three different settings: a toy model that highlights the failure modes of other synthetic likelihood methods, a series of benchmark datasets for SBI, and real life neuroscience model.
Experimental details AUNLE and SUNLE are implemented using jax (Frostig el al., 2018). We approximate expectations of AUNLE's joint EBM using 1000 independent MCMC chains with a Langevin kernel parameterised by a step size o, that automatically update their step size to maintain acceptance rate of 0.5 during a per-iteration warmup period, before freezing the chain and computing a final particle approximation. Additionally, we introduce new method which replaces the MCMC chains by a single Sequential Monte Carlo sampler (Chopin et al., 2020; Del Moral et al., 2006), which yields a similar performance as the LCCC approach discussed above, but is more robust for lower computational budgets (see Appendix A.2). The particle approximations are persisted across iterations (Tieleman, 2008; Du & Mordatch, 2019) to reduce the risk of learning a "short run" EBM (Nijkamp et al., 2019; Xie et al., 2021) that would not approximate the true likelihood correctly (see Appendix B.2 for a detailed discussion). All experiments are averaged across 5 random sceds (and additionally 10 different observations To for benchmark problems). We provide all code! nceded to reproduce the experiments of the paper. Training and inference are computed using single RTX5000 GPU. For benchmark models, single round of EBM training takes around minutes on a GPU (see Appendix B.4).
4.1 A TOY MODEL WITH A MODAI LIKELIHOOD
First, we illustrate the issues that SNLE and SMNLE can face when applied to model certain distributions using a simulator with a bi-modal likelihood. Such a likclihood is known to be hard to model by normalizing flows, which, when fitted on multi-modal data, will assign
'nttps://gi htt///gtthuub..t com/ anon- autors-a a- wwnle-ineiee iclr/unle
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Two Moons
SLCP
Lotka Volterra
Gaussian Linear Uniform
A-UNLE (Ours)
1.0
1.0
1.0
1.0
SMNLE
0.8
0.8
0.8
0.8
NLE
0.6
0.6
0.6
0.6
S-UNLE
(Ours)
0.8
0.8
0.8
0.8
SNLE
0.6
0.6
0.6
0.6
10a
10
10a
10a
10
16
10
10
10
10a
10
10
Num, Simulations
Num. Simulations
Num. Simulations
Num. Simulations
Figure 2: Performance of AUNLE (resp. SUNLE) compared with NLE and SMNLE (resp. SNLE), using the Classifier Accuracy Metric (Lueckmann et al., 2021) (lower is better). AUNLE and SUNLE exhibit robust performance across wide array of problems. Additional details on the experimental setup can be found in Appendix B.5.
high-density values to low-density regions of the data in order to "connect" between the modes of the true likelihood (Cornish et al., 2020). Moreover, multi-modal distributions are also poorly handled by score-matching, since score-matching minimizes the Fisher Divergence between the model and the data distribution, a divergence which does not account for mode proportions (Wenliang & Kanagawa, 2020). Figure shows the likelihood model learned by NLE and SMNLE on this simulator, which exhibit the pathologies mentioned above: the score-matched likelihood only recovers a single mode of the likelihood, while the flow-based likelihood has distorted shapc. In contrast, AUNLE estimates both the likelihood and the posterior accurately. This suggests that AUNLE has an advantage when working with more complex, possibly multi-modal, distributions, we confirm later in Section 4.3.
4.2 RESULTS ON SBI BENCHMARK DATASETS
We next study the performance of AUNLE and SUNLE on 4 SBI benchmark datasets with well-defined likelihood and varying dimensionality and structure (Lueckmann al., 2021):
SLCP: A toy SBI model introduced by (Papamakarios et al., 2019) with a unimodal gaussian likelihood ()) The dependence of p(=(00) 0 is nonlinear, yielding a complex posterior.
The Lotka- Volterra Model (Lotka, 1920): An ecological model describing the evolution of the populations of two interacting species, usually referred to as preys and predators.
Two Moons: A famous 2-d toy model with posteriors comprised of two moon-shaped regions, and yet not solved completely by SBI methods.
Gaussian Linear Uniform: A simple gaussian generative model, with a 10-dimensional parameter space.
These models encompass a variety of posterior structures (see Appendix B.1 for posterior pairplots): the two-moons and SLCP posteriors are multimodal, include cutoffs, and exhibit sharp and narrow regions of high density, while posteriors of the Lotka- Volterra model place mass on very small region of the prior support. We compare the performance of AUNLE and SUNLE with NLE and its sequential analogue SNLE, respectively: NLE and SNLE represent the gold standard of current synthetic likelihood methods, and perform particularly well on benchmark problems (Lueckmann el. al., 2021). We use the same set of hyperparameters for all models, and use a 4-layer MLP with 50 hidden units and swish activations for the energy function. Results are shown in Figure 2.
While some fluctuations exist depending on the task considered, these results show that the performance of AUNLE (and SUNLE when targeted inference is necessary) is on par with that of (S)NLE, thus demonstrating that a generic method involving Energy-Baut -ased models can be trained robustly, without extensive hyperparameter tuning. Interestingly, the model where UNLE has the greatest advantage over NLE is Two Moons, which is the benchmark that exhibits likelihood with the most complex geometry; in comparison,
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the three remaining benchmarks have simple normal (or log-normal) likelihood, which unimodal distributions for which normalizing flows are particularly well suited. This point underlines the benefits of using EBMs to fit challenging densities. Finally, remark that SMNLE, which addresses only amortized inference Pacchiardi & Dutta (2022) struggled in practice for the toy problems investigated here.
4.3 USING SUNLE IN A REAL WORLD NEUROSCIENCE MODEL
We investigate further the performance of SUNLE by running its inference procodure on a simulator model of a pyloric network located in stomatogastric ganglion (STG) of the crab Cancer borealis given an observed an neuronal recording (Haddad & Marder). This model simulates 3 neurons, whose behaviors are governed by synapses and membrane conduct ances that act as simulator parameters of dimension 31. The simulated observations are composed of 15 summary statistics of the voltage traces produces by neurons of this network (Prinz et al., 2003; 2004). Amortized SBI methods require tens of millions of samples, while currently, the most sample-efficient targeted inference method for this problem is SNVT (a variant of SNLE that replaces the MCMC- MCMC-powered posterior sampling by a variational inference step Glöckler al. 2021) which uses 30 rounds simulating each 10000 samples.
SNVI
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Figure 3: Inference with SUNLE on a model of the pyloric network. Left: simulations obtained by using the final posterior mean and maximum posteriori (MAP) parameter. Center: the empirical observation To: arrows indicate the summary statistics. Top-right: fraction of simulated observations with well-defined summary statistics (higher is better) at each round for SNVI and SUNLE, with dashed lines indicating the maximum fraction for cach method. Bottom-right: performance of the posterior using the Energy Distance.
We perform targeted inference on this model using SUNLE with a MLP of 9 layers and 300 hidden units per layers for the energy Eg,, and perform doubly intractable MCMC to draw new proposal parameters across rounds. All inference and training steps are initialized using previously the available MCMC chains and EBM parameters. We report in Figure 3 the evolution of the rate of simulated obvservations with valid summary statistics, indicative of posterior quality as well as the Snergy-Scoring Rule (Gneiting & Raftery, 2007) of SUNLE and SNVT's posteriors across rounds. The synthetic observation simulated using SUNLE's posterior mean closely matches the empirical observation (Figure 3, Left vs Center). As shown in Figure 3, SUNLE matches the performance of SNVI in only 5 rounds, reducing by the simulation budget of SNVI to achieve a comparable inference quality. After 10 rounds, SUNLE's poterior significantly exceeds the performance of SNVI in terms of number of valid samples obtained by taking the final posterior samples as parameters. The total procedure takes only 3 hours (half of which is spent simulating samples), 10 times less than SNVI.
metric
Conclusion The expanding range of applications of Simulation-Based Inference poses new challenges to the way SBI algorithms model data. this work, we presented SBI methods that use expressive Energy-Based Model their inference engine, fitted using Maximum Likelihood. We demonstrated promising performance on synthetic benchmarks and on a real-world neuroscience model. In future work, we hope to see applications of this method to other fields where EBMs have been proven successful, such as physics (Noé et al., 2019) or protein modelling (Ingraham al., 2018).

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SUPPLEMENTARY MATERIAL FOR THE PAPER Maximum Likelihood Learning of Sneegy-Basse Models for Sumulation-Based Inference
The supplementary materials includes the following materials:
A discussion of the computational rationale motivating the tilting approach of AUNLE.
We propose a training method for EBM which uses the family of Sequential Monte Carlo samplers to efficiently approximate expectations under the EBM during approximate likelihood maximization. We show that using these new methods can lead to increased stability and performance for fixed budget.
An experiment in Appendix B.3 that suggest that the uniformization of AUNLE's posterior holds in learned AUNLE models.
A discussion Appendix B.2 about the (absence of) manifestation of the short-run effect [Nijkamp et al., 2019] in UNLE.
A detailed computational analysis in Appendix B.4 of AUNLE, which proves highly competitive over alternatives.
Figures in Appendix B.1 of UNLE's posterior sapmles for benchmark and the pyloric network's problems.
Finally, we provide additional details in Appendix B.6 on the results of SUNLE on the pyloric network: we provide an estimation of the pairwise marginals of the final posterior, which contains patterns also present in the pairwise marginals obtained by Glöckler al. [2021].
A METHOLOGICAL DETAILS
A.1 ENERGY-BASED MODELS AS DOUBIN-INTRACTTBBER JOINT ENERGY-BASRA MODELS
AUNLE learns likelihood model qu (z|0) by minimizing the likelihood of a tillted joint EBM p(0)e Fgle.s
Z+(e) While the gain in tractability arising in AUNLE's posterior suffices to motivate the use of this model, another computational argument holds. Consider the non-tilted joint model:
(x10) (0) ())
Expectations under this model can be computed by running MCMC chain implementing Metropolis- Within-Gibls sampling method as in Kelly & Grathwohl [2021], which uses:
any proposal distribution (z(0) x (z)0), such a MALA proposal
an approximate doubly-intraetable MCMC kernel step for (0)x) x T e (r.0) which is doubly intractable.
Zz(0)
However, running the approximate doubly intractable MCMC kernel step requires sampling from (=r0),, incurring an additional nested loop during training. Thus, naive MCMC-based Maximum-Likelihood optimization of untilted joint EBM is prohibitive from a computational point of view.
A.2 TRAINING EBMs USING SEQUENTIAL MONTE CARLO
The main technique to compute particle approximations of the EBM iterations (returned by the generic make_ wwlpppalllappppppeeeelpllpapaaap
when training an EBM using Algorithm to run
N MCMC chains in parallel targeting the EBM Song & Kingma [2021]: aggregating the final samples y; of each chain yields a particle approximation q = & E. 5y, of the EBM in question. In this appendix section, we describe alternative make ebm approx which
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efficiently constructs EBM particle approximations across iterations of Algorithm through a Sequential Monte Carlo (SMC) algorithm [Chopin et al., 2020; Del Moral et al., 2006]. In addition to its efficienty, this new routine does not suffer from the bias of incurred by the use of finitely many steps in MCMC-based methods. We apply this routine within the EBM training step of AUNLE's, and show that the learned posteriors be more accurate than MCMC methods for a fixed compute power allocated to training.
A.2.1 BACKGROUND: SEQUENTIAL MONTE CARLO SAMPLERS
Sequential Monte Carlo (SMC) Samplers [Chopin et al., 2020; Del Moral al.; 2006] are family of efficient Importance Sampling (IS)-based algorithms, that address the same problem as the one MCMC, namely computing a normalized particle approximation of a larget densily q known up to normalizing constant Z. The particle approximation Üswe computed by SMC samplers (consisting of NN particles y", like in MCMC methods, but weighted non-uniformly by some weights w²) is produced by defining set of L intermediate densities (vi)_o bridging between the target density 11=9 and some initial density Po- for which a particle approximation ny Σ; = I = = yb are readily available. The geometric interpolation between vo and
intermediate densities are often chosen to PL, i.c. Vi x (vo)-=((n)) +, so that ii are also known up to some normalizing constant. SMC samplers scquentially constructs approximation ef : Σ yf to the respective density in at time I, using previously computed approximations of v at time -- 1. At each time step, the approximations are obtained by applying three successive operations: Importonce Sampling, Resampling and MCMC sampling. We provide a vanilla SMC sampler implementation in Algorithm 5, and refer to this algorithm as fiike__sm_parriclle aapproo
Algorithm 5 SMC(g, v
1: Hyper-parameters: Number of particles N, number of steps T., re-sampling threshold A E 1).
2: Input: Target density 9, initial density vo, particle approximations v and ve
3: Output: Particle approximations to q.
4: Construct geometric path (ui)f= from vo and q-
5: for l= 1,.....,, do
6: Compute IS weights wf and Wi
7: Draw N samples ( from (V/1) according to weights (W)),, then set Wi= N
8: Sample Yi K((Y),))) using Markov kernel Kr.
9: end for
10: Return approximation d ( -
Importantly, under mild assumptions, the particle approximation constructed by SMC provides consistent estimates of expectations of any function f under the target q:
Σ of sw'') F. Ey-al(()))
We briefly compare the role played by the number of steps and particles in both MCMC and SMC algorithins:
Number of particles SMC samplers differ from MCMC samplers their origin of their bias: while the bias of MCMC methods comes from running the chain for finite number of steps only, the bias of SMC methods comes from the use of finitely many particles.
Number of steps While it is usually beneficial to use high number of iterations within MCMC samplers to decrease algorithm bias and ensure that the stationnary distribution is reached, the number of steps (or intermediate distributions) in SMC is beneficial to ensure a smooth transition from the proposal to the target distribution: however, the variance of SMC samplers function of the number of steps is not guaranteed to be decreasing
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even if variance bounds that are uniform in the mumber of steps be derived by making assumptions on K2 Chopin et al. [2020]. When applying SMC within AUNLE's training loop, we find that using more SMC samplers steps usually increase the quality of the final posterior.
In the next paragraph, we describe how to use SMC routine efficiently to approximate EBM expectations within Algorithm 2.
A.2.2 EFFICIENT OF SMC DURING AUNLE TRAINING USING OG-SMC
A naive approach which uses the SMC routine of Algorithm 5 within the EBM training loop of Algorithum 2 would consist in calling the SMC al every training iteration using a fixed, predefined proposal density yo and associated particle approximation and ân such as one from a standard gaussian distribution. However, as training goes, the EBM is likely to differ significantly from the proposal density 9o: requiring the use of many SMC inner steps to obtain a good particle approximation.
A more efficient approach, which propose, is to use the readily available particle umnor- malized EBM densily fg=-1 and associated particle approximation q² computed by SMC al the ileration k-1 as the input to the call to SMC targeling the EBM qut at iteration k. Algorithm implements this approach.
Algorithm 6 SMC-powered ML training of EBMs
Input: Training Data {x(") Jim1 Initial EBM parameters Vo
Output: Density estimator (1)
Initialize 4vo (I) xx e Beg (x) Q-1 = vo, 10:4-1 = Do
for = O,..., maa_max_tttet iter - do
# q = make_ caike_particee_pppeeee (922 9)
Tk. = SMC(g 01-
Ix Iva
G Σ V 2) Eup, (x²) v.
k = ADAM Ukk G)
end for
Return UUR
In practice, we find that using 20 SMC intermediate densities (with 3 steps of K.)) in each call to SMC yields a similar performance as a MC0- steps EBM training procedure. By considering more constrained budget, using only 5 SMC intermediates densities outperforms a 30-steps MCMC EBM training procedure.
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Figure 4: Performance of AUNLE, using her a MCMC-powered particle approximation routine, or SMC: using 30 MCMC steps or 5 SMC steps
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Figure 5: Performance of AUNLE, using either MCMC-poweree particle approximation routine, or a SMC: using 200 MCMC steps or 20 SMC steps
B ADDITIONAL. EXPERIMENTAL AND INFERENTIAL DETAILS
B.1 POSTERIOR PAIRPLOTS ON BENCHMARK PROBLEMS
We report the ground truth estimated posterior pairplots on benchmark problems. AUNLE and SUNLE exhibit satisfying mode coverage, and are able to capture complex posterior structures.
B.2 MANIFESTATION OF THE SHORT-RUN EFFECT IN UNLE
It was shown in Nijkamp el al. [2019] that EBMs trained by replacing the intractable expectation under the EBM with an expectation under a particle approximation obtained by running parallel runs of Langevin Dynamics initialized from random noised and updated for a fixed (and small) amount of steps can yield an EBM whose density is not proportional to the true density, but rather a generative model that can generate faithful images by running few steps of Langevin Dynamics from random noise on it. Our design choices for both training and inference purposefully avoid this effect from manifesting itself UNLE. estimate the intractable expectation using persistend MCMC or SMC
During training, chains, e.g by initializing the MCMC (or SMC) algorithm of iteration & with the result of the MCMC (or SMC) algorithm at iteration & - 1, yielding different training method than short-run EBMs. At inference, the posterior model is sampled from Markov Chains with a significant burn-in period, contrasting with the sampling model of short-run FBMs. Figure 7 compares the density of UNLE's posterior estimate for the two-moons model (a 2d posterior which can be easily visualized) with the true posterior. As the Figure 7 shows, AUNLE and SUNLE's posterior density match the ground truh very closely, demonstrating that UNLE's EBM is not a short-run generative model, but a faithful density estimator.
B.3 VALIDATING THE (2,0) UNIFORMIZATION OF AUNLE'S POSTERIOR IN PRACTICE
Proposition 1 ensures that the normalizing constant v) present AUNLE's posterior is independent of 0 provided that the problem is well-speeifted, and that vo wh*, the optimum of AUNLE's population objective. In practice, these conditions will hold exactly, and the uniformization of AUNLE's posterior thus only holds approximately: To assess the loss of precision associated with using a standard MCMC posterior in the context of approximate uniformization, we compare the quality of AUNLE's posterior samples obtained using a standard MCMC sampler (which valid only uniformization holds), and using a doubly intractable MCMC sampler, which handles non-uniformized posteriors. We that approximation error of doubly intractable samplers by using a large number of steps (1000) when sampling from the likelihood using MCMC. As Figure 8 shows, there is gain is using a doubly intractable sampler for inference in AUNLE, suggesting that the uniformization property of AUNLE holds well in practice.
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SUNLE
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Figure 6: Posterior marginal (empirical) pairplots for SUNLE's posterior (first column), AUNLE's posterior (second column) and the ground truth posterior for the 4 studied benchmark problems. Each row outlines a separate benchmark problem.
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Unnormalized Density
KDE of MCMC samples
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SUNLE
Ground Truth
Figure 7: Normalized density of AUNLE and SUNLE for the two moons model. Left: manually normalized posterior density both AUNLE and SUNLE using a discretization of the posterior over grid. Middle: kernel density estimation of the MCMC samples obtained from AUNLE and SUNLE's posterior. Right: Ground Truth posterior. AUNLE and SUNLE's posterior densities match closely the true density, showing that these method indeed learn density estimator, and a generative model [Nijkamp et al., 2019).
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UNLE (D. MCMC)
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Figure 8: Quality of AUNLE's posterior samples (measured in classifier accuracy) obtained using Standard MCMC sampler (S. MCMC) and doubly intractable sampler (D. MCMC). The results show no gain in using a doubly intractable sampler, justifying the use of standard samplers for AUNLE.
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B.4 COMPUTATIONAL COST ANALYSIS
Training unnormalized models using approximate likelihood is computationally intensive, as it requires running & sampler during training at each gradient step, yielding a computational cost of (2TTT,,) where Ti is the number of gradient steps, T2 is the number of MCMC steps, and N is the number of parallel chains used to estimate the gradient.
To maximize the efficiency of training, we implement all samplers using jax Frostig et al. [2018], which provides a Just-In- In-mmme compiler and an auto-vectorization primitive that generates efficient, custom parallel sampling routines. For AUNLE, we the introduced a warm-started SMC approximat ion procedure to estimate gradients, yielding a competitive performance with as little as 5 intermediate probabilities per gradient computation. For SUNLE, we warm-start the parameters of the EBM across training rounds, and warm-start the chains of the Doubly Intractable sampler accross inference rounds, which significantly reduces the need for burning steps and long training. Finally, all experiments are ran GPUs. Together, these techniques make AUNLE and SUNLE almost always the fastest methods for amortized and sequential inference, with total per-problem runtimes of for AUNLE and 15 minutes for SUNLE on benchmark models (which is significantly faster than NLE and SNLE on their canonical CPU setup Lueckmann el al. 2021) and less than 3 hours for SUNLE on the pyloric network model (with half of this time spent simulating samples). The latter is 10 times faster than SNVI (30 hours) on the same model. A breakdown of training, simulation and inference time is provided in Figure 9. We note that (S)NLE was ran on a CPU, which is the advertisod computational setting [Lucekmann et al., 2021], since (S)NLE deep and shallow networks that do not benefit much from GPU acceleration.
We note that the time spent performing inference is negligible for AUNLE, which uses slandard MCMC for inference thanks to the lilting trick employed in its model. On the other hand, the runtime of SUNLE, which performs inference using a doubly intractable sampler is dominated by its inference phase. This point demonstrates the computational benefits of the AUNLE's tilting trick. Note that SUNLE performs inference in a multi-round procedure, and requires thus R training and inference phases (where R is the number of rounds), as opposed to 1 for AUNLE. We alleviate this effect by leveraging efficient warm-starting strategics for both training and inference, which allow to amortize these steps across rounds.
B.5 EXPERIMENTAL SETUP FOR SNLE AND SMNLE
The results reported for SNLE are the one present in the SBI benchmark suite [Lucckmann al., 2021], which reports the performance of both NLE and SNLE on all benchmark problems studied in this paper.
SNLE
The results reported for SMNLE were obtained by running the implementation referenced by Pacchiardi & Dutta [2022]. SMNLE comes in two variants: the first variant uses plain Score Matching [Hyvärinen & Dayan, 2005| to estimate its conditional EBM, while the second variant uses Sliced Score Matching |Song et al., 2020], which yields significant computational speedups during training. For both methods, we train the model using 500 epochs, and neural networks with hidden layers and 50 hidden and outputs units. To optimize the inference performance, we carry out inference using our own Doubly Intractable Sampler, which automatically tunes the all parameters of the doubly intractable samplers except. for the number of burn-in steps, and initializes the chain at local posterior modes. We carry out a grid search over the learning rates 0.01 and 0.001, and leave other training parameters to their default. Figures in the main body only report the performance of the Sliced Score Matching variant, which perform better in practice and runs order of magnitude faster. Figure 10 reports the performance of both variants for completeness. We used GPU both to train and inference using SMNLE, yielding similar or higher training compared to AUNLE for the sliced variant, and much longer training times for the standard variant.
SMNLE
B.6 NEUROSCIENCE MODEL: DETAILS
Pairwise Marginals We provide the full pairwise marginals obtained after computing kernel density estimation on the final posterior samples of SUNLE. We retrieve similar
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Figure 9: Runtime of UNLE: Analysis and Comparisons.
First row: time (in minutes) spent training, inferring, and simulating for AUNLE. Second row. Second row: time (in minutes) spent training, inferring, and simulating for SUNLE. Third row: runtime comparison between ANLE and NLE (in log-scale). Forth row: runtime comparison between SUNLE and SNLE.
Two Moons
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Figure 10: Comparison of AUNLE, SMNLE with Sliced Score Matching (SSM), SMNLE with Score Matching (SM) and NLE on a set of benchmark problems.
20

---

Under review as a conference paper at ICLR 2023
patterns the one displayed in the pairwise marginals of SNVI samples. We refer to Glöckler et al. [2021] for more details on the specificities this model.
Figure 11: Pairwise marginals of SUNLE's posterior estimate on the C. borealis simulator model.
Use of Calibration Network Due to the presence of invalid observations, proceed as in Glöckler et al. [2021] and fil. a calibration network that allows to remove the bias induced by throwing away pairs of (parameters, observations) when the observations do not have well defined summary statistics. We use a similar architecture as in Glöckler et al. [2021].
21


Reviews:
[{"review_id": "nikLAA3Xn4i", "reviewer": "Reviewer_m5HY", "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": "", "summary_of_the_paper": "This paper proposes a method for learning synthetic likelihoods in the likelihood-free setting of simulation-based-inference (SBI).  The method fits a particular tilted version of the joint distribution of the parameters and observations.  Under certain assumptions, the solution of this optimization problem is proportional to the conditional density of the data given the parameters.  As such, the authors then use this unnormalized conditional density to estimate posteriors using standard approaches (e.g., MCMC).  The authors also present a sequential version of this approach.  The authors apply these methods to 4 toy datasets and compare to other SBI posterior inference methods, where they obtain better performance on one of the toy problems.  Finally, they apply their method to a neuroscience dataset, where they can obtain a good approximate posterior using fewer simulations than a previous method.", "summary_of_the_review": "Overall, I think this is a well-written interesting paper providing a promising approach for simulation-based-inference.", "strength_and_weaknesses": "Strengths\n* The proposed approach is straightforward, conceptually neat, and very interesting.\n* The paper is well-written and the method and results and presented exceptionally clearly.\n* I appreciate that in the toy models the strengths _and_ weaknesses of the present approach relative to existing methods (e.g., normalizing flows) are presented.  I think that the gains of the present approach on complex posteriors are sufficiently interesting (and well-explained and well-motivated) to more than outweigh the comparable (or slightly worse) performance to existing methods on simpler toy problems.\n\nWeaknesses:\n* The key assumption of AUNLE is $(Z, \\theta)$-uniformization.  This allows one to interpret the learned energy model as a likelihood up to an unknown normalizing constant that is independent of $\\theta$.  If this uniformization does not hold, then the normalizing constant will depend on $\\theta$, resulting a double-intractable problem.  Given that uniformization only holds if 1) the energy model is sufficiently flexible such that the true joint distribution lies within the space of learnable models; and 2) the optimization problem can be exactly, globally solved, it seems like in practice uniformization will always be violated.  Such assumptions are frequently made in SBI (e.g., it is common to plug approximate posteriors into various formulas involving the true posterior) so I don't think that this is deeply problematic (and indeed such assumptions are often justified by good empirical performance), but I would be interested to see some empirical results about how much uniformization is violated in practice and how much that affects downstream inference.  One possibility would be to run double-intractable methods on the learned energy model (without assuming that the normalization is independent of $\\theta$) for an extremely long time to obtain a \"ground truth\" and compare to the output of AUNLE.  Another possibility would be to estimate different normalizing constants for the implied likelihoods for different values of $\\theta$, and see the extent to which they differ.  In any case, I think it should be made more explicit in the main text that this uniformization is unlikely to _exactly_ hold in practice, and the rest of the AUNLE methodology hinges on it holding  _approximately_.\n* For all results it would be good to show some degree of replicability across initializations and random seeds.  How different are the different methods relative to differences in neural network initialization and randomness in the simulations?  Are the exact same simulations used to train the different methods, or are different random sets used?  E.g., the difference between using MALA and SMC within AUNLE results in comparable differences on SLCP as compared to the difference between AUNLE and NLE.\n\nTypos:\n* \"making naive Bayesian inference impossible\" --> \"making standard Bayesian inference impossible\"\n* \"each of which endowed with\" --> \"each of which is endowed with\"\n* Species names are never capitalized: \"_Cancer Borealis_\" --> \"_Cancer borealis_\"\n* \"given an observed an neuronal recording\" --> \"given an observed neuronal recording\"\n* \"We show that using this new methods can\" --> \"We show that using these new methods can\"\n* \"does not suffer from the bias of incurred by the\" --> \"does not suffer from the bias incurred by the\"\n* \"the number of steps (or intermediate distributions) is SMC is beneficial\" --> \" the number of steps (or intermediate distributions) in SMC is beneficial\"\n* \"When applying SMC to within AUNLE’s training\" --> \"When applying SMC within AUNLE’s training\"\n* \"using more SMC samplers steps usually increase the quality\" --> \"using more SMC samplers steps usually increasea the quality\"\n* \"comparing multirond sequential methods\" --> \"comparing multiround sequential methods\"\n* \"posterior estimate on the _C. Borealis_ simulator\" --> \"posterior estimate on the _C. borealis_ simulator\"", "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": {}}, {"review_id": "jaLBeCmA-l2", "reviewer": "Reviewer_kgMn", "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": "", "summary_of_the_paper": "This work proposes new methods for Simulation Based Inference. The goal is to perform Bayesian inference on model parameters given data for situations where the likelihood function is not known in closed form and can only be accessed through a simulation mechanism. The central challenge of SBI learning is \"doubly intractibility\" because the normalizer of the posterior distribution depends on the model parameter. This work proposes two energy-based methods for SBI. The first bypasses the doubly intractibility by defining a joint EBM over $X$ and $\\theta$ so that the normalizer does not depend on $\\theta$. This allows amortized posterior sampling for any observed data. To tailor the posterior inference to a specific observed datapoint, the work further proposes to use an EBM to learn the conditional distribution and performs doubly intractible inference. Results show competitive performance with score based SBI models for toy datasets and a crab biology dataset.", "summary_of_the_review": "The most interesting part of the paper is the tilting of the normalizer in the AUNLE method, although unfortunately this method likely will not be able to achieve the performance of models tailored towards inference for specific data. The lack of novelty of the main SUNLE method and sparse experimental results lead me to recommend not accepting this paper at this time.", "strength_and_weaknesses": "STRENGTHS\n\n1. The AUNLE formulation provides a nice way of dealing with a major limitation of double intractibility by tilting the normalizer. There are clear methodological and computational benefits to this approach, but it still remains limited in its ability to fine-tune the learned posterior to a specific observed datapoint.\n2. Experimental results show that AUNLE and SUNLE can outperform score-based SBI for multimodal datasets, and that SUNLE can outperform score-based SBI for complex biological data.\n\nWEAKNESSES\n\n1. The presentation of equations and algorithms is not always clear. In Algorithm 2, should the dataset be $(X_i, \\theta_i)$ instead of just $(X_i)$? In Equation 5 and Algorithm 4 is the data variable in the intractible term meant to have the index on the $X_i$, or should it be $X$ like in Algorithm 2, since it is a generated sample and not a data sample?\n2. The novelty of the SUNLE method is somewhat limited because it simply accepts the doubly intractible limitation and proceeds with standard EBM learning. Standard methods for flow and score models have already been used for a nearly identical purpose. \n3. It would be interesting to see other real-world applications similar to the crab synapses, although benchmarking and evaluation are likely difficult.", "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": "WqGBcOiDzxO", "reviewer": "Reviewer_VwrC", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "The authors propose two new synthetic likelihood methods for simulation based inference using energy based models, UNLE and AUNLE.\n\nThey introduces a tilting trick and an amortized sequential model to perform posterior inference with intractable likelihoods and improve both modeling quality and inference budget.", "summary_of_the_review": "The authors propose a rigorous and interetsting spin on SBI using EBMs.\n\nThe proposed extensions make sense and should be useful in the field, and the experiments appear well justified -if comparatively simple.", "strength_and_weaknesses": "Stregths:\n- the authors propose an interesting \"tilted\" inference procedure which performs well and is a neat trick\n- the utilization of conditional EBMs in SBI makes sense and is well executed\n- the authors propose a useful sequential version of their system\n\nWeaknesses:\n- like in many papers on this topic, the experiments are quite toy and not as ambitious as they could be, but they are clean and well-executed.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "I6JweumCOx", "reviewer": "Reviewer_jDWH", "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": "", "summary_of_the_paper": "The paper introduces (adapts) ML training of EBMs for SBI problems.\nThe main problem in SBI is that one cares about sampling from posterior $q(\\theta|x)$, and in the case of using EBMs, sampling from posterior results in doubly intractable inference since the partition function depends on $\\theta$ as well. \nThe paper suggest parameterizing the joint model $q_\\psi(\\theta, x)$ as $\\pi(\\theta)\\exp(-E\\psi(\\theta, x))/Z_\\psi$ and show that the optimal posterior also have similar form:  $q(\\theta | x) = p(\\theta)\\exp(-E\\psi(\\theta, x))/Z_\\psi$. This is important since they now can directly sample from the posterior without worrying about the partition function. The main limitation of this approach is that they need to have an analytical form p(\\theta) as prior, which is not always available, especially for the sequential setup, in which the prior is defined using posteriors on the previous rounds. Therefore, the sequential version of their approach (SUNLE) just trains an EBM using contrastive divergence and uses doubly-intractable MCMC for inference.\n\nThe authors compare AUNLE and SUNLE with NLE and SNLE, which use normalizing flow, on four toy datasets and one real-world example.\n", "summary_of_the_review": "The paper is well-motivated and discusses an important application. However, the main novelty of the paper is the introduction to AUNLE, however, the amortized version is not very useful in practice since using sequential modeling reduces the number of required samples. \nI am not seeing any novelty in the sequential version of their work except only training the EBM using CD in their setup.\n\n", "strength_and_weaknesses": "AUNLE is well-motivated, the approach is interesting and novel, and its correctness has been backed up by Proposition 1.\nHowever, SUNLE is simply training EBMs with CD, I am not seeing any novelty there.\n\nI don't understand why the authors didn't use SMNLE as a baseline. They ruled out SMNLE for being double-intractable, but they are already using double-intractable MCMC for their real-world setup. The comparison in Figure 1 is not enough to rule out SMNLE from the baselines. Moreover, SMNLE shows a better likelihood ratio than NLE in Figure 1.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Gozo8CIT01e", "reviewer": "Reviewer_mHnK", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper try to solve the Simulation-Based Inference (SBI) task using the energy-based model (EBM). In SBI task, one assume a known simulator that can generate samples x from interesting distribution parameterized by $\\theta$. The task is then estimate $p(\\theta | x)$. The authors using EBM to parameterize $q_\\phi(x|\\theta) $ or $q_\\phi(x, \\theta)$ and then inference can be done using bayes rule. In the paper, the authors propose two algorithms. AUNLE use a parameterization with tilting trick to make the normalizing constant of EBM independent of $\\theta$ and thus reduce the computational cost in the previous SMNLE method. And AUNLE replaces the normalizing flow model used in previous method with EBM. Experimental results seem to support that the proposed models outperform its baselines.", "summary_of_the_review": "In general, I like the idea of this paper of using EBM to solve the Simulation Based Inference (SBI) problem. And I think the theory part is OK for me.  But since I'm not an expert in SBI, I would like to listen other reviewers' opinion on how solid the experiments are.", "strength_and_weaknesses": "I mainly work on EBM instead of SBI. So my opinion my focus more on EBM part.\n\nFor the shining point of this paper, I think I like idea of this paper of using EBM to model the joint or conditional distribution of the paper. On one hand, EBM can be more expressive than normalizing flow given similar number of parameters and on the other hand, it makes sense to me that by cleverly design the EBM term, one can avoid the approximate doubly intractable estimation needed in previous method.\n\nHowever, I also have some questions.\n1. As pointed out by [2] and [4], EBM trained with MCMC as a sampler and MLE as objective may have the \"short-run\" effect. That is, instead of learning a correct energy function, the model converges to a point that the Langevin sampler becomes a noise injection generator. Since in the task of SBI, one actually needs a correct energy term to infer the likelihood. I'm wondering whether this problem exists in the current training process, especially for those complex tasks.  \n\n2. The authors mentioned that their model can be more effective comparing with previous methods. Thus, I think it is necessary to add a complexity analysis part for both AUNLE and SMNLE, e.g. list different models' (both the new models and baselines) accuracies together with their training/testing/inference time and make a comparison.\n\n3. I'm not very sure whether the SBI experiments results listed here are strong enough to make conlusion that the new proposed method outperforms the previous ones. Would like to listen to other reviewer's opinions.\n\n4. Some important EBM literatures are missing. Recommend the authors to add the following [1] [2] [4] papers.\n\n[1]  A Theory of Generative ConvNet. ICML 2016\n\n[2] Learning non-convergent nonpersistent short-run mcmc toward energy-based model.  NeurIPS, 2019\n\n[3] Cooperative Training of Descriptor and Generator Networks. PAMI, 2020\n\n[4] A Tale of Two Flows: Cooperative Learning of Langevin Flow and Normalizing Flow Toward Energy-Based Model. ICLR 2022", "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:

Decision: Reject