Learning Likelihood-Free Reference Priors
Authors: Nicholas George Bishop
Daniel Jarne Ornia
Joel Dyer
Ani Calinescu
Michael J. Wooldridge
Venue: ICML 2025 poster (2025)
Keywords: simulation-based inference
objective Bayes
reference priors
variational approximations
mutual information
Primary area: probabilistic_methods->bayesian_models_and_methods
TL;DR: We propose two new methods for learning reference priors that do not require access to the model's likelihood function.

Abstract: Simulation modeling offers a flexible approach to constructing high-fidelity synthetic representations of complex real-world systems. However, the increased complexity of such models introduces additional complications, for example when carrying out statistical inference procedures. This has motivated a large and growing literature on *likelihood-free* or *simulation-based* inference methods, which approximate (e.g., Bayesian) inference without assuming access to the simulator's intractable likelihood function.  A hitherto neglected problem in the simulation-based Bayesian inference literature is the challenge of constructing minimally informative *reference priors* for complex simulation models. Such priors maximise an expected Kullback-Leibler distance from the prior to the posterior, thereby influencing posterior inferences minimally and enabling an ``objective'' approach to Bayesian inference that does not necessitate the incorporation of strong subjective prior beliefs. In this paper, we propose and test a selection of likelihood-free methods for learning reference priors for simulation models, using variational approximations to these priors and a variety of mutual information estimators. Our experiments demonstrate that good approximations to reference priors for simulation models are in this way attainable, providing a first step towards the development of likelihood-free objective Bayesian inference procedures.

Learning Likelihood-Fres Reference Priors
Nicholas Bishop" Daniel Jarne Ornia Joel Dyer"
Anisoara Calineseu' Michael Wooldridge
Abstract
et al., 2021) to economics (Wiese et al., 2024; Dyer et al., 2024a) and robotics (Todorov et al., 2012), these models can be used to explore intricate dynamics, test hypotheses, and make predictions about real-world phenomena. However, these models often lack an analytically tractable likelihood function for the outputs they produce. This intractability prevents the direct application of classical statistical infer- ence methods. For this reason, a set of simulation-base inference (SBI) techniques has emerged, including Approxi- mate Bayesian Computation (Beaumont, 2019) and modern methods based on density and density ratio estimation (see, e.g., Thomas et al., 2016; Papamakarios & Murray, 2016; Hermans et al., 2020; Greenberg et al., 2019). These meth- ods approximate common statistical inference procedures using only the ability to simulate from the simulator, rather than the ability to evaluate the model's 's likelihood function.
Simulation modeling offers a flexible approach to constructing high-fidelity synthetic represen- tations of complex real-world systems. How- ever, the increased complexity of such models introduces additional complications, for example when carrying out statistical inference procedures. This has motivated large and growing litera- ture on likelihood-free or simulation-basele infer- ence methods, which approximate (c.g.. Bayesian) inference without assuming access to the simu- lator's intractable likelihood function. A hith- erto neglected problem in the simulation-based Bayesian inference literature is the challenge of constructing minimally informative reference pri- ors for complex simulation models. Such priors maximise an expected Kullback-Leibler distance from the prior the posterior, thereby inftuencing posterior inferences minimally and enabling an "objective" approach to Bayesian inference that does not necessitate the incorporation of strong subjective prior beliefs. In this paper, we propose and test a selection of likelihood free methods for learning reference priors for simulation models, using variational approximations to these priors and variety of mutual information estimators. Our experiments demonstrate that good approxi- mations to reference priors for simulation models are in this way attainable, providing a first step towards the development of likelihood objec- tive Bayesian inference procedures.
Given a simulator with implicit density pa, prior distribu- tion π over model parameters 0 E θ C Rd and observed iid data 2 2im == (ZI,..., x,), z, E X, the goal of simulation- based Bayesian inference methods, is to approximate the posterior distribution given by Bayes's theorem:
1i (0) x Pe(
(I)
The prior, n, is intended to encapsulate pre-existing knowl- edge or beliefs about the plausible parameter values, while Tai.. encapsulates the updated beliefs about plausible val- ues for 6. However, in many practical scenarios, strong prior information regarding likely values for the model param- eters may be unavailable or unreliable. Even when prior beliefs exist and can be readily expressed in the form of a prior distribution, they may not be confidently held, or the modeller might fear that they are unduly influencing the re- sults of their Bayesian analyses by incorporating strong prior beliefs into т. The modeller may therefore wish to carry out Bayesian inference in a way that minimises the influence of their own prior beliefs on the resulting inferences, and determine what posterior beliefs would arise when minimal prior information is encoded into the inference procedure.
1. Introduction
Simulation models have played crucial role across diverse scientific disciplines, offering a powerful framework for understanding complex systems. From epidemiology (Kerr
'Equal contribution Uniiversity of Oxford. Correspon- www.sbissshhss......oo dence Nicholas Bishop htw.ammmmmiieeesoo............. Daniel Jarne Ornia ddaniel.aaneoomiaaeee <joel.dyer@cs.ox. www.darreee..o.............
Such considerations have motivated the development of objective Bayesian methods, which offer a principled ap- proach to constructing priors that are minimally informative and, correspondingly, posteriors that are maximally data- driven. This has often been solved through reference priors (Bernardo, 1979; Berger & Yang, 1994; Bernardo, 1997:
ac uk>,, Joel Dyer
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Learning Likelihood-Free Reference Priors
Berger al., 2009), defined to be a prior distribution x², within some class IT of distributions on O that maximizes the expected information gain from the data and minimises the influence of the prior on the posterior:
where liI [0] Eg |- log [(] is the entropy of 0 when generated from and
my (21m)) = EBa=x [po(Tin)]
(4)
log TT Zin (0)
is the marginal likelihood associated with prior x. Equation (3) makes it apparent that RPs aim to maximise the initial uncertainty in the prior while minimising, in expectation, the posterior uncertainty.
T arg max FEll 21
E Arw
I (0)
The right-hand side of Equation (2) can be recognised as
the mutual information T (11m;; 0) between 2i:. and 0 ~ æ, Such reference priors (RPs) have additional desirable prop- erties, such as invariance under reparameterization, good fre- quentist coverage (Consonni et al., 2018), and they asymp- totically achieve the minimax entropy risk when IT is the class of continuous priors on O (Clarke & Barron, 1994).
By specifying a maximally vague initial state of knowledge, RPs provide useful tool for conducting prior sensitivity analysis: they provide the modeller with a way to assess what posterior inferences would ensue when those infer- ences are dominated by the observed data, rather than by the modeller's 's own subjective prior (Bernardo, 1997). In this way, the modeller can assess the relative influence of their own subjective prior on the final inference.
Solving Equation (2) is, however, difficult except for in the simplest of cases. As described above, it is generally not the case that the model's 's likelihood function is tractable, let alone the mutual information (MI) between the model's s parameter and output. Identifying a prior TA that solves Equation (2) for given simulator is therefore challenging.
It is typical (Bernardo, 1979; Berger & Bernardo, 1992; Lafferty & Wasserman, 2001) to distinguish between the n-reference prior, defined by Equation (2) for finite n, and the reference prior, defined to be
In this paper, we address this problem by proposing and testing selection of likelihood-free approaches to learning RPs for implicit simulation models. A subset of these ap- proaches allow for the simulator to be differentiable, while others address the case of non-differentiable simulators. To learn these RPs, we leverage recent advances in informa- tion theory and machine learning to estimate and maximise I 1m 6). Specifically, we consider both non-parametric entropy estimators, such the Cozachenko-Leonenne es- timator (Kozachenko & Leonenko, 1987; Kraskov et al., 2004), and more recent powerful neural MI estimators (Oord
lim (1)
π (0)
(5)
72030 TT (0n)
where 8, E θ is an arbitrary reference value for @ for which T, (0)) > 0 for all n. This latter definition provides instead a measure of the total information that is missing about the value of 8, while circumventing issues resulting from the fact that I& (Zine, 9) often diverges with infinite data.
A known result in the objective prior literature is that reference priors are often only supported on finite, discrete set of atoms in θ (Berger et al., 1988; Zhang, 1994). This restriction can be undesirable, and in conflict with the aim for a minimally informative prior. To counteract this, is sometimes preferred to consider only continuous, positive priors that maximise the objective in Equation (2) (Nalis- nick & Smyth, 2017); we will assume this approach when describing the methods under consideration in Section 4.
et al., 2018; Song & Ermon, 2020; Letizia et al., 2023). By maximizing the estimated MI, we approximate the RP without requiring explicit access to the likelihood function or the MI between and Zimn Through experiments, we demonstrate the effectiveness of our approach and compare methods across series of benchmarks, showcasing their potential for enabling objective Bayesian inference in simu- lation modelling.
A further motivation for restricting attention to continu- ous priors that solve Equation (2) is provided by Bernardo (1979); Clarke & Barron (1994). Here, is established that - asymptotically (i.e., as n - oc) and under regularity conditions (such as asymptotic Normality of the posterior distribution; see Clarke & Barron (1994) for details) Jef- freys' prior, which has density
2. Background
2.1. Reference Priors
Reference priors (RPs) were introduced in Bernardo (1979)
as approach to designing prior distributions representing a vague initial state of knowledge. Such priors aim to be minimally informative about the parameter 0, in the sense of maximising the missing information to be gained from the data. This can be seen more explicitly by rewriting Equation (2) as a difference between the prior and posterior entropies:
my (0) x (det(F(0)))
(6)
'The term "objective" has been historically adopted for meth- ods giving rise to "minimally informative" priors, according to different definitions of "minimally informative". We use this term to align with existing literature, but note that "objective" is in some senses a problematic term; see, e.g., Bernardo (1997).
... = arg max Hr [0]- - E E m. H TEL. [0],
(3)
2

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Learning Likelihood-FFii Reference Priors
is solution, and is the unique solution among continuous positive priors. Here, F()) is the Fisher information matrix
(2009) then propose evaluate the right-hand side of Equa- tion (8), which we denote below with #*, pointwise using Mote Carlo estimates of the expectations in Equation (8)
F(0)= = Eg pp Vo log po (#1m)) @ Vo log Po po(21s)) )
and an arbitrary positive prior e: 6 + R4:
with determinant (Ff)). and @ denotes an outer product. Clarke & Barron (1994) prove that the continuous positive prior asymptotically solving Equation (2) - i... Jeffreys' prior, Equation (6) induces marginal likelihood function my, that asymptotically minimises the largest distance be- tween the space M of all densities on A and members of the model family {pe 0 E }]: that is, as n + 0c,
R
po po(a 1im (c()
(0) R exp
log
R
T. + Pacts (r))
with Fin Pa. B0) c, and large. In contrast, Figure of Lafferty & Wasserman (2001), describes an iterative MCMC algorithm inspired by the Blahut-Arimoto algorithm (Blahut, 1972; Arimoto, 1972) for sampling from a model's RP. In each of these procedures, knowledge of and the abil- ity to easily calculate (exactly or numerically) the likelihood function corresponding to the model under consideration is assumed. More recent and more closely related work by Nalisnick & Smyth (2017) consider the problem of learning variational approximations to RPs, once again by assum- ing knowledge and tractability of the likelihood function associated with the model. Gao et al. (2022) consider how n-reference priors can be used to pre-train deep neural net- works in classification settings. In contrast to these prior works, we do not assume that the modeller enjoys access to the likelihood function associated with their simulator, expanding the scope of the literature on objective priors in Bayesian inference to the case of likelihood free simulation models that are either differentiable or non-differentiable.
())
log Po (11mm))
m,y = arg inf
sup E rop BE0
(7)
mE."
m (3mmm)
That is. asymptotically, RPs identified in the space of contin- uous positive distributions also satisfy the alternative notion of uninformativeness captured by (7).
2.2. Simulation-baseer Inference
Approximate, simulation-based inference (SBI) procedures for performing parameter inference for complex models have been in development for multiple decades. Beginning with approaches such as Approximate Bayesian Computa- tion (Diggle & Gratton, 1984; Pritchard et al., 1999; Beau- mont, 2019; Dyer al., 2024b) and synthetic likelihood (Wood, 2010) methods, the past decade has seen multi- ple innovations in SBI methods from the machine learning community, such as through the use of neural density (e.g., Papamakarios & Murray, 2016; Greenberg et al., 2019) or density ratio (e.g.. Thomas et al., 2016; Hermans et al., 2020) estimation techniques. This literature has, however, primarily focused on the problem of estimating the poste- rior density for given prior and simulator, rather than on the problem of finding priors with different properties of interest. Our work addresses this gap.
4. Methods
In this section, we provide details on the methods we pro- pose and test for learning (n- (n-)reference priors in likelihood- free settings for simulation models. Each of the methods we consider entails optimising a (lower bound on a )n estimate î for the MI between Zizu and 0. The main structure of the optimisation loop is shown diagrammatically in Figure 1.
Throughout, we assumee the use of parameterised varia- tional family II == { I Φ } of proper priors (see Appendix 1. with tractable density function, such as nor- malising flows (Rezende & Mohamed, 2015). We will at times omit the dependence on 6 for case of notation. Such continuous approximations to (n- (e-errefeeence priors can be ad- vantageous, as discussed in Section 2.1. Further, by chang- ing the number n of independent replications of the simula- tor output, the modeller can decide whether to approximate an n-reference prior for smaller values of n, or approximate the RP (5) by taking larger n.
3. Related Work
Early work on RPs have proposed procedures for pointwise estimation generating samples via, c.g., Markov chain Monte Carlo (MCMC). An example of the former is Algo- rithm I of Berger et al. (2009), which exploits the following form of xt as obtained through a calculus of variations argument (Bernardo, 1979; Berger& & Bernardo, 1992):
π (0) x exp (EzI.~ [log T2 (0))
(8)
The posterior appearing in the exponent on the right-hand side of this expression is typically taken to be the limiting posterior as n + do, which is (under certain regularity conditions, see Van der Vaart (2000) independent of the (continuous, positive) prior that generates it. Berger et al.
We also assume that data samples Ii.n are generated from a simulator with implicit likelihood function pe that depen- dent on parameters 0. By sequentially sampling parameters O and data Tiaa po can construct dataset D& = (af (ri)) generated from the joint distribu-
3

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Learning Likelihood-Frie Reference Priors
4.1.1. GENERATIVE ENTROPY DIFFERENCE ESTIMATOR
Flow Te
Simulator pe
21 ................................ Z.
Making use of (11), we devise an approach for estimating the mutual information. Recall that we can obtain estimates H. [0] directly from (13). Therefore, it only remains to find an estimate for Exy,... ~-m H (0). Taking inspiration from Pichler et al. (2022), a practical approach is to construct pa- rameterised estimators for the conditional distribution Tei.. denoted as # (for parameters &b E V), and use these to directly estimate this term using Monte Carlo sampling as in (13). Pichler et al. (2022) propose doing so by assuming the parameters & to be a function of the conditioning variable (the observed data). This yields the estimator:
MI Estimator
(1), Fnn
Encoder spp
Figure 1. A schematic of the overall pipeline for our methods.
tion (Ii:s; f) h,, where
( 0) = (0((1=))<(())
(10)
is the joint distribution of (m):m; 0) under prior ग,, and dis- card the ((r) to produce samples from my.
B
(o:)(Ds) = H. 10] +
# log
(s (0(0)).
(14)
B b=1
"in
4.1. Difference of Entropy Estimators
An important caveat is that in our case, the data marginal My depends on prior π that changes iteratively (as the parameters e evolve). Then, such an approach would entail repeating the following iteratively: (1) For the current prior r,, sample Da from h (2) Build the estimator fy (3) Compute 1 (Dp). (4) Update 6 by SGD on V /. (D&).
From the definition of MI between model parameters and outputs, we obtain the following equivalent expressions (see Appendix A.2):
1 12 0) H [0] ExI. ~m, H [0]
(11)
= H4,, [0] + Hm., [_1=]] - H, Im1m 0].
(12)
As one of the main approaches, we consider a set of methods that rely on estimating MI through a difference of entropy re- lation, either through estimating the entropy of the prior and the posterior (11), or through estimating the prior, marginal and joint entropies (12). The problem of estimating (and maximising) MI now becomes problem of effectively esti- mating the entropy of these different distributions.
Step 2 above requires possibly learning a new estimator each time we update Φ, which can be sampling and com- putationally inefficient. Instead, we adopt a two time-scale approach (Borkar, 1997). We consider the prior т and the estimator a..... to pertain to the same parameterised model class (particularly, a normalizing flow). We then update the conditional density estimator Tai... with faster learning rate than the prior n.
Given that we have assumed a variational prior family with tractable density π (0). the first term in equations (11) and (12) can be directly estimated without bias through Monte- Carlo entropy estimate for B prior samples:
Generative Entropy Difference (GED) Therefore, to learn RP using a parametric MI estimator, we propose defining π and æ to belong to some param- eterised class of conditional density estimators, and use fe. (D,) (14) as an estimate of the MI. We then update both #z.... at each iteration through maximum likelihood on the sampled data, and π through SGD on the MI estima- tor, with slower learning rate. For further implementation details, see Appendix B.6.
MI Estimator:
B log π (0(0)).
10
1 B
Σ
(13)
Then, to estimate the terms Ex. mm, H ..... (0). Hm,, [+1m]] and HH,, [Tim. 0] in likelihood free SBI settings there ex- ist different parametric (Pichler et al., 2022) and non- parametric (Kraskov al., 2004) approaches. In non- parametric approach, can follow Jarne Ornia et (2024) and exploit the differentiability of entropy estimators (c.g.. the Kozachenko- -enennenno-Leonenkoo estimator, Kozachenko & Leo- nenko (1987)) and of differentiable simulators to propagate gradients backwards from outputs to prior; a parametric approach would instead construct density estimates of the corresponding distributions and compute the entropy di- rectly. We propose the use of a parametric estimator in the interest of stability and flexibility".
4.2. Variational Lower Bound Estimators
When the simulator is differentiable, we may also exploit variational lower bounds on the mutual information to learn an approximate RP. A wide range of variational lower
"hhe formal two time-scale argument by Borkar (1997) requires lcarning rates axt) and (tt) decay to zero for t + ac, have infi- nite sum and finite sum squares. Then, for the resulting stochas- tic approximation to guarantce convergence, lim B(e) must hold. We instead adopt an approximated argument, and choose fixed B > a. This proved sufficient to learn the RPs correctly. See appendix for further details.
a(t) 0
2 parammaaa,,rric, sample based entropy estimators usually high variance and numerically unstable (Bonachela et al., 2008).
4

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Learning Likelihood-Free Reference Priors
bounds have been proposed in the literature (see Poole al. (2019) for thorough overview). Many such bounds rely on the following variational characterization of the KL-divergence (Donsker & Varadhan, 1975) between two distributions P Pnd Q:
3. Update the parameters & and j via stochastic gradient ascent using Vo. (o.))D).
The procedure outlined above relies on differentiability of the simulator, as gradients must backpropagate through the critic and then the simulator before reaching the variational prior. We show experiments that this requirement, while apparently stringent, can be reliably circumvented using
DxI - sup Ep[T] log Egle"I, (15) TE
where Leo is the space of essentially bounded measurable functions. In particular, Belghazi et al. (2018) exploit Equa- tion (15) by proposing MINE, which parameterises T with a neural network (commonly referred to as a critic) so that a reasonably tight lower bound on the MI can be learned via stochastic gradient ascent. Due to the second term of Equa- tion (15), which corresponds to the log-partiin peg-patttiion function of Q, MINE suffers from high variance, especially when the MI is large (Song& & Ermon, 2020; McAllester & Stratos, 2020). To mitigate this issue, Song & Ermon (2020) pro- pose SMILE, which clips empirical approximations of the log- parition function to lie in the range [-T, r]. where r > 0 a hyperparameter controlling the variane trade-off associated with truncating the log-partition function.
surrogate gradients (Blondel & Roulet, 2024).
Additionally, note that the critic and the variational prior are updated simultancously. As result, the critic is optimizing a non-stationary objective. If the variational prior changes too rapidly the MI estimate provided by the critic may de- grade, in turn causing the performance of the variational prior to degrade. In practice, we stabilize training through a two time-scale scheme, as described in Section 4.1.
In experiments, we adopt both SMILE and InfoNCE as vari- ational objectives, for several reasons. Many real-world simulators produce high dimensional outputs such as time series, leading to the possibility of high variance. SMILE is naturally suited to this setting due to the hyperparame- ter r that enables explicit management of the bias-variance trade-off. Meanwhile, InfoNCE typically exhibits low vari- ance, since the optimal critic does not depend on batch size (Poole et al., 2019). This property is especially important for expensive simulators, since only smaller hatch sizes can be used under limited simulation budgets.
In the context of contrastive predictive coding, the InfoNCE objective was proposed by Oord et al. (2018) for the purpose of density ratio estimation:
a log I EEje'
T(r:; M6)
sup E (x.(
(16)
n
T (2xi U5)
Oord et al. (2018) observes that InfoNCE implicitly max- imises a variational lower bound on the MI. Song & Ermon (2020) further show that MI estimation methods derived from the Garber-Akov inequality (Barber & Agakov, 2003) may be reformulated as an optimization over density ratios. That is, from estimate of the density ratio one may esti- mate MI and, by optimising a variational lower bound the MI, one obtains an estimate of the density ratio.
4.3. Performing Simulation- Inference
It is worthwhile highlighting that, with each of the methods described in the previous subsections, the ability to perform SBI for the implicit model and learned RP *" comes at no further training cost. In the case of GED, the method entails the construction of amortised (in 21...) estimator ay (0) for the posterior density 0 that results from the use of the learned prior 7* in Bayes's 's theorem, which can be imme- diately reused to generate samples from 2 through, for example, Markov chain Monte Carlo (MCMC) or forward passage through the network defining the normalising flow model. Similarly, in the case of the approaches outlined in Section 4.2, which are based optimising variational lower bound to the mutual information between &inn and 0, the learned discriminators Í' estimate function w of the density ratio has (I1m 0)/ //mm (#1:)) f (0). (For exam- ple, in the case of SMILE, w is the identity.) Since we have assumed that the variational RP density 5" is tractable through the use of, for example, a normalising flow model for the variational family, estimates log # (0) of the log- posterior density log - (0) resulting from the use of the true RP x¹ may then be obtained as
Given a variational lower bound on the MI, we may jointly train a critic T and a variational prior to learn a RP. The variational family responsible for producing prior that induces high MI between Fi: and 0 by maximising the variational lower bound. Meanwhile, the critic is tasked with keeping the variational lower bound tight so that the variational family has a good approximation of the MI to benchmark against. Both networks may be simultancously updated via stochastic gradient ascent. For instance, using InfoNCE, a critic T, with parameters μı, and a flow Te we may proceed as follows:
1. Sample D& ~ has
2. Compute the variational lower bound using critic Ty:
B
Ty (= (b) B(b)
(Do) = B b=1
log
B Eafb Tu (1)) O(0))
(6)
log A* Bim
(0) = log w 1 (f 0)) + log 4" (0)) (17)
5

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Learning Likelihood-Free Reference Priors
Garetan
Eqpmentia
Thosgule
AR(L)
GED SMILEE 2ad phar
GED 1000 SMILEE
GEED
GED
1.
Redtooo pehrt
Redeosse puRs
SNILLS
0.8
Sifiran 2012
SMILE
Theiser
0.66
(a) Gaussian Scale Model
(b) Exponential Model
(c) Triangular Model
(d) AR(1) Model
Figure 2. Comparison of proposed methods for learning reference priors on tractable models.
Equation (17) can then be used in, e.g., an MCMC procedure to generate samples from the posterior # In both cases, Bayesian SBI can immediately be performed with no further training of density ratio or posterior estimators.
where 0 E (0, 1) is a free parameter and dh (O, 1) is random variable distributed uniformly on [0, 1]. While the RP for 0 in this model is not available analytically, it is known (Berger et al., 2009) to be approximated well by Beta(1/22 1/2) distribution. Additionally, although the derivative Tr with respect to for fixed u, 0 almost ev- erywhere when defined in the usual sense, we may nonethe- less define approximate surrogate gradient through, e.g., the straight-through gradient trick (Bengio et al., 2013) in order to backpropagate through Tg. In this way, we may con- tinue to apply the methods described in Section 4.2, which require differentiable simulator.
5. Experiments
Here, we present series of experiments* to assess the RP- learning methods described in Section 4. In the first instance, we consider their ability to recover the RPs for a collection of simulators with known RPs, before considering more complex simulation models whose RPs are unknown.
5.1. Tractable Examples
Autoregressive Time-series Model. Finally, we consider the standard autoregressive time-series model of order I (AR(I)). Using ", (N, t ,, n, this model generates time-series F1, ................................ En
5.1.1. MODEL OVERVIEWS
Gaussian Scale Model. The first tractable example we consider simulates Gaussian random variables. For this model, n samples are generated iid from NV(,, as It = = + Oug, with 4 ~ (O, 1), and where μ E is known, > is free parameter, and ((a, b²) is a Normal distribution with mean and variance b2, Here, the RP (Yang& & Berger, 1996) for 0 is T (0) x 1/0.
Ti ou,, and 34 Or, tou for = 2, 2....,,. n, (19)
where i fixed and [-1, is free parameter, can be shown (Berger & Yang, 1994) that the corresponding
RP for 0 E [-1, is (0) x &(1 02) 1/2
Exponential Rate Model. We next consider an Exponen- tial model, whose generative process is as follows: for t I.....7 n, we generate random variables V, from an Exp(O) density as Ie = log (1 u4) /0, where u, U0, 1), 0 > 0 is parameter, and U (a, b) is uniform distribution on [a, 6). As with the Gaussian scale model, the RP for 0 is known (Yang& & Berger, 1996) to be x (0) x 1/0.
5.1.2. RESULTS
Figure shows the obtained RPs for each method in Section alongside their ground-truth counterparts. Each prior is generated from N = 10 samples and building histogram (with fixed number of 30 bins). In general, the GED and variational lower bound (VLB) methods obtain accurate approximations of the ground truth RPs. However, bigger discrepancies can be observed especially near asymptotic parameter limits: both the Gaussian and Exponential models have asymptotic densities at 0 = the AR(1) model has asymptotes at 0 1-1,1} and the Triangular model at @ E 1). These asymptotes are in general difficult to reproduce for generative models; for example, we see that InfoNCE and SMILE slightly overestimate densities close to the boundaries in Figure 2c. However, the proposed architecture of bounded generative flows are overall able to reconstruct these densities and approximate the true priors
Triangular Model. In this example, we generate random variables from a Triangular distribution on [o, 1]. Here, iid data is generated for / = I,,..., as
21 [u S 0) V V.
(18)
+ (1 I [4 1 81) (1-V(I- 41) (1 (1-8)
01)
available joel ammyerr/1_rreerrence priors.
*Code
httpa://github.com
6

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(SIR) baset- Learning Likelihood-Frie -Fee Reference Priors
Table 1. Performance metrics (mean [standard deviation] from 5 repeats) for different prior methods. Bold indicates best performance.
WASSERSTEIN INFONCE
C2ST
TASK
NUMERICAL.
SMILE
GED
NUMERICAL.
INFONCE
SMILE
GED
GAUSS.
6.95 [0.64]
7.08 [0.56]
6.77 0.533
1.66
[0.20]
0.90 [0.01]
0.62 [<0.005]
0.63 [0.01]
0.49 [0.01]
EXP.
4.25 [0.39
2.43 [0.61]
3.83 0.75
2.04 0.33
0.91 00001
0.58 (0.04
0.61 0.00
0.59 00006
TRI.
0.08 00.01]
0.04 [<0.005]
0.05 [0.01]
0.02 00.01
0.64 00001
0.54 [0.02]
0.55 [0.02]
0.51 [.002]
AR(1)
0.14 [0.01]
0.06 [0.02]
0.05
[0.03]
0.06
[0.02]
0.62 [0.01]
0.50 [0.02]
0.50 [0.01]
0.51 [0.01]
relatively accurately.
marginal distributions for the pa(0) to be uniform in the range [-3,3], and the marginal distribution for det. E(O), where det E()) is the determinant of the model's covariance matrix, to decay rapidly in det E() (Yang & Berger, 1996).
To provide quantitative analysis, we compare each learned prior to the corresponding ground truth RP using two com- mon metries in the SBI literature: the Wasserstein distance and classifier two-sample test (C2ST) (Lueckmann al., 2021). Lower values are preferred in both cases, and indi- cate closer match to the ground truth RP. As a baseline, we use the numerical method described in Berger et al. (2009), which generates pointwise evaluations of the unnormalised RP using (9). Full results are provided in Table 1. From this we see that, for the Exponential, Triangular, and AR(1) mod- els, all of our proposed methods perform consistently well. Indeed, each outperforms the numerical method from Berger et al. (2009) according to both metrics. For the Gaussian Scale model, InfoNCE marginally underperforms compared to this numerical method in terms of the Wasserstein dis- tance to the true RP; this is likely due to the difficulties of capturing asymptotic behaviour with proper densities.
In Figure 3, we plot the marginal distributions for the priors learned via the InfoNCE lower bound in Sec- tion 4.2 and the GED objective in Section 4.1; we see that both methods have been able to recover this general shape, with InfoNCE doing more successfully than GED. This is in line with our expectation that GED will underperform in this case, given that SL.PP-D is known to generate complex posteriors that can be difficult to approximate accurately.
Results
To conclude this experiment, we consider how the quality of the estimated RPs might be assessed when the form of the RP is unknown, as will be the case in practical applications. Given that the purpose of the RP is to maximise the MI between @ and model output, a simple method for assessing the relative performance of a procedure for learning RPs is to perform a classifier two-sample test. In particular, we train a binary classifier to distinguish between samples pairs (Iim 0) drawn jointly from hus and drawn from the product of the marginal distributions, To and mng- Such metrics for assessing performance are common in SBI (see, e.g., Lueckmann et al., 2021).
5.2. Complex Examples
5.2.1. SIMPLE LIKELIHOOD, COMPLEX POSTERIOR
We next consider the popular SBI benchmark task SL.PP- D (Lueckmann et al., 2021), based on the experiment first introduced by Papamakarios et al. (2019). This simulator has 5 parameters, 0- (01, 0z), taking values in θ = [-3, 3 and parameterising the following data- getta-generatig process: z4 ~ ~N((())) E(0)), i = I,...,ki, where
In Table 2, we report classification accuracies for this classi- fication task under four different choices for priors: Uni- form distribution on θ, and the approximate RPs estimated via InfoNCE, SMILE, and GED. From this, we see that the RPs estimated through fication accuracy than the Uniform distribution a common default choice of prior - with the variational lower bound (VLB) methods outperforming the GED method in this in- stance. In summary, this practical method for assessing the quality of the learned RP corroborates our expectations that Uniform distribution is poor approximation the RP for this model, while GED generates better approximation that is inferior to those generated by the VLB methods.
(0) = 02), Σ(0) = (00g 01 003 03
p0z 02 01
(20)
methods produce higher classi-
and p tanh 0s. Distractor variables 8; are also drawn from a mixture of Student t-distributions, with the final model output constructed as x; = (2,, 5,). Further details are provided in Appendix C. The RP for @ is not analyti- cally known. Further, this model has higher dimensional parameter space than the experiments in Section 5.1, and is known to generate complex posteriors; this makes this experiment more challenging than those in Section 5.1.
5.2.2. AN EPIDEMIC AGENT-BASED MODEL
While the RP for 0 in the SLCP model is unknown, it should be expected that the RP for the transformed parameter 0 = (p()), I()) should adhere to the shape of the RP for a multivariate Normal. In particular, we should expect the
As a final example of a complex simulation model, we consider a snucepible-infeceee-reoe model of disease spread in a population of N interacting
7

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Learning Likelihood-Fres Reference Priors
1,00
1.00
6060
IubNCE
0.75
0.75
GED
1000
SMILE
0.50
0.50
2000
0.25
0.00
0.00
-2
-2
0
2
10
2
10e
10
(1(
Py(6)
Cuv. Des. [E()]
Figure 3. Marginal distributions of the priors learned SLCP with distractors (SLCP-D) simulator.
outcomes while retaining low conditional entropy of individ- ual parameter outcomes; in contrast, while the conditional entropies are still relatively low for the Uniform prior, the diversity in the sampled trajectories from across all param- eter values is visibly lower. This can be especially useful for model designers, as with very few runs RPs produce specific (and diverse) outcomes.
Table 2. C2ST metrics (mean [standard deviation] from 5 repeats) for different priors in the SLCP-D experiment (Section 5.2.1). Higher values are better, bold denotes best performance.
UNIFORM
INFONCE
SMILE
GED
0.55 [0.01] 0.99 [<0.005] 0.98 [<0.005]
0.69 [0.10]
individuals (the "agents"). Agents in the model interact via a network, through which disease transmission occurs. Specifically, at time = I,..., T, the state ži, of agent is
Finally, we demonstrate our claim in Section 4.3 that SBI can be performed at no extra cost using the posterior and critic networks trained during GED and VLB-based methods. In particular, compare inferences obtained by sampling directly from the posterior network trained in GED; per- forming MCMC using the critic network produced from the InfoNCE objective; sampling directly from neural poste- rior estimator (NPE) trained as in Greenberg et al. (2019); performing MCMC using a neural density ratio estimator (NRE) trained as in Miller et al. (2022). For NPE (resp. NRE)
2,,t = T( 0) Mu,, S (1- (1 -)hh.) + - (T S 7 + [ 2],
(21)
where the states 21, = 0, 2,2 represent, respectively, that agent i is susceptible, infected, recovered time t. In the above, Ug, ~ (0, 1). Th, counts the number of agent i's neighbours that are infected at time I, and B, E (0, 1) are parameters that determine the rate of infection and recovery. The initial states 2,,0 ~ Bermoull((dn). where in E (0, 1) determines the proportion of individuals who are infected at time O. Further details on the model and its implementation are given in Appendix C.2. Finally, we extract the total number of infected agents at each step, such that
the RP learned via GED (resp. InfoNCE) as the prior density, and use 104 simulations. In Figure 5, we show that samples from the posterior predictive distribu- tions for each method are comparable, demonstrating that SBI can be performed accurately at no additional training cost once RPs are learned via the GED and VLB methods we consider.
N
12 = 1).
6. Discussion
Main Results In this paper, we study and test multiple approaches to learning RPs for arbitrary simulation models. Through experiments on tractable and intractable exam- ples, we have shown that these methods can to some extent construct good RPs and attain properties of interest: they have been able recover the overall shape of known RPs, in many cases being indistinguishable from the ground truth ac- cording to standard two-sample tests, and produce behaviour that can be seen visibly as achieving high MI between model parameters and model output in comparison to common "default" priors used in Bayesian inference (e.g., the Uni- form distribution). Using these methods for constructing approximate RPs may therefore be useful in practice for complementing subjective, modeller-specified priors during
Taking io 0.1, 50, and N 200. we learn n-reference prior for the parameter 0 = (8,7) E (0, 1)?, where in this case n is the number of iid length-T trajectories generated at each 6.
To illustrate the impact of RPs in simulation mod- els, we plot in Figure trajectories generated from the marginal likelihood functions corresponding to each of the learned priors for the SIR model. From each learned prior, we sample 10 values for 0, and for each of these we generate 10 trajectories. Trajectories simulated from the same param- eter value share the same colour. Visually, it is apparent that the priors that have been trained to maximise MI (via InfoNCE and GED) produce much more diverse set of
Results
8

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Learning Likelihood-Frie Reference Priors
140
15
375
300
Tea: AC7
Time stop
Time star)
(a) InfoNCE SIR results
(b) GED SIR results
(c) Uniform SIR results
Figure 4. Prior predictive simulations for the SIR model, as induced by (a) the InfoNCE prior, (b) the GED prior, and (c) a Uniform prior.
Price
Obsevved Data
IAANNE
CED
NRE
SPB
Figure 5. Posterior predictive samples for SIR model conditioned on synthetic data sampled from learned RPs. The top rows displays samples generated using the density ratio/posterior estimator trained whilst learning each RP, whilst the bottom row shows samples generated using density ratiofposterior estimators seperately trained via NRE/NPE.
real-world Bayesian analyses involving simulation models: they can allow modellers to assess the degree to which their inferences differ from the case where "minimal" informa- tion- as measured by the MI between 0 and data is built in to their subjective priors. This is, believe, the main practical benefit of our work.
reasonable estimates, training can also be unstable, exhibit- ing sensitivity to learning rates. However, GED methods can be applied to non-differentiable simulators, while cur- rently the VLB methods we consider are only applicable simulators for which derivatives of the model output with respect to the input parameters can be constructed through, e.g., reparameterisation tricks (see, e.g., Jang et al., 2016), 6). Further, through GED, posterior samples can be generated rapidly via a forward pass through the posterior network, while for VLB posterior samples must be generated via MCMC. Thus, since MCMC can be more time consuming, GED methods may be preferred when fast posterior sam- pling is also required. In general, however, applying both methods where this can be done, since it may in general be difficult to predict beforehand which approach will perform best for any given problem.
Method choice and the complexity of learning reference priors From technical perspective, learning (good) RPs proved a relatively difficult task even with the chosen gener- ative models and (deep learning) heuristics employed: RPs not necessarily proper, which means that estimating them via generative models introduces inevitable biases and approximation errors. Further, we have on occasion seen that multiple runs of the same method can produce approx- imated RPs that differ substantially. This may reflect the fact that multiple (local) optima may exist; as discussed in Section 5.2.1, metrics such as classifier two-sample tests may be useful for identifying this if it occurs.
recommend
Limitations Our work naturally suffers limitations. In this paper, we focus attention to scenarios in which all components of the parameter vector are of interest to the modeller, with no nuisance parameters present. Nui- sance parameters introduce additional complications, and constructing RPs in their presence requires a more nuanced treatment (Bernardo, 1979; Berger & Bernardo, 1992). Fi- nally, RPs are only one possible approach conducting "objective" Bayesian inference, and other considerations can lead to alternative objective priors in Bayesian analysis (see, e.g., Consonni et al., 30. The present paper is a first step towards enabling objective Bayesian inference for implicit simulation models; developing likelihood-fii methods for estimating other classes of objective priors would also con- stitute interesting future work.
With respect to choosing between methods, number of fac- tors are to be considered. VLB methods seemed faster and more stable, but do not produce precise estimates of MI. Fur- ther, VLB methods involve only one density estimation task that estimating the prior density while GED involves two- learning both the prior and posterior density and for this reason VLB can be less computationally expensive and more accurate when in settings with complex posteriors. This is reflected to some degree in our experiments with the SLCP model in Section 5.2.1; however, the fact that GED has nonetheless produced good approximate RP even in this case demonstrates that this may not be as limiting a fac- tor. Additionally, we have seen that while GED can produce

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Learning Likelihood-Frie -Feee Reference Priors
Acknowledgements
Bernardo, M. Reference posterior distributions for Bayesian inference. Journal the Royal Statistical So- ciety Series B: Statistical Methodology, 41 1979.
NB, DJO, JD, AC, and MW acknowledge funding from a UKRI AI World Leading Researcher Fellowship awarded to Wooldridge (grant EP/W002949/1). MW and AC also acknowledge funding from Trustworthy Al Integrating Learning, Optimisation and Reasoning (TAILOR), project funded by the European Union Horizon2020 research and innovation program under Grant Agreement 952215.
Bernardo, J. M. Non-informative priors do not exist. Jour- nal of Statistical Planning and Inference, 65(1):159-189, 1997.
Blahut, R. Computation of channel capacity and rate- distortion functions. IEEE transactions on Information Theory, 18(4):460-473 1972.
Impact Statement
This paper presents work whose goal advance the field of machine learning and simulation modelling. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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Learning Likelihood-Fres Free Reference Priors
A. Further Background
A.I. Proper and Improper Priors
An improper prior π on space θ is one for which the quantity
J T (0)
(22)
diverges (see, e.g., Berger et al., 2009). In contrast, prior is said be proper if (22) is finite, such that corresponds to (or can be appropriately normalised to correspond to) probability measure. While not corresponding to probability distributions in their own right, improper priors are sometimes still used in Bayesian analysis if the posterior distribution they induce via Bayes' theorem corresponds to a probability distribution; that is, the posteriors they generate are proper.
A.2. Relationships between Mutual Information and Entropies
We provide some basic relationships between MI and entropies for the unfamiliar reader. With MI under prior defined as
Tx (0)
(Z1:n. 0) = E O~r
log
(23)
T0)
we can separate out the logarithms to obtain
T (Fim. 0) = E, Oww
[log
7 Tiim (0) log x
(24)
Fi Pip
= D Fi.n Pop
[log t am
(0)] E
OMT
[log (0))
(linearity of expectation)
(25)
21 mie
-E ome
| log Trin (0)
BaNT [- log (0))
(26)
--Brin
[H [H [0] + H
(27)
revealing the relationship stated (11). Finally, the relationship observed (12) is obtained by recognising that
Em m.
12 [0]]
= E a~5
[log TTr.... (0)]
(28)
Pian
[log he (F 0)
E Planre
aur
m (Ii:n)
(Bayes" theorem)
(29)
E Out Pimm-o
[lg h (I1m0) 0)]
0~T [log m. (I1m)
(30)
(eLn 0) |- log h (x1:n, 0)] Ea |- log m2 (x1:a))
(31)
Hh, [11m 0] + Hm [x15]]
(32)
B. Experimental Details
In this section, provide additional details regarding each of our experiments including the neural architectures used to implement variational families and mutual information estimators. We also provide a more in depth overview of our training procedures.
B.1. Variational Prior Families
To describe the variational family II in each of our experiments we use normalizing flows. Each flow is composed of sequence of blocks. Each block is comprised of masked affine autoregressive layer (Papamakarios et al., 2017) and an LU permute block (Durkan et al., 2019). If required, we use a sigmoid flow layer at the end of the flow to ensure that the flow's output are bounded to lie within the range support by the simulator of study. Details about the sigmoid flow layer are presented in Section B.2.
B.2. Sigmoid Flow Layer
The variational family must constrained to produce parameters within the range supported by the simulator. As previously mentioned, this is achieved via sigmoid flow layer. Let my and M, respectively define the minimum and
13

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function for sets of fixed size. As a result, set-encoders are well-suited to encoding the independently and identically distributed outputs {-(th) generated by simulator under parameter ((6), We also use TRANSFORMER to refer to the transformer-based set encoder architecture proposed by (Lee et al., 2019). The transformer used is the same Learning Likelihood-Fre. Reference Priors
Algorithm 1 Flow Pretraining Procedure pretrain
Hyperparameters:
Learning rate [3, Epochs R. Pretrain epochs Rpre Input: Variational prior T ==
for 1. Rpre do Sample (0(b) LB bb=1 ~ & Compute loss I B EB log 2 (0
Update prior parameters @ + AD,Ms(0, V VoLo) end for
maximum permissible values of the parameter 0, Given an output E Rd from the previous flow layer, the sigmoid flow layer performs the following forward transformation:
0. m, + (M; - mm))o(2)) Vi 1,2,.....4 .d
(33)
where (2,) is the logistic sigmoid function. Note that 0, is now guaranteed to lie in the range [m, M.]. The corresponding inverse (or backwards) transformation is given by:
2 = log (6, - m;) - log (M, - 0.).
(34)
Meanwhile, the log-determinants of the forward and backwards transformations can be readily computed:
08
log det
log(M, - m,; + logo + log (1-o(z))))
0z
(35)
06
log
det
(log(M, m,) log + log (1 (zi)) ozg(11((z))).
0z
B.3. Notation for Network Architectures
Throughout the remainder of the Appendix, we will use the following notation to describe network architectures. We use MLP(-) to refer to fully connected multilayer perceptrons (MLPs). For example, MLP(10, 16, 32, 64, 1) refers to a three layer MLP with input dimension 10, hidden layer dimensions 16, 32, and 64, and final output dimension 1. Likewise, we use LSTM, (-) to refer to stacked Long Short-Term Memory units (LSTT) with hidden state dimension For example, LSTMg (4, 2) denotes stacked LSTM unit with input dimension 4 that is comprised of two LSTMs each with hidden dimension Similarly, we use FLOW() to refer to a normalizing flow. For example, FLOW(4, 10, 16) refers to a normalizing flow with input dimension 4 composed of 10 blocks (as described Section B. each with 16 hidden neurons. Enclosed brackets are used to denote concatenated network architectures For example (LSTMs (4, 1), ML, 16, 1)) denotes a network which passes time-series of 4-dimensional inputs through LSTM, before passing the 8-dimensional final hidden state of the LSTM to MLP, which produces final 1-dimensional output.
Similarly, we use -ET(..)) to describe the set encoder architecture proposed by (Zaheer et al., 2017). More specifically, (ET(p, Φ) first applies the network ø to each individual element of a set { before passing their sum to the network P. which produces the final output. As shown by Zaheer et al. (2017), such encoders are capable of representing any permutation-innariie
across all experiments and consists of two ISAB blocks and one PMA block, each with four attention heads and four inducing points.
B.4. Pretraining Procedure for Variational Lower Bound Methods
In all of our training procedures, we first pretrain the variational prior to maximise entropy over the parameter space. Experimentally, we found that this helped prevent the reference prior from getting stuck in local optima, and reduced the
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Learning Likelihood-Fre -Fee Reference Priors
Algorithm 2 Training Procedure with Variational Lower Bounds
Batch size B, interval I, Learning rates ot, 3, epochs R. pretrain epochs Ape Inputs: Variational prior TT. Critic Tu Simulator pa
Hyperparameters:
< pretrain(o. 8, Rgte)
==1
for t & R do
Sample {0(6)) ~ T
Simulate (8)
2i. ~ Pamn for forb=11 I...., B. DØ - ((0()) (10) 1=
Compute (Ds) via SMILE or InfoNCE
Update critic parameters ip ADAM, ((v, V (D())) if (mod I) = 0 then
Update prior parameters e + (D, (D())
end if t+1 end for
total number of epochs required for training to converge. As we use normalizing flows to implement the variational family IT in each of our experiments, the density of the variational prior can be easily evaluated. We exploit this fact to compute a plug-in estimate of the entropy and pretrain the prior using stochastic gradient descent. This pretraining loop is described in Algorithm 1.
B.5. Training Procedure for Variational Lower Bound Methods
Here, we outline the training procedure for learning reference prior using a variational lower bound, such as SMILE or InfoNCE. The general procedure is outlined in Algorithm 2. On each epoch batch of B samples {0(*) 1o=1 Ts sampled from the variational prior. Each ((h) is run on the simulator times to generate input-output pairs of the form D3 {(0(0) 21
(b) )
These pairs are used to estimate a variational lower hound on the mutual information (Ds), which relies on a critic form T. = ( to score input-output pairs. We refer to as s. as the encoder as is responsible for encoding outputs into dower-dimension representations, whilst we refer to fo as the score network as it is responsible producing scalar value scoring given parameter and its corresponding encoded output. In other words, for any input-output pair (0, 11m) the score computed T.. (0, )i:n)= (5 (f1:),, 0).
As mentioned in the main body, we use two variational lower bounds in our experiments; InfoNCE and SMILE. The InfoNCE objective is given by
1
T Ty (2 g(h))
(Do)
log
(36)
B
B Eazo T Tayp ( A(o))
(b)
6=1
while the SMILE objective is given by
jo-1
1 B
Σ
T (1 0(6)) - -log B(B B 1)
cl clip(e (i (0) ,0
(37)
where el ip denotes the clipping function clip(a, = min(max(a, b), c). Given (D,)(Ds), (D&), we then update o, y and y using standard gradient ascent rule such Adam (Kingma, 2014) using Vo.e. fv (Do). Note that the parameters of the variational reference prior are only updated every | epochs. This provides the critic Tr.p time to adjust so that the variational reference prior has good lower bound to train against. Additionally, for some complex simulators, we normalize the gradient to unit length before performing gradient update.
The neural architectures used for each experiment are display in Table 3, whilst the training hyperparameters used are described in Table 4. The variational prior and the critic are updated with learning rates a and B respectively.
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Learning Likelihood-Frie -Fee Reference Priors
Table Neural Architectures used each SMILE and InfoNCE: experiment.
SIMULATOR
VARIATIONAL FAMILY
ENCODER
SCORE NETWORK
GAUSSIAN SCALE
FLOW(1,
TRANSFORMER
MLP(9, 64, 64, 64, 1) MLF (2, 32, 1)
EXPONENTIAL
FLOW( 1, 4,8)
TRANSFORMER
TRIANGULAR
FLOR(1, 16)
SET (MLP (1, 16, 16, 1), (L, 8, 1)) (1STM1c(1, 2), MLP (16,
ML, 16, 16, 1)
AR(I)
FLOW(1, 1. 6, 16)
MLP(2, 8, 8,
SLCP
FLOW(5, 6)
SMIPML((100, 64, 32), MLP MLP(32, 64, 32))
MLF (37, 7, 64, MLP(K 128, 1) MLF (L0( 16, 8, 1)
G-AND-K
FLOW(4, 4, 16)
X SET((1STMa(3) 2). MIP (8, 8, 16, 8)), MLE (8,
SIR
FLOW(2, 16)
Table 4. Hyperparameter settings for InfONCE and SMILE experiments.
SIMULATOR
Ryer
EPOCHS
GRADIENT NORM
B
/
GAUSSIAN SCALE
200
3000
YES
256
50
10
5 10
5 x 10
EXPONENTIAL
200
2000
YES
256
50
00
5 10
5 10
TRIANGULAR
200
2000
NO
256
50
10
5 10
5 10
AR(1)
200
2500
YES
256
5
5 10
10
SL.P
1000
1000
YES
256
100
100
5 10
5 10
G-AND-K
1000
1000
YES
256
100
5
10
10
SIR
250
2000
NO
10 64
5
5 10
5 X 10
B.6. GED
The following sections include details on our implementation of the GED method described in Section 4.1.
The GED method requires learning an estimator # for the conditional distribution p(0 Ii:n). For consistency,
Architectures
the same general architecture" for this estimator for the proposal but normal distribution as the base distribution, and use an encoder 8, X + W that maps the sampled data I.n to some latent variable w. As a result, the base distribution Go for the normalizing flow # is a conditional Gaussian denoted as (o(2 & (T1m))) The way to apply this conditional is by having 8y (TTm))))) E R22d where the number of parameters in the model. Then,
conditional
qo(2 (z(1m))) ((p s (Z11)))))))
In other words, the encoder is trained to output the mean and the variance of the base flow distribution.
B.6.1. REFERENCE PRIOR TRAINING
The main training loop follows Algorithm We set a >> 33 to induce a two time-scale learning dynamic. Both update steps are computed through an Adam optimizer (Kingma, 2014). As section B.5, each parameter simulated n times to ensure that the asymptotic posterior can be well approximated. The architectures and hyperparameters used are presented in Tables and The same architectures were used for both the prior and the conditional density estimator per experiment.
3.6. REFERENCE PRIOR TRAINING WITHOUT SIMULATOR GRADIENTS
For each experiment discussed in the main body, GED was used in conjunction with pathwise simulator gradients in order to provide more like-fo--like comparison with VLB methods. Next, we show how GED can be applied to arbitrary simulators by providing an an estimator for network gradients that does not rely on differentiability of the simulator.
To begin, we restate the objective function for GED:
I p (Ds) LEg~, | log NH (0) Ee-ryy Ba--p |- log t x)]
(38)
The derivative of the first term with respect to Φ is easy compute. Since the variational prior family has reparameterisable
'While assume that the prior and posterior estimators helong the variational family in experiments, in the interest of simplicity, this is not an essential aspect this method, and general they may belong to different variational families.
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Learning Likelihood- -Fee Reference Priors
Algorithm 3 Flow Pretraining Procedure pretraiin-coditiions
Hyperparameters: Learning rate a, Pretrain epochs Ape. Batch size B Input: Variational prior Tg. Density Estimator #
==1 for 1. Rpre do Sample {B(b) 1:1
(b) B liT
Compute loss I 2hml log 2t (0(0)
8p (a(())))
Update estimator 1/ c ADAM V 1V) end for
Algorithm Training for GED
Hyperparameters: Batch size B, Interval I, Learning rates o, B, epochs R, pretrain epochs Ppre Input: Variational prior Tæ. Simulator ps, encoder Su,, density estimator #
+
pretrain(o. 3, Rgee)
pretrain-conditional
a, Rpre)
t =
for 1. < R do
Sample (16) x. repeat each sample k times (b = 1...., legg b 1, ,,. x k.
......... B x k).
Simulate x 2in
(b) ((0)
for S < / do
Update & + ADAMa e - log t (b) (0(6)))
end for
Update d ADAM3 (0 [0] + 1+1 end for
IiI xx11n 10)
sampling paths, we have 0 = 92(2) where Io is a differentiable function (such as flow) and 2 ~ is random sample from base distribution independent c. Using the Law of the Unconscious Statistician (Grimmett Stirzaker, 2001), we obtain
V Volena |- log (0)] VoBemil log π. (o)(()))
(39)
= E Ex-d- V log π (gx(
(40)
V a log R (go (1)))
(41)
Meanwhile, the derivative of the second term with respect to & can be estimated follows:
Volann Ernpo [- log Te x)] = EBNTT [V & log T Ez~po |- log ( x)]
(42)
EB Exaogo
V log (0) log क. ( x)]
(43)
N
v a log
Tz (0(m)) log (0(m) (m))
(44)
1=1
where (0()),(()) ((m) are drawn jointly by first sampling ((n) from na and then forward-simulating. In practice, Equation (44)
can be evaluated (Paszke al., 2019) by applying a stop_op stop_grade gradgaddennt method (e.g.. .detach () in PyTorch ((n) after it is sampled from the prior. Summarising, we may compute the full gradient in PyTorch follows:
Σ #=1
- log To (O(n)) + log To (0(m)) detach ()) log o (0(m) detach () |x(x) d))ach (45)
Figure shows the results of applying the above described gradient-free free GED method to three the models studied. As can be seen quality of the learned priors differs slightly when compared to the differentiable models, but still representative
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Learning Likelihood-Frie -Fee Reference Priors
Table 5. Neural Architectures Used in Each GRD experiment.
SIMULATOR
VARIATIONAL FAMILY
ENCODER
GAUSSIAN SCALE
FLCW(1, 16)
SET(ID, MLF MLP(64, 128, 2))
EXPONENTIAL
FLOW(1,
SET MLP (64, 128, 2))
TRIANGULAR
FLCW(1,8,
SET MLP (64, 128, 2))
AR(1)
FLOW(1,8, 16)
SET(L (1, 2), MLP MLP(32, (32, 64)), (64, 128, 2))
SLCP
FLCW(1,8, 16)
SET (100, 64, 64), MLF (64, 128, 2))
G-AND-K
FLCW 4, 16)
MLP (8, 16, 8)
SIR
FLCW(3, 16)
(ET (3, 2). MLP(M. 16, 6))
Table 6. Hyperparameter settings for GED.
SIMULATOR
Reus
EPOCHS
GRADIENT NORM
/
B
B
GAUSSIAN SCALE
100
1000
YES
2
256
5 x 10
1 x 10
EXPONENTIAL
100
1000
YES
2
128
x 10
x 10
TRIANGULAR
100
1000
YES
2
256
8
5 x 10-3
1 X 10
AR(1)
100
2000
YES
2
128
5x10-1
1 10
SLCP
1000
3000
YES
4
256
100
5 5x 10-3
1 10
G-AND-K
100
2000
YES
4
256
128
5 10
1 1 10
SIR
100
2000
YES
4 64
x 10
5 X 10
of the true underlying reference prior. As qualitative note, training reference priors without model gradients seem to yield higher variance learning processes, which (in this case) was compensated by slightly increasing the batch sizes and learning rates.
B.7. Entropy Estimation for Models with Tractable Likelihood functions
For simulators with tractable likelihood functions, we are able obtain Monte Carlo estimates for the between Zim generated from the model and 0 using the decomposition
T( 0) m., [3 Egg pg [fr
(46)
Using the the Law of the Unconscious Statistician (Grimmett & Stirzaker, 2001) and the model's likelihood function, the first term in Equation (46) may be estimated follows:
H
[I1m]]
E Er ~ome cme |- log Egrwn [po (I1:n)ll
(47)
R
log
Patei (b)
((()
(0) rT, Iin mx
(48)
B
This provides (biased) estimator for Hm, [21.]] The second term can be estimated this straightforwardly as follows:
Eg Hp, [21=]]
Egn Eninppo (og
(49)
log Paros (2 (1))
Tins
(b) (0)) hr
(50)
C. Further Details Complex Experiments
C.1. Simple likelihood, complex posterior with distractors
Our implementation of SLCP with distractors follows that detailed Appendix T.4 of (Lueckmann et al., 2021). In particular, distractors are sampled in from a mixture of L-distributions
(6, (6
tu4, E'),
(51)
20
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Learning Likelihood-Frie Free Reference Priors
Triangular
Gaussian Scale
Exponential
0.25
GED (grad free)
0.25
GED (grad, (rre)
GED (grad. Itee)
GED
GED
0.20
Reference prior
0.20
Referetuee prioe
Redermnce peio
Donetyy
Donsiyy
Demiity
0.15
0.10
0.10
0.06
0.05
3.00
0
20
30
60
50
100
0
20
40
60
80
200
0.0
0.2
0.1
0.6
02.5
1.0
(a) Gaussian Scale Model
(b) Exponential Model
(c) Triangular Model
Figure 6. Reference prior learned for each model, with and without gradients, for the GED method.
where je¹ ~ (N E, ((0, k. Ens = 3,,,, where a ~ NO, 1), and j,K = otherwise.
C.2. The SIR agent-based model
We use an SIR simulator similar in nature to the model supplied as part of the BlackBIRDS software package (Quera- Bofarull et al., 2023). In order to make the model differentiable, we make use of surrogate Gumbel-Softmax gradients (Jang et al., 2016) to backpropagate through Bernoulli random variables used to determine the discrete transitions between agent states. Each simulation involves simulating 200 agents for 7 50 time steps on a Watts-Strogatz random network (Watts Strogatz, 1998), initialised with 10 edges per node and rewiring probability of 0.1.
C.3. The g-and-k model
The g-and-k model appears frequently as a benchmark case study for SBI methods (see, e.g., Fearnhead & Prangle, 2012). Inference is challenging for this model due its ability to produce a wide range of data distributions from relatively few parameters. The generative process is as follows: for ž₁ NNO0 1) and parameters 0 b,g, E O =0,55 output Iz R is generated as
exp(-92() =a +b 1 c exp(- (1 2?) Zt:
(52)
where it is customary take 0.8 fixed. In this case, the true (n-)reference prior isn't available. However, we can expect two features to be present in the g-and-k reference prior based the roles the parameters play in the data-generating process, (52): since plays the role of location parameter, 1996); in contrast, since b is scale parameter determining the magnitude of the contribution from the second term in (52), we might expect the reference prior to decay as approximately 1/b (Yang & Berger, 1996).
expect the reference prior to be flat in (Yang & Berger,
The RPs obtained by GED, SMILE and InfoNCE for the same random seed are plotted in Figure 7. We emphasise that each method failed to generate consistent prior estimates across random seeds, suggesting that the asymptotic RP associated with the g-and-k model is particularly difficult estimate. We conjecture that this is due the heavy tails of the g-and-k distribution. The development of RP estimators that perform consistently the g-and-k model forms interesting open challenge.
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Learning Likelihood-Fei Free Reference Priors
GED InfoNCE SMILE
b
a
k
Figure 7. Learned RPs for the g-and-k model.
20


Reviews:
[{"review_id": "5WmjzOCoZA", "reviewer": "Reviewer_RgwU", "summary": "This paper addresses the problem of constructing reference priors for simulation-based\ninference (SBI). Unlike most SBI research, which focuses on posterior or likelihood\nestimation given a user-defined prior, this work tackles the challenge of developing\n\"uninformative\" or \"reference\" priors in a principled way when strong prior knowledge is\nunavailable or undesirable.  The authors formalize the problem in the context of SBI,\nwhere the likelihood is intractable, and propose several approaches for learning\nreference priors using normalizing flows. These methods, adapted from the existing\nreference prior literature, are based on variational approximations and mutual information\nestimators. The paper demonstrates and compares these methods on several benchmark\ntasks, including toy examples with analytically tractable reference priors and more\ncomplex, intractable examples. The goal is to enable \"objective\" Bayesian inference in\nSBI, minimizing the influence of subjective prior beliefs on the posterior.\n\n## Update After Rebuttal\nI thank the authors for their detailed rebuttal and the inclusion of additional experiments. This effectively addressed all my questions and concerns. With the promised changes incorporated, I believe the paper represents a valuable contribution to the field, consistent with my initial positive evaluation.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "Thank you for the valuable feedback. We have included a set of new experiments, see the reply to Reviewer UJGD for a detailed explanation.\n\n## Questions\n\n1. We will include explicit technical definitions for, and a discussion of, proper and improper priors. Briefly: an improper prior is an \"unnormalisable\" prior, i.e. one with infinite integral over the support, while a proper prior can be normalised to have integral 1; improper priors do not strictly correspond to probability measures for this reason, but can sometimes still produce proper Bayesian posterior distributions.\n2. We will use some of the additional space to include a discussion on these 4 points at the end of our revised paper. Briefly:\n    - The first of these points we discuss in bullet point 3. directly below this one. \n    - Using a user-specified (e.g., Uniform) prior may be cheaper since it involves no learning of a prior; however, we are not recommending that our learned reference priors replace any subjective priors the modeller believes in, but rather that they complement such subjective priors by enabling a prior sensitivity analyses. Our methods allow modellers to demonstrate the degree to which their inferences differ from the case where \"minimal\" information (as measured by the MI between $\\theta$ and $x$) is built in to the prior, and so we envision that our methods generate priors that are used _alongside_ the modeller's subjective prior.\n    - To assess whether a reference prior is sensible (whether it serves the purpose it intends to serve) the modeller can estimate the MI between $\\theta$ and $x$ using the different networks trained in the VLB and GED methods, or e.g. via sample-based entropy estimators [1]. However, note that estimating MI from finite data is a challenging problem (see [2], [3]). Alternatively, one can measure divergences between prior and posterior distributions as an estimate of the MI to assess these.\n    - We outline what we believe is the main practical and conceptual benefit in the second bullet point above, and will further emphasise this in the revision.\n### Question 3: Differences between VLB and GED\nWe will include detailed heuristics in the paper. VLB methods maximise a variational lower bound for the MI which relies on learning critic networks to (essentially) construct density ratio estimates; GED methods construct density estimates from samples of the marginal and conditional distributions of $(\\theta, x)$ and maximise directly the estimated MI.\n- GED methods do not require differentiability so they are preferred for problems with non-differentiable data collection, but (for equally computationally complex simulators) VLB methods are less computationally expensive than GED methods (in the sense that they only need to learn one density estimator instead of two).\n- GED methods rely on constructing an estimate for $\\pi(\\theta \\mid x)$. In problems prone to yield very complex posterior distributions, this may hinder the efficacy of the method. In these, VLB methods might be preferred (when applicable), since this entails only one density estimation task on $\\Theta$ (i.e., learning the prior), while GED involves two (i.e., both the prior and posterior).\n- When successfully learning a reference prior $\\pi$ via GED, we also get (at no extra cost) an estimator for the posterior $\\pi(\\theta \\mid x)$ under $\\pi$, which we can directly use for e.g. SBI. We did this in the new rebuttal experiments (see Figures 2 and 3). \n\n## Weaknesses \n1. We will add detail to Figure 1 in the revision for clarity and place it earlier. \n2. We agree that practitioners will often have prior constraints, and we support the incorporation of such beliefs into their Bayesian analysis. As we discuss above, we do not argue against the use of such subjective priors, but instead intend to provide practitioners with tools to assess how much information they have built into their prior, by equipping them with methods that let them find \"minimally informative\" priors. Our methods can of course also be used when no strong subjective prior beliefs exist, or in situations in which the practitioner prefers to minimise their own influence on the Bayesian analysis for whatever reason.\n\n## General comments\n\n- **On the connection to GBI**: We’ll use some of the extra space to briefly discuss connections to GBI.\n- **Line 198**: Thank you, we'll correct this typo.\n\n## References\n[1] Kraskov, Alexander, Harald Stögbauer, and Peter Grassberger. \"Estimating mutual information.\" Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 69.6 (2004): 066138.\n\n[2] Song, Jiaming, and Stefano Ermon. \"Understanding the limitations of variational mutual information estimators.\" arXiv preprint arXiv:1910.06222 (2019).\n\n[3] McAllester, David, and Karl Stratos. \"Formal limitations on the measurement of mutual information.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2020.", "claims_and_evidence": "The primary claim – that reference priors can be learned in the SBI context using the\nproposed methods – is well-supported. The paper provides clear derivations of the\nrelevant algorithms, adapting them from established reference prior literature.  The\nexperiments on various benchmark tasks demonstrate the feasibility of learning\napproximations to reference priors.  The theoretical grounding is solid.\n\nA secondary, implicit claim is that learning reference priors with these methods enables\nstandard SBI via subsequent posterior estimation or density-ratio estimation, followed\nby MCMC sampling. While this is logically sound (given the learned prior, which provides\neither a likelihood estimate or a posterior/ratio estimate), the paper lacks a\ncomprehensive empirical demonstration of this end-to-end practical application. While\nthe authors briefly mention a comparison to the uniform prior predictive in the SIR\nexample, this is insufficient to fully showcase the utility.  What's missing are\nexperiments that systematically compare posterior inference (including both posterior\ndistributions and predictive performance) using a learned reference prior versus a\nstandard, hand-crafted prior (e.g., uniform or a domain-informed prior). These\ncomparisons should highlight the practical advantages, if any, of using the learned\nreference prior in subsequent SBI tasks. Crucially, this demonstration should emphasize\nthat once the reference prior is learned, it can be readily reused for standard SBI\nwithout requiring further training, which is a key potential benefit of the proposed\napproach. This expanded demonstration would significantly strengthen the practical\njustification for the methods. This is not a major flaw, but a missed opportunity.", "methods_and_evaluation_criteria": "Yes, the experimental design is generally appropriate. Using toy examples with\nanalytically tractable reference priors provides a crucial validation of the learned\npriors.  The more complex examples add further evidence, although a more in-depth\ndiscussion of the choices made, or the use of a well-established, real-world SBI\nbenchmark (e.g., from a field like neuroscience or astrophysics, where SBI is commonly\napplied), would increase the practical relevance and impact.", "theoretical_claims": "I did not check the proofs in detail, focusing on the conceptual soundness and experimental validation.", "experimental_designs_or_analyses": "Yes, I reviewed the experimental design and analyses, and they appear sound in their current form (apart from the additional experiments and demonstrations mentioned below).", "relation_to_broader_scientific_literature": "The paper makes a valuable contribution by addressing a largely unexplored area within\nSBI: the principled construction of reference priors. While reference priors are\nwell-established in Bayesian inference, their application to SBI, with its intractable\nlikelihoods, is novel.  This work connects to the ongoing discussion about the\nfaithfulness of SBI methods (e.g., Hermans et al., 2023, concerning posterior\ncalibration and overconfidence) by highlighting the often-overlooked role of the prior\nin the overall inference pipeline. It addresses the need for \"objective\" Bayesian\ninference in SBI, where minimal prior information is incorporated.", "essential_references_not_discussed": "The paper provides a good discussion of prior work on learning reference priors. A\nconnection to generalized Bayesian inference (or \"post-Bayesian\" inference) could\nstrengthen the contextualization.  This related field, with applications in SBI (e.g.,\nMatsubara et al., 2021; Gao et al., 2023; Jävernpää et al., 2025), offers alternative\napproaches to handling uncertainty and model misspecification. While this paper focuses\non a different approach (the prior), discussing the relationship and contrasting it with\ngeneralized Bayesian inference would provide a more complete view of the research\nlandscape.", "other_strengths_and_weaknesses": "### Strengths\n\n- The paper is generally very well-written and clearly structured.\n- The problem and motivation are clearly articulated, and the necessary background is\n  introduced thoroughly yet concisely.\n- The contributions are clearly stated.\n\n### Weaknesses/Suggestions\n\n- Concept Figure: Figure 1 is quite minimal and sketch-like. A more prominent and\n  informative figure, appearing earlier in the paper, would help convey the overall\n  approach more effectively.\n- Practical Context for SBI: While the paper argues for the use of reference priors, it\n  would benefit from a more thorough discussion of the practical context within SBI. The\n  introduction mentions that the modeler might want to minimize the influence of their\n  prior beliefs. However, in many practical SBI applications, practitioners do have\n  prior constraints (e.g., bounds on interpretable simulator parameters). This\n  difference between the \"fully objective Bayesian\" ideal and the typical SBI use-case\n  should be explicitly addressed.", "other_comments_or_suggestions": "1) line 198: \"update update\"", "questions_for_authors": "1) Proper vs. Improper Priors: On page 6, the paper mentions that the Exponential and\n   Scale Gaussian models have improper reference priors. However, this distinction\n   between proper and improper priors wasn't introduced earlier in the paper. Could you\n   clarify this point and perhaps provide a brief discussion of the implications of\n   using improper priors in this context?\n2) Practical Application: Could you elaborate on how the proposed methods would be\n   applied in a practical SBI scenario? For instance, consider a well-known SBI\n   benchmark like the Hodgkin-Huxley model, which is often used with uniform priors.\n   - How would one design and learn a reference prior in this case?\n   - What are the expected computational costs compared to using a uniform prior?\n   - How would a practitioner evaluate whether the learned reference prior is \"good\" or \"sensible\"?\n   - What would be the conceptual and practical benefits for the practitioner?\n3) Method Selection: The paper proposes several methods for learning the reference prior\n   (e.g., VBEM, direct optimization, sequential learning). How should a practitioner\n   choose among these options in a given application? Are there any heuristics or\n   guidelines that can inform this choice, based on factors like the complexity of the\n   simulator, the dimensionality of the parameter space, or the computational resources\n   available? If no such heuristics currently exist, could you outline a potential\n   research direction for developing them?", "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": "", "extra_scores": {}}, {"review_id": "RM4shNfl7M", "reviewer": "Reviewer_gyn2", "summary": "The paper proposes a way to approximate a reference prior for a Bayesian analysis from a flexible family of priors in the SBI (simulation based inference) context, where the likelihood is intractable. The primary contribution here is the SBI context, in which various estimators of entropy are required to be specified. The paper uses methods adapted from those in the literature. The simulation results show that, in principle, this approach has some credibility.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your feedback. We have included a set of new experiments [here](https://github.com/ICML-7582/rebuttal_plots/blob/main/plots.pdf), and see reply to Reviewer UJGD for a detailed explanation.\n\n## Figure 2\n- The bumps in panels a) and b) were an artefact of the network architecture. Sigmoid layers in normalizing flows seem to induce unexpected tail behaviours in the resulting densities. Having experimented further, they have now disappeared, see Fig. 5 in the rebuttal pdf.\n- The stopping point of $\\theta = 100$ was largely arbitrary, but not going higher can be justified by the fact that the prior falls off as a power law for panels a) and b).\n- Neither the reference prior theoretical foundations nor the VLB or GED methods require compact parameter sets. It happens to be that most simulators chosen require parameters to be in some compact set.\n- The question of where to truncate the prior for a potentially unbounded parameter space is important, but is a general problem when specifying priors and not specific to our methods. We instead address the problem of \"given a parameter space, how could a minimally informative (proper, continuous, positive) prior be constructed?\"\n- The y axes in panels c) and d) must necessarily be truncated due to the presence of asymptotes at the boundaries. We considered these plots to already adequately show that our methods have imperfections, such as InfoNCE and SMILE slightly overestimating densities close to the boundary in panel c), and GED exhibiting a slight left-skew in panel d).\n- Re. question on SBI effects: We now include a gold standard baseline for the reference priors in Fig. 2 of the accompanying pdf using the algorithm in [1] which uses knowledge of the tractable likelihood functions of these models. Plots and KS test statistics for these can be seen in Figure 6 in the rebuttal plots, and table of additional metrics requested by Reviewer V6ZV are given in our response to them below. In general we perform better, but in some cases similarly.\n\n## Figure 4\n- We agree that diversity in one statistic $\\neq$ diversity in another. We chose $x$ to be curves of the proportions of individuals who are susceptible, infected, and recovered through time. A good reference prior should -- since the MI is a difference between the entropies of the marginal likelihood $H(x)$, and the likelihood function in expectation over the prior, $H(x\\mid\\theta)$ -- maximise $H(x)$ (encouraging diversity _in statistic_ $x$) while minimising $H(x\\mid\\theta)$ (discouraging diversity _in statistic_ $x$ from individual likelihood functions, on average). Panels a) and b) in Fig. 4 aim to show this. In the accompanying rebuttal plots we include similar plots obtained by finding reference priors from GED and VLB with $x$ as the _infection curve only_. In this case, the infection curves are much more diverse.\n- Panel d): We agree that the features we expect are approximately, but not perfectly, recovered from our prior learning methods. In the revision, we will use some of the extra space to discuss reasons why we believe our methods have struggled for this g-and-k simulator. Our other experiments, in addition to our new SLCP experiments (see Fig. 1 in the rebuttal pdf), do however demonstrate that our approaches can learn useful approximations to reference priors. \n\n## Methods and Evaluation Criteria\nPlease see our discussion of our comparison to the method from [1] and about diversity above.\n\n## Relation to Broader Literature\nWe'll use some of the extra space to give more details on Nalisnick & Smith (2017) & how we differ from this prior work.\n\n## Other Strengths and Weaknesses\nWe hope our new experiments that compare the performance of our methods to the gold standard tractable-likelihood approach from [1] (see above & response to Reviewer V6ZV) and demonstrate the ability of VLB & GED to perform SBI accurately at no extra cost (see Fig. 2 & 3 in accompanying pdf) address your concerns.\n\n## Other Comments\nWe'll fix the typos. We'll also: say \"minimally informative\", not \"uninformative\", in abstract; use \"Jeffreys' prior\"; write \"optimise an estimate of, or a lower bound on, the MI...\"; ensure \"VLB\" is defined on first use; and write $c$ in Eq. 18 and state that $c=0.8$ is a common choice that we also use. We'll explain the (missing) narrative link to Jeffreys' prior; the point of this, and Eq. 7 in particular, was to highlight that continuous, positive distributions satisfying Eq. 2 also (asymptotically & under regularity conditions) satisfy an alternative notion of \"minimally informative\" given in Eq. 7. Finally, we'll state these regularity conditions.\n\n## Questions\nOur use of a prior and posterior estimator that are of the same model class is just for ease of hyperparameter tuning, and not a requirement of GED. On the comment re. continuous approximations: Please see \"Other comments\" above.\n\n[1] Berger et al., \"The Formal Definition of Reference Priors\", The Annals of Statistics (2009)", "claims_and_evidence": "Overall the principle of the method seems credible. However, there are a number of rough edges on the results, described below.\n\nComments on Experiments (Section 4).\n\nIt's good to see the initial focus on the known univariate reference prior examples. However the \"accurate approximations of the ground truth reference priors.\" is slightly generous. The overall shape is there. However, in Fig 2:\n\n* Panel (a,b) there is clearly some density estimated to exist at theta=100, which definitely should not be there. That is, there seems to be some kind of undesirable \"edge effect\".\n\n* Panel (a,b) why is theta limited to 100. What happens if we consider theta larger than this? Does the method require the parameter space to be compact? How does the user know where to truncate it, if it needs to be truncated?\n\n* Panel (c,d) there is some truncation on the y axis here. While I appreciate keeping the focus on the detail, it would be good to see just how off the approximation is on the boundaries instead of hiding it.\n\nFig 4:\n\n* Panels (a-b) are claimed to have higher diversity than Panel c. Not sure about the argument being presented here. The diversity of various quantities depends on the parameterisation of the statistic being shown. Diversity in one statistic can correspond to non-diversity in a different function of the same statistic. E.g. here the top plots (statistic I) seem more diverse under the uniform prior. Though I'm sure one could compute a different statistic and have the opposite conclusion. So I'm not sure what these panels really demonstrate.\n\n* Panel d:\n\n- It seems quite generous to interpret the a marginal prior as uniform, under any method. Also, there seem to be finite bounds on this (unbounded) parameter. What are they?\n\n- Its hard to judge if the marginal for b is even close to the 1/b rate as everything is so tiny. Please add the 1/b line to the plot so we can judge this more clearly. Similarly include a line on the marginals for g and k that indicate where the Gaussian case is (in which a 1/b rate might be expected). That is, the text claims that \"we might expect the reference prior to decay as approximately 1/b\", which may be true for a Gaussian, but is likely untrue for distributions with large skew and kurtosis.\n\n- There are large differences among the methods. Which are we to believe? The range of differences here could be quite influential on any posterior inference.\n\n- Similarly, what are we to make of the vastly different joint distributions? Some pairwise marginals exhibit very strong dependence, and others, very little.\n\nFig 2 again:\n\n* As these models have tractable likelihoods, we can compare the outcome of the results of the current paper (the SBI focus) against other papers that have approximated reference priors with tractable likelihoods. Which of the features that are presented here are SBI-method effects, and which are not?", "methods_and_evaluation_criteria": "Because no comparisons with other (tractable likelihood) methods are performed, the authors come up with some odd ways of justifying that the results are good/credible.\n\n* The KS tests of differences between the actual and estimated reference priors are only really a way to compare the relative performance of the different methods, in that one can pass or fail any test based on ones choice of number of samples. But then statements like (p.6. last para) \"all of our proposed methods score consistently low [test statistics], avoiding the red regions where the null hypothesis is rejected.\" seem slightly misleading, as it seems fairly clear that if the number of samples is increased (they are mysteriously truncated at the low low value of 200 in Figure 3), then the KS statistics will fairly quickly go into the rejection region. Similarly in the discussion (p.8) the claim \"in many cases [estimated priors] being indistinguishable from the ground truth according to standard two-sample tests\" seems to be pushing credibility past acceptable bounds.\n\n* As discussed in Claims and Evidence, it's not clear what point the agent-based-model example is providing. One can generate \"diverse\" or \"non-diverse\" statistics simply by choice of statistic.", "theoretical_claims": "No theoretical claims made. It's all empirical.", "experimental_designs_or_analyses": "See other sections.", "relation_to_broader_scientific_literature": "The theoretical and methodological links to work like Nalisnick & Smith (2017) (Section 5) needed closer attention, as the contribution in this paper is purely the SBI setup.", "essential_references_not_discussed": ".", "other_strengths_and_weaknesses": "The strengths are that the use of reference priors is something that more researchers and analysts should be using, and methods and techniques that empower them to do so, particularly in the era of SBI, are highly valuable.\n\nThe weaknesses are perhaps slightly not fully convincing simulations, and a lack of comparison to tractable-likelihood approaches to emphasis the performance of the SBI aspect more clearly.", "other_comments_or_suggestions": "Typos etc.\n\nAbstract \"uninformative reference priors\". All priors are informative in some way. The authors even cite Bernardo (1997)'s paper entitled \"Non-informative priors do not exist\".\n\np.2 col 1 l.-7 \"missing information to be gained\" Should \"missing\" be \"expected\"? It doesn't make sense otherwise.\n\np.2. and elsewhere. \"the Jeffreys prior\" -> \"Jeffreys' prior\". No \"the\", note the punctuation. It's the prior of Mr Jeffreys: Jeffreys' prior.\n\np.2. Discussion of Jeffreys' prior as \"a further motivation\" for continuous priors. It's not clear what the narrative link to these priors is, nor why it's a further motivation.\n\np.3 col 1 \"a (lower bound on a )n estimate\". Please write better sentences that don't play such tricks.\n\np.6 col 2. Is \"VLB\" defined anywhere?\n\np.7 equation (18). The value 0.8 here is actually a user specified parameter (typically labelled \"c\"). This is how mis-understandings propagate in the literature.\n\np.7 last para \"Early work on reference priors HAS\". Also \"Mote\"->\"Monte\"", "questions_for_authors": "Questions are in \"Claims and Evidence\" and everywhere else (why are there so many sections in this review form?), and also:\n\n* p.4, col 1, paras 2 and 3 notes that the prior and the posterior estimator are assumed to be of the same model class (so that both can be obtained by a different choice of parameters), such as a normalising flow. This is proposed for computational efficiency, but there is no discussion of the potential weaknesses or negative implications of this if, for example, the model class is unable to approximate these distributions well.\n\np.3 col 1. Section 3 para 2. \"Continuous approximations to reference priors are advantageous for a number of reasons\" we are told \"as discussed in Section 2.1\". Section 2.1 mentions the avoidance of a technical issue (second last paragraph), and an unclear link to Jeffrey's prior. So the merit of this claim is unclear.", "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": "", "extra_scores": {"ethical_review_concerns": "."}}, {"review_id": "jSE5sWy28N", "reviewer": "Reviewer_V6ZV", "summary": "This paper focuses on learning an objective prior in the context of likelihood-free inference (SBI), where the likelihood function is intractable. Within the mutual information (MI) estimation framework, the authors propose three methods: GED, InfoNCE, and SMILE. These methods are systematically compared through various toy examples in simulation studies and further applied to more complex models, including the g-and-k model and an SIR model.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "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": "Thank you for your helpful feedback. We have included a set of new experiments [here](https://github.com/ICML-7582/rebuttal_plots/blob/main/plots.pdf), and see reply to Reviewer UJGD for a detailed explanation.\n\n## Methods and Evaluation Criteria\n\n- On inconsistencies between methods: It is reasonable that there may be inconsistencies between methods since there may be multiple local optima that give approximately the same mutual information between the model parameters and output but correspond to different priors. We will use some of the extra space to discuss that a sensible approach in practice may be an \"ensembling\" approach, namely to perform multiple runs of the prior learning procedure to account for any such variation. Such ensembling approaches are common in SBI to account for variations of this kind, see e.g. [1, 2] below. \n- On choosing between methods: Please see our discussion on this question in our response to Reviewer RgwU ('Question 3'). We will add this discussion to the final version.\n- On the performance of GED given that it learns a posterior as well as a prior: Indeed, estimating the posterior is not straightforward for some simulators. This is a possible disadvantage of GED. However, the method has produced reference priors in almost all cases matching either the ground truth or the other approaches. Other approaches might be preferable depending on problem specifications. We believe our methods (the first to address approximating reference priors in likelihood-free settings) constitute a useful baseline upon which future work can likely improve.\n- On lines 196--198: We'll use some of the extra space to give further details on the underlying methodology: we'll move the algorithm from Appendix A.6 into the main body, and specify that taking the prior and posterior estimators to be of the same model class is not necessary (the main reason for this was to reduce tuning and method complexity).\n- On higher-dimensional parameter spaces: Please see discussion on SLCP below.\n- On metrics: We have now used these metrics and present them in the tables below (format: _mean (standard dev)_ from 5 repeats). We'll use some of the extra space to discuss in the main body that we perform generally better but often similarly to/a bit worse than a likelihood-based method (\"Berger\") from [3].\n\nWasserstein:\n| **Task** | **Berger** | **InfoNCE** | **SMILE** | **GED** |\n| -------- | -------- | -------- | -------- | -------- |\n| **Gaussian**     | 6.95 (0.64) | 7.08 (0.56) | 6.77 (0.53) | 1.66 (0.20) |\n| **Exponential**  | 4.25 (0.39) | 2.43 (0.61) | 3.83 (0.75) | 2.04 (0.32) |\n| **AR(1)**        | 0.14 (0.01) | 0.06 (0.02) | 0.05 (0.03) | 0.06 (0.02) |\n| **Triangular**   | 0.08 (0.01) | 0.04 (0.00) | 0.05 (0.01) | 0.02 (0.01) |\n\nC2ST:\n| **Task** | **Berger** | **InfoNCE** | **SMILE** | **GED** |\n| -------- | -------- | -------- | -------- | -------- |\n| **Gaussian**     | 0.90 (0.01) | 0.62 (0.00) | 0.63 (0.01) | 0.49 (0.01) |\n| **Exponential**  | 0.91 (0.01) | 0.58 (0.04) | 0.61 (0.02) | 0.59 (0.06) |\n| **AR(1)**        | 0.62 (0.01) | 0.50 (0.02) | 0.50 (0.01) | 0.51 (0.01) |\n| **Triangular**   | 0.64 (0.01) | 0.54 (0.02) | 0.55 (0.02) | 0.51 (0.02) |\n\n- On Equations 2, 3, 9, and 10: Thank you; we'll derive these relationships in the appendix for clarity.\n- On performing SBI: We include a demonstration of how SBI can be performed at no additional cost using our methods, and a comparison against the results of running NPE and NRE in Figs. 2 & 3 of the rebuttal pdf. We'll use the extra space to include these results in the revised paper.\n\n## Experimental Designs or Analyses\n1. Please see comment above on the new test metrics. For further comments and discussion on the use of the KS test, please see reply 1. to Reviewer UJGD below.\n2. We have included new experimental results for models with a higher number of parameters (SLCP, for description see reply to UJGD). The true reference prior for the parameters should be uniform in $\\mu(\\theta)$ whilst the prior density for the determinant $|\\Sigma(\\theta)|$ should decay rapidly in $|\\Sigma(\\theta)|$. The priors learned by VLB and GED recover this behaviour. This highlights that our methods are approximately invariant to reparameterisation, since we learn reference priors in terms of $\\theta$ but recover the correct reference prior in terms of $\\mu (\\theta)$ & $\\Sigma(\\theta)$. VLB methods outperform GED; GED approximates a (complex) posterior during training, while VLB methods don't. More generally, the tradeoffs between VLB and GED are analogous to those between NRE and NPE.\n3. Please see our responses above on how to decide between/combine methods.\n\n[1] Cannon et al., \"Investigating the impact of model misspecification in neural simulation-based inference\", arXiv preprint (2022)\n\n[2] Elsemüller et al., \"Sensitivity-aware amortized Bayesian inference\", TMLR (2024)\n\n[3] Berger et al., \"The Formal Definition of Reference Priors\", The Annals of Statistics (2009)", "claims_and_evidence": "This work is novel, as it may be the first attempt to obtain reference priors in the context of simulation-based inference. The paper reframes the problem of finding reference priors as a mutual information (MI) estimation task and explores solutions using various machine learning techniques. The presented simulation studies are particularly interesting, making this article an enjoyable read.", "methods_and_evaluation_criteria": "1. Inconsistency among the three methods\n\nFrom a practical perspective, it is unclear which of the three methods should be preferred.\n \nAccording to the experiments in Section 4, the GED method appears to be the most favorable. However, in Section 4.2.1, GED fails to produce a flat prior for the first parameter, $a$ whereas the other methods do. Additionally, for the parameter $g$, the support varies significantly among the methods, leading to minimal overlap in the joint distributions of GED and InfoNCE (Figure 4(d)). This discrepancy could result in entirely different Bayesian posterior inferences.\n\nGiven that the paper aims to establish an objective prior as the author illustrates in Introduction Section, these inconsistencies undermine the article’s central goal.\n\n\n2. Methodological questions on GED\n\nIn Equation (12), obtaining the MI estimator requires estimating the posterior distribution \\( \\pi_{x_{1:n}} \\), but this step is not straightforward. I believe the accuracy of posterior estimation plays a crucial role in determining the performance of the optimization in Equation (12). \n\nEven with a parameterized prior \\( \\pi_\\phi \\), obtaining the exact posterior distribution remains challenging. Modern ML-based SBI methods \\cite{papamakarios2016fast, greenberg2019automatic, lueckmann2021benchmarking} and references therein rarely achieve exact posterior estimates in many cases.\n\nSpecifically, in lines 196–198, the authors state: ``we propose defining \\( \\pi \\) and \\( \\hat{\\pi}_{x_{1:n}} \\) to be of the same parameterized model class, and use \\( \\hat{I}^{\\phi, \\pi}(\\mathcal{D}_\\phi) \\) (12) as an estimate of MI.” However, this explanation remains somewhat unclear. I believe this aspect should be elaborated further in the main text to clarify the underlying methodology.\n\n\n3. Simulation studies\n\n The overall structure of the simulation studies could be improved for better clarity and organization.\n    \n Many challenges in SBI arise from high-dimensional parameter spaces, as seen in models described in Section 4.2.1. Since multi-parameter models are a common concern among practitioners, extending the analysis beyond Section 4.1.1 to include such cases is strongly recommended, even if only proper priors are considered. \n\n In these cases, using classification two-sample tests (C2ST) or sliced Wasserstein distance, instead of KS statistics, would enhance consistency with the SBI literature. This would provide a more systematic comparison among the methods.\n\n3. Minor comments and questions\n\n Additional derivations for clarity: The article states that Equation (2) implies (3) and that \\( I_\\pi(x_{1:n},\\theta) \\) is equivalent to Equations (9) and (10). Providing explicit derivations for these statements would improve clarity for the reader. \n    \n Comparison with standard SBI methods (Section 3.3): This section demonstrates the ability to perform SBI at no additional cost using the learned reference prior. However, how does the posterior distribution obtained through this approach compare to those from standard SBI methods?", "theoretical_claims": "There are not many theoretical results in this paper.", "experimental_designs_or_analyses": "1. Choice of Test Metrics:\n\nThe authors used the Kolmogorov–Smirnov two-sample test (KS test) for quantitative comparisons between the learned priors and the ground truth reference priors.\n\nThe KS test is less common and may not be ideal for assessing differences in higher-dimensional distributions or distributions with pronounced tails or boundary behaviors.\n\n Adopt metrics more established in SBI literature, such as classification two-sample tests (C2ST) or sliced Wasserstein distances, which can better handle high-dimensional distributions and provide clearer insights into distributional differences relevant to SBI.\n\n2. Dimension and Complexity of Simulation Studies:\n\nThe paper primarily evaluates methods on relatively simple, low-dimensional examples. While the authors briefly consider multi-parameter models (e.g., g-and-k and SIR models), the depth of analysis for these higher-dimensional cases is somewhat limited.\n\nGiven practitioners' frequent need to handle higher-dimensional parameter spaces, it would greatly strengthen the paper to explicitly include additional high-dimensional simulation studies. Even if restricted to proper priors, this would better highlight practical applicability.\n\n3. Stability and Consistency of Results:\n\nThere are notable inconsistencies among the GED, InfoNCE, and SMILE methods across different experiments (e.g., Section 4.2.1). While authors briefly acknowledge this, the practical implications for choosing one method over others are not adequately explored.\n\nProvide explicit guidance or decision rules (possibly via additional experiments) on selecting among methods based on specific contexts or performance indicators.\n\nComparison with Standard SBI Methods:\n\nThe paper highlights that the proposed approach facilitates SBI at no extra cost but does not quantitatively compare these posterior distributions to those obtained by existing standard SBI methods.\n\nIncorporate explicit quantitative comparisons of posterior inference accuracy or consistency with established SBI benchmarks to demonstrate added value or trade-offs.", "relation_to_broader_scientific_literature": "This paper's key contributions lie at the intersection of simulation-based inference (SBI), objective Bayesian methods, and machine learning-based mutual information estimation.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See above", "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": "", "extra_scores": {}}, {"review_id": "xY9RR83OPD", "reviewer": "Reviewer_UJGD", "summary": "This paper proposes the learning of reference priors via simulation-based approaches and normalizing flows. I am very short on time for ICML reviews. Apologies for my reviews being a bit short.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "We thank the reviewer for their useful comments. We have included a set of new experiments, described at the end of this review.\n\n# Reviewer Comments\n\n1. On the KS test performance: The KS tests for the Exponential and Gaussian models do reveal discrepancies. Both reference priors have asymptotes at $\\theta=0$. Our failure to fully capture the shape of the reference prior results from having to constrain the output of the variational prior to a subset of the real line. Currently, the final layer of the normalising flows we use is a sigmoid layer, and it must therefore be the case that the flow's density $\\to 0$ at the boundaries of the region (since it cannot assign mass to infinitely low values in logit space), meaning the current network design cannot capture an asymptote accurately. This can passing a KS test hard. Yet, aside from the asymptote, the learned and true priors are similar over the majority of the support (see updated Fig. 2 in pdf below). Note that this problem would persist even when modelling $\\log(\\sigma)$: either we'd leave the space unconstrained, in which case we'd fail to fully capture the uniform, improper prior on the unconstrained space, or we'd constrain to a bounded domain through a sigmoid layer again, leading to the same problem as above. We will discuss this in the revision, along with the use of alternative variational families that can better capture asymptotes.\n2. On learning priors in unconstrained spaces and learning flat priors over the reals: Please see previous bullet point, which we hope addresses this.\n3. On experiments being low dimensional: To demonstrate our methods' abilities to perform well for higher-dimensional parameters, we include in the pdf below (Fig. 1) new experiments on the SLCP model (description below, and further discussion in our reply to Reviewer V6ZV), which is higher-dimensional. Both methods recover expected priors; GED is less successful since it does posterior estimation, and SLCP often has complex posteriors. We'll discuss this in the revision.\n4. On the additional references: Thank you for these suggestions. These are relevant to the problem of determining prior distributions but focus on eliciting expert priors (arguably the opposite of our problem).\n5. On applications & practical implications of reference priors in SBI: It is possible for certain regions of parameter spaces to produce unreasonable/nonsensical outputs from simulators, and that reference priors might assign significant mass to such regions. However, the modeller can still restrict attention to regions of the parameter space $\\Theta$ that do not produce absurd behaviours even when using our methods; we simply address the problem of how to learn a reference prior on a parameter space once it is given (which may be a subset of $\\Theta$). We provide examples in the new experiments (Figures 2 and 3 of accompanying pdf) on doing SBI using the learned reference priors (and posteriors), and we can further point to our answer to Question 2 for reviewer RgwU, where we provide additional discussion on the use of reference priors for SBI.\n\n# New experiments\nWe have included a set of new experiments here, https://github.com/ICML-7582/rebuttal_plots/blob/main/plots.pdf, including:\n- Fig. 1: Experiments on a new simulator (SLCP-D [1]) that possesses a higher dimensional parameter space. The SLCP-D model has five parameters which parameterise the mean $\\mu(\\theta)$ and covariance $\\Sigma(\\theta)$ of a 2D Gaussian. The output of SLCP-D consists of four iid samples from said Gaussian and 48 distraction vectors sampled from a mixture of Student's $t$-distributions that are independent of the parameters. SLCP-D has a relatively **s**imple **l**ikelihood that produces **c**omplex **p**osteriors. The true reference prior should be uniform in $\\mu(\\theta)$, and for the determinant $|\\Sigma(\\theta)|$ should decay rapidly in $|\\Sigma(\\theta)|$. VLB & GED recover this behaviour. This also highlights that our methods are approximately invariant to reparameterisation: we learn reference priors for $\\theta$ but recover the correct behaviour of $\\mu (\\theta)$ and $|\\Sigma(\\theta)|$. VLB outperforms GED for SLCP-D; intuitively this is because GED approximates a (complex) posterior during training, while VLB methods do not.\n- Figs. 2 & 3: SBI experiments using reference priors from InfoNCE (a VLB method) & GED. Fig. 2: NRE posterior vs. ratio estimator trained during the VLB reference prior training; Fig. 3: NPE posterior vs. posterior density estimator trained during GED training. Inferences are almost identical, demonstrating our point about being able to perform SBI with no additional cost. \n- Fig. 4: New plots for SIR model outputs when only the infection data is used to find reference prior.\n- Fig. 5: An updated version of Figure 2 in the paper.\n- Fig. 6: An updated plot of the KS test.\n\n[1] Lueckmann et al., \"Benchmarking Simulation-Based Inference\", AISTATS 2021.", "claims_and_evidence": "They claim that they can learn reference priors via simulations. Indeed, we do see evidence that this can be done in a few simple and two more complex examples. It is a bit confusing to me that the approach is not able to learn the reference prior for two of the simple models well enough that the KS test already detects differences for small sample sizes. Did you try to learn reference priors on an unconstraint scale (e.g., log SD)?", "methods_and_evaluation_criteria": "They seem appropriate.", "theoretical_claims": "Only basic theory of reference priors is used which for my understanding is correctly interpreted and applied", "experimental_designs_or_analyses": "Experiments are reasonable but only very low dimensional. What would happen if the normalizing flow would attempt to learn a flat reference prior over the reals, i.e. for the mean of a Gaussian model? Would that work reasonable at all?", "relation_to_broader_scientific_literature": "This paper fits well into the literature on reference priors.", "essential_references_not_discussed": "Outside of reference priors, other approaches exist what use simulations to learn priors, partially even with normalizing flows. E.g., see https://arxiv.org/abs/2411.15826 , https://arxiv.org/abs/2308.11672 , https://arxiv.org/abs/2410.08710 perhaps these lines of literature should be cited too?", "other_strengths_and_weaknesses": "see questions", "other_comments_or_suggestions": "nothing major", "questions_for_authors": "One issue I have is that, since reference priors are typically very wide, they are likely to create trouble when applied in an SBI setting to estimate the posterior. This is because, for parameters in the tails of a reference priors, simulated data is often so unreasonable that learning the posterior in the space of realistic data becomes quite hard. Can the authors comment on how applicable they see their reference priors (or reference priors more generally) in actual SBI settings beyond the few examples they provided in the paper?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper studies the topic of simulation-based inference, which has the challenge of specifying a prior. This paper considers the problem of specifying an uninformative reference prior, and proposes and tests likelihood-free methods for learning reference priors by maximizing mutual information between parameters and simulated data, using variational approximations and both classical and neural mutual information estimators.

Reviewers were overall positive about this paper. There has not been much work on constructing reference priors in the SBI literature, and so reviewers thought this was a valuable contribution to the literature that lays the groundwork for objective Bayes in SBI. There were some initial concerns about the experimental results, including baselines. The authors provided a detailed rebuttal that included additional experiments. Several reviewers said that the promised changes in the rebuttal addressed their primary concerns.
Decision: Accept (poster)