Truncated Marginal Neural Ratio Estimation
Authors: Benjamin Kurt Miller
Alex Cole
Patrick Forré
Gilles Louppe
Christoph Weniger
Venue: NeurIPS 2021 Poster (2021)
Keywords: simulation-based inference
likelihood-free inference
implicit likelihood
posterior
marginal posterior
likelihood-to-evidence ratio estimation
neural ratio estimation
empirically testable inference
Bayesian inference
inverse problem

Abstract: Parametric stochastic simulators are ubiquitous in science, often featuring high-dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present a neural simulation-based inference algorithm which simultaneously offers simulation efficiency and fast empirical posterior testability, which is unique among modern algorithms. Our approach is simulation efficient by simultaneously estimating low-dimensional marginal posteriors instead of the joint posterior and by proposing simulations targeted to an observation of interest via a prior suitably truncated by an indicator function.  Furthermore, by estimating a locally amortized posterior our algorithm enables efficient empirical tests of the robustness of the inference results. Since scientists cannot access the ground truth, these tests are necessary for trusting inference in real-world applications. We perform experiments on a marginalized version of the simulation-based inference benchmark and two complex and narrow posteriors, highlighting the simulator efficiency of our algorithm as well as the quality of the estimated marginal posteriors.

Truncated Marginal Neural Ratio Estimation
Benjamin Kurt Miller University Amsterdam b.k.mille wimmlerruuaa.. nl
Alex Cole
Patrick Forré
University of Amsterdam e.e. cow.euva......... ml
University of Amsterdam p fod.forreSuva.ir nl
Gilles Louppe University of Liège g louppe@ullege.
Christoph Weniger
University of Amsterdam
v.wnngerruua...
Abstract
Parametric stochastic simulators ubiquitous in science, often featuring high- dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present neu- ral simulation-basee inference algorithm which simultancously offers simulation efficiency and fast empirical posterior testability, which is unique among modern algorithms. Our approach is simulation efficient by simultaneously estimating dimensionn marginal posteriors instead of the joint posterior and by proposing simulations targeted to an observation of interest via prior suitably truncated by indicator function. Furthermore, by estimating a locally amortized posterior our algorithm enables efficient empirical tests of the robustness of the inference results. Since scientists cannot access the ground truth, these tests are necessary for trusting inference in real-world applications. We perform experiments on marginalized version of the simulation-bon posteriors, highlighting the simulator efficiency of our algorithm as well the quality of the estimated marginal posteriors. Implementation on GitHub.
inference benchmark and two complex and narrow
Introduction
Parametric stochastic simulators ubiquitous in science 3) and using them to solve the Bayesian inverse problem is of general interest. Likelihood methods like Markov chain Monte Carlo (MCM)) 4 5 or nested sampling [6] are applicable when the likelihood tractable. It is equally common that the likelihood is only implicitly defined by the simulator or is inefficient to compute. For this so-called likelihond-free or simulation-basel inference, the traditional approach is Approximate Bayesian Computation (ABC) 7 [
See [9] for a reference.
Simulation-based inference (SBI) is closely connected to ABC and has been open research topic since as early as the 1980s [10. Deep learning has accelerated progress in the field [1]
12
[3 [4].
Proposed algorithms that learn the likelihood [33 or the posterior [14 15 16) utilize density
estimator. The likelihood-to-evidenee ratio [ can be learned via a classification-bat -base technique.
Refs. M and [44 were brought into a unified framework by [17].
fideliiy simulators often have many parameters and/or an intractable likelihood function, which can make inference notoriously difficult. Practitioners are usually faced with observational data and an expensive stochastic simulator, without access to the ground truth posterior. They want a testably accurate posterior estimate without extreme simulation expense. With existing methods, the
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35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Table 1: Comparison of SBI methods, including our proposed TMNRE, along with select properties. Properties listed are intended to showcase TMNRE and do not necessarily reflect the most desirable properties in every inference setting. For example, if cost were not a inhibiting factor tractable joint
distribution may be more appealing than targeting marginals directly. Similarly, fully amortized posterior estimate more flexible than a targeted one but remains, often, prohibitively expensive.
Property / Method
Likelihood based
ABC
NRE
NPE
SNRE
SNPE
TMNRE
Targeted inference
X
J
V
Simulator efficient direct marginals
X
/
X
X
(Local) amortization
X
X
/
/
X
X
practitioner must choose between increased accuracy per simulation (so-called sequential methods 13 18) or efficient empirical testability (so-called amortized methods [I].. We provide method
which offers both simultancously with a balance that can be tuned by a hyperparameter. Three attributes contribute to this goal:
Targeted inference. Focusing simulations on the parameter regions that are most relevant for the inference problem and target data is more efficient. This is particularly true when most posterior density concentrated compared the prior's density.
Marginal posteriors instead of the joint. Scientific insight is often based a low dimensional marginalization of the posterior with nuisance parameters removed [19]. The additional information of the full joint posterior might not be worth the additional cost afforded. Targeting marginals directly, by estimating only the marginal for the parameters of interest, is simpler and sufficient for many scientific, parameter estimation, and bounding purposes.
Consistency checks through local amortization. Practitioners are interested in testing the quality of inference methods 20 21 [22]. One such test is to compare the empirical and nominal contained mass of estimated credible regions 23 Amortized methods learn the posterior for any data, generated by any parameter, facilitating empirical study of the nominal credible regions on fabricated data. Still, learning an amortized posterior is excessive if only a small subset of parameters are consistent with a target observation.
We propose the concept of local amortization learn the posterior on said subset, combining simulator efficiency of targeted inference with the testability of amortization. Both are critical components for enabling trustworthy scientific results.
Our contribution. We propose an algorithm that simultaneously achieves all three of the above aspects: Truncated Marginal Neural Ratio Estimation (TMNRE). It approximates the marginal likelihood-to-eviddnee ratio in a sequence of rounds and shines when the joint posterior is prohibitively costly. As a basis, we adopt fikelihood-to-evidenee ratio estimation proposed in [T.. although our truncation scheme is applicable to other neural simulation- based inference methods which estimate the posterior or likelihood [13 17. Our iterative scheme is loosely inspired by likelihood-bain based
nested sampling 66 24] 25 since we generate training data drawn from a nested sequence of truncated priors in multiple rounds. Our algorithm (a) preferentially generates simulations in relevant regions of the parameter space, (b) allows estimation of all marginals of interest simultaneously and in parallel from the same training data, and (c) yields posteriors that are locally amortized in constrained region around the posterior, enabling empirical self-consistency test of the inference results.
Related work. In Table compare the properties and features of selection of deep-learning based simulation- inference methods that are directly relevant for our work. Sampling from regions of highest probability density is baked into most likelihood-baat based methods 5 24 25 26 27 28). Amortization is generally not available with these methods because they sample from a particular posterior. Approximate Bayesian Computation (ABC) is a rejection sampling technique where proposed samples from the generative model are accepted based on user defined distance criterion comparing generated data the observation of interest. Two important methods include REJ-ABC [9] where the proposal distribution is simply the parameter prior and SMC-ABC 22 30
where the proposal is iteratively refined. Blum and François (31 introduce an ABC distance criterion weighting mechanism to tune the posterior sampler as well as a proposal prior which draws from truncated region of true prior. It estimates the support of previously-aceeptee samples via support vector machines [32] and samples from this region with rejection.

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Likelihood-frie free inference can be cast as conditional density estimation problem targeting either
the posterior directly [14 T5 T6 17 or the likelihood [3] 33 This technique requires density
normalizing flow 35 36 Neural Likelihood Estimation (NLF) performs well on benchmark tasks but must learn a density representation of the data in an unsupervised setting-difficult difficult for complex data. Modern variants of Neural Posterior Estimation (NPE) [7 have become effective enough to offer an alternative estimation method for scientific practitioners [37].
estimator, normally implemented as mixture density network 34
Amortized Approximate Ratio Estimators Neural Ratio Estimation (NRE): Binary classification allows estimation of the likelihood ratio between two hypotheses [12 38
42 and was most famously applied Generative Adversarial Networks [43] Ref. [ noted that naive application in the likelihood- -reee setting was unsatisfactory because the mathematically arbitrary choice of reference hypothesis significantly affected empirical MCMC results. Comparing likelihoods from jointly drawn (ar, 0) ~ p(a,, f) and marginally drawn (az, 0) ~ p(z))p(f) (f) samples, where æ and O refer respectively to simulated data and simulator parameters, addresses the issue. Ref. [17 cast NRE and NPE in a unifying framework by adapting the loss function to contrast several possible hypothetical parameters. In this paper we refer to the algorithm described in []] as NRE or NRE_A while the likelihood ratio algorithm described in 117 is referred to as NRE_B.
39
40
41
Directly estimating the marginal posterior distribution has been mentioned [1] and explored 44 45
Moment networks 45 produce the (central) moments of the posterior distribution, without calculating the density explicitly, via hicrarchy of neural networks trained on regression problem. Ref. 45 also introduced a method which learns marginal posteriors with normalizing flows but does not address targeted inference or testability of estimated posteriors.
Sequential Methods: The neural likelihood-free methods generally offer a so-called sequential formulation that targets the posterior of a particular observation æ, [T]]3] 17 Rather than drawing samples from the prior, the simulation budget is divided between rounds and the previous round's posterior used as the new proposal distribution for the next round. This method increases simulation efficiency, but does not allow for amortization. Importantly, sequential methods can become highly inefficient when targeting multiple marginal posteriors directly because the previous round's marginal posterior does not update beliefs about the other parameters. A full parameter vector is necessary to run the simulator, thus defeating the purpose for all nuisance (marginalizeed-over) parameters.
2 Method
We aim estimate any marginal posterior of interest using approximate marginal likelihood-to- evidence ratio. Although we normally compute every one- and two-dimensional marginal posterior for visualization purposes, our method is not limited to this restriction. We define the object of study...
Let parametric stochastic simulator g be a nonlinear function that maps a vector of real parameters 0 (0I...., 0n) and a stochastic latent state E RRN N₂ E N to observation - 00, z). We consider factorizalle prior p(0)- = p(0,) ................... (((0))..p(pp))) over the parameters. The joint posterior is given via Bayes' rule p(0
p(æ p)p(() 0))(()))) where p(x 0) is the intractable likelihood (or implicit distribution 10, 46 and p(æ) is the evidence. Our goal is to efficiently compute arbitrary marginal posteriors, p(O 0 x). Here, 2 are the parameters of interest, and denote all other (nuisance) parameters by 17 such that = (0, The marginal posterior is obtained from the joint distribution p(0,1 n x) == p(0 x)) by integrating over all components of 17.
p x=) / p(0 n x)) dn p(x 0 0,n)p(n) dn p(0) p(ar 9
p(x) p(1).
(1)
p(x)
where we used Bayes' rule, prior factorizibility, and defined the marginal likelihood p(æ: 1).
2.1 Marginal Neural Ratio Estimation (MNRE)
This paper considers the set of one- and two-dimensional marginal posteriors and their corre- sponding parameters of interest. Given parameter vector E IRD define the set of all pa- rameters associated with the one dimensional marginal posteriors by θ == (O1...., do something similar, up to symmetry, for all two dimensional marginal posteriors O₂ = {(0,, 0,) 2.2 i ii=,...., D, = i 1,..., D}. We set marginals of interest {0x) =
We
3

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1 UU O2 but in the general case, { can be any set of marginals that the practitioner desires. For every 9g we use NRE estimate the corresponding marginal seclihood-to-evideee ratio
rk(2 0x) p(æ: 0x)) p(a, p(0x x)) v(0x)
p(x)) p(z))p(0x)
To this end, we train binary classifiers ( (z. 0x) to distinguish jointly drawn parameter-simulation
pairs (xz, 02,) ~ p(ær, 2xx) from marginally drawn parameter-simulation pairs (a, 22,) ~ p(z))((@))
where Φ represents the parameters of the classifier. A Bayes optimal classifier Pk would recover the
density pi (x (x)) pla. Then the ratios of interest can be estimated by
r(a,0x)
(2, 0x)
p(æ, 0x)
fx( 0x)) = 1- Pk (x, 1k,)
2
r (x)..
(3)
(k))
We train each ratio estimator (x 9) using Adam
47 to minimize the binary cross- (BCE)
/ 1o( (0) In ( 02) + p(=))p(0) In (1 pk, 0x)) dx de.
lk =
(4)
In practice, we concatenate æ with &g as the input to Ôv.ó- Since each classifier trains independently, it is trivial train them all in parallel using the same underlying (zx, 0) pairs.
Practically, we parameterize the classifier by Pk.o (x, 2x) = o fk.o(2, 22), where o is the logistic sigmoid and f is a neural network. The connection in Eq. (3) between the estimated ratio and the classifier implies that log 1% (x xx) = (a. xx). We call the above technique MNRE.
When training data is limited, we found empirically (see Sec. 3.2 below) that the MNRE approach typically leads to conservative (i.e., not overconfident not narrower than the ground truth) likelihood- to-evidence ratio estimates, provided early stopping criteria are used to avoid over-fitting of the classifier. At its core MNRE solves simple, supervised binary classification task rather than a complex, unsupervised density estimation problem. Classification tasks generally easier to train [2), and can rely on battle-tested network architectures.
2.2 Truncated Marginal Neural Ratio Estimation (TMNRE)
MNRE and NRE estimate a (marginal) Akelihoood-oo-evidenee ratio agnostic to the observed data or parameter 0, so-called amortized estimate. In other words, MNRE is suitable when æ e {g(0, =) 0 E Ω, E IR-Va), } where Ω is the support of the prior. We propose an extension of this algorithm that enables targeted simulation of parameters relevant to given target observation Vo, and locally amortizes posteriors such that it enables empirical tests of the inference results. Local amortization implics that our proposed method is suitable when æ E (0, xx) e I, TeI,ZERN where the parameter region I C. Q is function of x. and will be defined below.
We observe that values of 0 which could not have plausibly generated E, evaluate to negligible posterior density, i.e. p(0 | æ.)) = O, which suggests that the corresponding parameters A do not significantly contribute the marginalization Eq. ( We denote a prior that suitably constrained to parameters with non-negligible posterior density p(e 0 x.)) by
(0 = V 1 (0))(()
(5)
where (0) is an indicator function that is unity on C Ω and zero otherwise, and V-1 is a normalizing constant (which can be interpreted as the fractional volume of the truncated prior). The subscript Dr denotes that arbitrary symbol is based on a prior truncated by indicator function 1r-
We define rectangular indicator function 1rw by discarding parameters that lie in the far tails of the one dimensional marginal posteriors of our target observation H.. using a thresholding 1, via
e E Ω Vd 1...., maxe, p(0a xx)
p(0a x.))
(6)
For Gaussian joint posteriors, this scheme leads to one dimensional marginal posteriors pr(Oa xx) that are truncated at their approximately +--2lne tail. In general, truncation will lead to an approximation error that can estimated as pros (0 n..) = p a:,) + O(e) maxo p(O m.),, see Appendix C Throughout this paper use = 10-6 which corresponds to 2.266o for a Gaussian

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Algorithm Truncated Marginal Neural Ratio Estimation (TMNRE)
Inputs: Simulator p(æ 0), factorizable prior p(0), real observation æu, max rounds M, training data per round N(m), threshold e, dimension of parameters D, mass ratio 8, classifiers Pi (x, 0) = too (2, 0a) and P2(0, 0) = foofo fa 0a) d= Parameterized classifiers P1 (z, 0) and (2 0). constrained region l
(D.D)
Outputs:
1: procedure MNRE(D, 0', pb)
2: while Pe not converged do
3: 2--0 OPTIMIZER (Ø, V Ex BCB( 0x), 1) BCE(Ao, 8)).
4:
return f.
Initialize: DD() < (t. ((O) t supp(p(0)). a t +0. m 1
1: procedure TMNRE
2:
while a (m-1) < 3 and S M do
3:
D (m-1) - (x(m), ((m)) D(n-1) Φ(x) rmm-1)
Retain data in region
(mm]
- N(m) [D] 1(m-1))
4:
simulate
D Calculate num. necessary simulations Sample for jointly distributed pairs b Simulate jointly distributed pairs
Nrm))
5:
< (0(m) pim 13 (0)p(())
( n==
6:
a - (n) p( ((())
simile
I==
7:
0' (0 1l 1p 1) (0)p(0)
N(
Sample for marginally distributed pairs
8:
D(m) + D p(m-")) ((x(1)) ((())
(mn)
imit
D Aggregate training data
1n=1
9:
Pi - MNRF(D(m) 0', P1)
p(m-1) vd: Pa.r (Pg(ma)
10:
p(m)_
max9, P Parr(m) (B2)wa)
Find constrained region
11:
a (m) pow (0)p(0)d0/ 1 (0)p(0)d
Update prior mass ratio Increment counter
m + m
13:
+ MNRE(D(m) (m) 0', P22)
14:
return P1. P2. ((m)
Those truncations do not affect the location of high-probability credible contours and have hence no practical effect on parameter inference tasks. We provide more exemplary error estimates for range of cases in Appendix C
posterior.
Our algorithm defines series of nested indicator functions 1r(m) whose regions have the property
Ω = p 5 (2) 5 ... (MM) 5 p
They iteratively approximate the indicator function 1 in multiple rounds 1, I,... M. This sequence is generated with the following steps:
We initialize ( = Ω == supp(p(0)), meaning that we start with the unconstrained prior.
Each round 1
M, train D, dimensional ratio estimators farrom) (z 0a) using data from within the constrained region, 0 E ((m), The estimated marginal posterior is Prim (0x x)) = farro (æ (a)Prow) (6x). To this end, do MNRE, setting 0₂ = 0R. d E {1,2...., D} using the constrained prior Prom) (0) with (m) training samples per round.
For each round m < M, we estimate the indicator function for the next round using the approximated posteriors, via
{ A (m) Yd
Prom) (Oa Fo) maXg, Prom (fa xo) c
(m++))
The last round determined either when m M or when a stopping criterion is reached. The stopping criterion is defined by the ratio of consecutive truncated prior masses. satisfied when the sequence of truncated priors have the property f 1rro (0)p(0)d0/ Lrim n (0))(())d B. We often set 3 = 0.8.
5

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Using the data from this final constrained region, we can approximate any marginal posterior of interest. In this paper, we estimate the one- and wwo-dimensional marginals necessary for corner plots. We emphasize that the data already generated during the truncation phase can be reused to learn arbitrary marginals of interest. When higher accuracy ratio estimates are needed, the user can simulate from the truncated region.
We briefly address failure modes. First, this algorithm relies the assumption that posterior estimates Prom) (6al σo) from MNRE provide a good approximation of p(Aa zo). An over-confident estimate would remove parameter ranges that are part of IT In practice, we have not observed this effect, although it is a concern with all simulation based methods [23]] We give credit to early stopping and conservative choice of e, and provide further illustration and support in Sec. bellow Second, since the truncated posterior only agrees with the ground truth up to corrections of order c, the iterative scheme will not converge Eq. 6 rather to similar expression where the right-hand side of the inequality in Eq. (6) receives additional (E)) corrections. Although these corrections mildly affect the truncations, they are of little practical relevance since we choose an which is very small. Both failure modes are diagnosed by checking whether high probability regions of the estimated posteriors intersect with the boundaries of the indicator function.
Algorithm| esstmmates the necessary marginal posteriors for corner plot visualization. We demonstrate the cost-effectivenee effectiveness of this algorithm in Section An important limitation of Algorithm is the inaccessibility of the posterior predictive distribution. This limitation is mitigated by training a ratio estimator all parameters within the truncated region; however, producing an accurate joint estimate may come with (sometimes significant) additional simulation costs.
Like sequential methods [7] the number of rounds M, the training data per round N(m), and any stopping criteria 3 are hyperparameters. For further discussion and default values see Appendix A for bound derivations and limitations see Appendix C and D We present TMNRE Algorithm
Properties of our algorithm. We discuss the properties of our algorithm in support of Table First, our algorithm performs targeted inference by successively focusing on regions of the parameter space that are compatible with an observation F.g- Second, since training data is always drawn from the prior, it is possible to efficiently train arbitrary marginal posteriors with the same training data generated for round M. Third, the algorithm trains locally amortized posteriors that valid for parameters 0 E p(m), facilitating empirical consistency checks of the estimated posteriors within this region. TMNRE's properties provide a favorable cost-benefi aatio-yielding marginal insight into the posterior without paying the price of accessing the joint posterior. The price of the joint is often inhibiting for expensive simulators. These aspects will be demonstrated in the experiments in the following section.
3 Experiments
First, we perform experiments to compare TMNRE to other algorithms on standard benchmarks from the simulation-based inference literature. Next, highlight useful aspects of our algorithm regarding targeted inference, marginalization and local amortization with two additional experiments. These experiments compare algorithms using performance metrics which access the ground truth posterior. For practitioners who normally cannot access the ground truth, TMNRE offers an empirical consistency check (see Section (3.3 to assess the quality of the estimated posterior. Such check is impractical for sequential methods and is one of the primary practical applications of TMNRE. Further experiments, including application on cosmology simulator, can be found in Appendix
3.1 Performance on standard tasks
We compare the performance our algorithm with other traditional and neural simulation- based inference methods on selection of problems from the SBI benchmark (18). Each task is defined by a simulator, ten observations, a simulation budget, and 10,000 samples drawn from corresponding reference and approximate posterior distributions. The reference samples enable quantification of algorithmic accuracy on range of performance metries. We evaluate performance on all tasks except the Bernoulli Generalized Linear Model because its prior is not factorizable. Details in Appendix A

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Since our method estimates every one- and two-dimensional marginal posterior. we compare samples from our approximate marginal posteriors with samples from the reference joint posterior, marginal- ized over nuisance parameters. We quantify the results using the Classifier 2 2-Sample Test 48 49
We train C2ST classifier for each of the (2) (2) possible one- and wwo-dimensional marginals, and report the mean values, and 95% confidence intervals, in Figure []] We call this averaged perfor- mance metric C2ST-ddm, see Appendix B for more detail. The results are presented grouped by
dimensionality since difficulty increases with dimension and is reflected in the C2ST-ddm scores.
For comparison, we computed the C2ST-ddm on the other benchmark methods' posterior samples. Unlike our method, which was trained on the marginals directly, the benchmark methods were trained estimate the joint posterior. We note that since TM- NRE trains neural network for every marginal (efficiently, in parallel), our method has many times more param- eters than any neural likelihood-free inference method that directly targets the joint. However, parameter count is not a scarce resource SBI. Train- ing hyperparameters Appendix[A] [
1-dimm 2-dim Marginal
C25T-ddm
TMNRE RE]ABC
NLE
NPE
NRE SMC
SNLE
SNPE
SNRE_B
0.9
0.5
1-dim/ 2-dim Marginal Gaussian Linear
C25Tddm c25 c25T
0.5
1-dim 2-dim Marginal Gaussian Linear Uniform
0.7
1-dim 2-dim Marginal SLCP
be found in
0.7
1-dim/ 2-dim Marginal SLCP Distractors
The maximum number of rounds be- fore meeting the stopping criteria var- ied across tasks from just one round with no truncation up to seven rounds on Gaussian Linear. Out of 240 runs, five did not converge for TMNRE.
C25Tddm c25 c251ddm
0.5
I-dim 2-dim Marginal Gaussian Mixture
0.5
As shown in Figure 1 our method outperformed REJ-ABC and SMC-ABC and offers increased efficiency com- pared to non-sequential methods on all tasks, except Gaussian Linear. On some tasks, TMNRE competitive with sequential methods. Generally, TMNRE performs best on narrow pos- teriors and large simulation bud- get. Benefits diminished on tasks with wide posteriors like Gaussian Linear, SLCP, and SLCP Distractors-2 a limi- tation of TMNRE. Based on these re- sults, ours is the only method, among sequential and non-sequential tech- niques, which offers both sufficient accuracy and local amortization.
1-dim
2-dim Marginal SIR
0.7
1-dim/ 2-dim Marginal Lotka-Volterra
C25Tdim
0.9
323
leq
105
le3
1o4
3e5
143
1ea
Set
Dat
Number of Simulations
Figure 1: Performance on marginalized posterior bench- mark tasks. Mean and 95% confidence intervals of classifi- cation accuracy (C2ST-ddm) for our method, TMNRE, and other SBI methods with 10 observations / budget. one- and two-dimensional scores are plotted. Lower scores imply bet- ter posteriors. Our simulation budget is approximate, see Appendix A The plot and tasks are derivative of (18).
3.2 Efficient targeted inference: a 3-dim torus model
We define a task which highlights the effectiveness of truncating the prior, namely a simulator with a very small torus shaped posterior. We present an ablation study of the truncation method along with hyperparameter scan of c. Task details and additional experiments are presented in Appendix A
We ran Algorithm| I which satisfied the stopping criterion after four rounds. We performed marginal likelihood- weweroood---vvvieneee ratio estimation on one- and two-dimensional marginals for each step the sequence of constrained regions, using the number of samples available that round. We also trained an estimator which used the same simulation budget but the samples were drawn from the unconstrained prior. We analyzed the prior volume, C2ST-ddm, and the sum of one dimensional KL divergences
7

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1c-5
Ground Truth
TMNRE
MNRE
CZST 300m0mmmmm SSmuutttoo
1.3
dm
9.2
3.1
81
82
61
61 -10
105 4
Figure 2: First through Third Punel: View of torus marginal posteriors as estimated by rejection sampling. TMNRE, and MNRE. In some dimensions, the posterior extends the full unit cube prior width, while in others it is very narrow. TMNRE easily finds the asymmetric details after constraining to the relevant region while MNRE does not. Fourth Panel: The results of hyperparameter scan of on the torus task. The C2ST-ddm per simulation is reported versus c. The mean and 95% confidence intervals are shown over five repetitions of the experiment. Lower values indicate better performance.
each round for both methods. The posteriors are shown in Figure and the performance metrics for the ablation study, shown in Figure
We found TMNRE very accurately approximated all marginals the maximum simulation budget. MNRE placed mass in the correct region but missed the shape of the posterior entirely. TMNRE improved simulation efficiency compared with MNRE as indicated by the slope of the C2ST-ddm. The normalome posterior estimates at every round are plotted in Figure 3 We note that given the limited training data in early rounds, our method predicts wider posteriors than the ground truth. These are called conservative posterior estimates and they are the preferred failure mode for practitioners [23]. Other SBI methods are tested on the torus in Appendix A
To determine the effects the hyperparameter we performed grid search between 10-10 to 10-1 using TMNRE on the same simulator. We reported the performance terms of the C2ST-ddm per simulation in Figure and repeated the experiment five times. We observe that the optimal value of was 10 since it was the most conservative value of e that optimized the metric.
10
mi
HDI Emotrccal Cremiiiity
HD1 Emoincal Owdenity pi
HDI Emvrnccll Crodiilitt
10-2
10-4
10-4
22
23 30
100
50
102
25
50
15
100
HPDI Nominal Crediilitt for 194]
HPDI Nomtnal Credialiity for 8 (94)
HPDI Nominll Credbillty for 8,
Figure 3: First panel: The ratio estimator for in the 3-dim rounds. The estimator is conservative with limited data because each round's 's estimate converges from above to the final results (red line); i.c the early rounds produce too-wide posteriors thus truncation decisions are made safely. Second to Fourth Panels: Empirical versus nominal credibility for the highest posterior density intervals (HPDI) for each 0, in the 3-dim torus. The line (above) the diagonal implies accurate (conservative or wide) nominal credible intervals. See section 3.3
example, for several consecutive
3.3 Empirical tests of inference results through local amortization
Our algorithm locally amortizes the posterior for parameters drawn from the constrained prior (r)) This opens the door for various experimental diagnostics to test the reliability of our trained inference networks with simulated data (which is also possible for NRE [50], but not for sequential methods that are exclusively targeting demonstrate this by comparing the empirical credibility to the nominal credibility for the highest posterior density intervals, similarly to (23) but estimate using the truncated prior.
specific observation rather than range of observations). We
For the 3-dim torus example, we draw 10000 samples (ar, 0a) ~ p(# ())r(() from the truncated generative model. For all samples, generate marginal posteriors, p(Oa) x). For those marginal

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posteriors, we then derive the frequency with which 1% highest density intervals contain the true value d.. The result is shown in Fig. [ It provides an immediate check of the reliability of our trained inference networks withoul knowing the ground truth, and provides a safeguard against overconfident statements, which critical for using the results of inference networks a scientific context.
This empirical test is designed to show the consistency of the estimated nominal credible intervals across realizations of fabri- cated data but it does not address a generative model mismatch or access whether the estimated posterior corresponds to the ground truth. When e is small enough, the effects of truncation will not significandly impact this empirical test because of the accurate posterior estimate. Large that trim the tails of the estimated posterior aggressively, render this test unreliable be- cause the estimated iikeiihoddtto-viddee curate highest density intervals. It possible to check whether is too large by observing high-density posterior equicontour intersecting with truncation bound, see Appendix
TMNRE
1-dim
MNRE
- 2-dim
C2ST-ddm Prior Volume
100
10-1
ratios will have inac-
1.0
0.8
3.4 Efficient marginal posteriors: the 10-dim eggbox
Sum KL
2-
We define posterior that, when plotted two dimensions, looks like top-down view of an 2 x 2 eggbox. Let 0, g E RDD and 0g denote the kith element of 0, then the simulator for this problem is defined 9& (0) s0 r). To fix the posterior shape, we set Oz.o = 3. k 1, 2...., D and æo = g(,)). The likelihood is determined by an additive noise model p(æ 0) (g)(0), o2 with o = 0.1 The total number of modes in our 10 (O-dimensional model is 2 typically feature such a regular mode structure, but this pattern enables comparison of the various algorithm's 's ability to handle multimodal data.
3633
18508
33384
Naimulation
Figure 4: Performance metrics on MNRE and TMNRE versus simula- tion budget. Budgets determined by truncation algorithm. T: Prior volume. M: According to C2ST- ddm, TMNRE produces more aceu- rate posteriors. L: KL divergence summed over 1-dim marginals; same result.
1024. Realistic models do not
Given 10,000 training samples drawn from the prior and D = 10 dimensional parameter space, we trained MNRE to estimate all one- and two-dimensional marginals, the SBI [51) implementation of NRE and SNRE on the joint, and finally a marginalized version of SNRE (SMNRE). In SMNRE, we divided the samples across 10 rounds and each round proposed samples according to the previous round's posterior distribution for the predicted marginals, but the initial prior for the nuisance parameters. Since, in general setting, SMNRE cannot samples from another marginal estimator, we divided the 10,000 training samples evenly among the 55, one- and two-dimensional marginal estimators, each estimator receiving 181 training samples. 25,000 samples from each reported posterior are visible in Figure 5 TMNRE recovered the structure of the ground truth marginal posteriors, providing empirical evidence that estimating marginals directly can provide high accuracy at low simulation budgets for complex high-dimensional posteriors. Experiments using other SBI methods are in Appendix along with discussion of rotated version of the problem.
Ground Truth
MNRE
NRE
SNRE
SMNRE
6
47
5
Figure 5: Posteriors from the 10-dim eggbox benchmark (only 4 parameters are shown for clarity). All methods received 10k training samples and produced 25k posterior samples. NRE and SNRE were trained jointly, while MNRE and SMNRE were trained marginally. SMNRE cannot share training samples between marginals so each estimator received an equal share of the total simulation budget, 181 samples. Our method recovered the structure of the ground truth, unlike NRE, SNRE, or SMNRE.

---

3.5 Practitioner's 's use case: cosmological inference with a simulator
Parameter inference commonplace in astrophysics and cosmology. We are interested in a simulator with six parameters that returns three angular power spectra, as they would be measured by an idealized Cosmic Microwave Background (CMB) experiment. A budget of 5,000 simulations is sufficient for MNRF to accurately estimate the corner plot, while NRE produces unconstrained and inaccurate marginal posterior estimates. Due to space constraints, we could not include this practical application of MNRE in the body of this paper. Please see Appendix[] details of the simulator, the inference technique, and the results. A complete study of the topic is currently in preparation [52]
Discussion and conclusions
We presented Truncated Marginal Neural Ratio Estimation (TMNRE), simulation-based inference algorithm based on NRE. The core idea of our algorithm is to focus on most probable parameter regions by truncating marginal posteriors in their very www-robabilltty tails. For Gaussian posteriors this is typically beyond 5a and does not significantly affect the higher density contours. In addition to performing on par or better than existing algorithms on standard benchmarks, TMNRE is better suited to the practitioner's needs than other algorithms because it offers simulation efficient marginal posterior estimation and the capacity perform efficient consistency checks through local amortiza- tion. These features are particularly desirable to scientists whose simulators are expensive and rife with nuisance parameters.
TMNRE uses a sequence of training and sampling rounds to automatically produce parameters with high posterior density, i.e. relevant to particular observation Eo. The output of this sequence is hyperrectangular approximation to a highest posterior density region, implicitly defined by hyperparameter c. That implies parameters from within this constrained region are more likely to produce data similar to x, than simulations from outside the region. Using data drawn from this constrained region, TMNRE estimates any marginal posterior of interest directly using marginal ikelibhod-to-eviddnee ratio; a simpler and more practical technique than estimating the entire joint posterior. Finally, by construction, our targeted inference method can accurately estimate posteriors of simulations from within the constrained parameter region. This freedom facilitates fast empirical studies of the nominal credible regions, which are of critical importance in real-life applications when there is no ground truth posterior to compare to.
On the SBI benchmark [18), found that TMNRE is on par with the most effective SBI algorithms, such as SNRE [11], as measured by the C2ST performance metric, Fig. [ We highlighted the benefits of TMNRE using two showcase tasks: a torus-shaped posterior and multimodal eggbox-shaped posterior. The torus featured a very narrow posterior that TMNRE found and accurately learned while simple MNRE failed to do so, Fig. We demonstrated validity of our iterative procedure, and the ability to perform important validation tests by testing the nominal credible intervals empirically, The eggbox's 's joint posterior was challenging for bRE-based methods but MNRE efficiently estimated the marginal posteriors. Fig. 5 Further torus and eggbox experiments are in Appendix A and cosmology posterior is estimated Appendix
Fig. 3
E
The presented algorithm is aimed at marginal posterior inference, which is a typical goal for scientific applications, but does not allow, e.g., to evaluate the posterior predictive distribution which requires the joint posterior. Furthermore, our algorithm particularly shines for high-dimensional problems with complex and/or narrow posteriors, whereas we expect that simpler problems may be better suited to other SBI methods. We address further limitations of our method in Appendix D
We note that the hyperrectangular indicator function, defined in Eq. ( is not optimal if some of the parameters are strongly correlated. However, it can be straightforwardly extended to more complex shapes. The challenge is to efficiently define the boundaries of the indicator function and sample from within it, a problem tackled by effective nested sampling algorithms [33]
This work is primarily foundational and the societal impacts, other than the cost of training machine learning models, would therefore be drawn from a hypothetical application. As this an inference method, it would be possible to apply it to biased simulators which could reinforce unethical patterns. In general, the societal impacts are closely tied to the implications of the simulators themselves.
10

---

Acknowledgments and Disclosure of Funding
This work uses numpy 54 scipy [55] seaborn [56]. matplotlib [57, altair [58] 59],
pandas
[60] 61] pytorch [62] and jupyter [6]] Benjamin Kurt Miller is funded by the University of
Amsterdam Faculty of Science (FNWI), Informatics Institute (Ivl), and the Institute of Physics (IoP).
We want to thank the DAS-5 computing cluster for access to their TitanX GPUs. DAS-5 is funded by the NWO/NCF (the Netherlands Organization for Scientific Research). We received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 864035 UnDark).
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15


Reviews:
[{"review_id": "pn2Fg7-Y21L", "reviewer": "Reviewer_opr4", "summary": "This study contributes with a new simulation-based inference method, Truncated Marginal Neural Ratio Estimation (TMNRE). It builds on a previous method, Neural Ratio Estimation (NRE), by combining NRE (for directly inferring marginals of the posterior distribution) and the sequential constraining of the prior distribution by truncation (targeted to a particular observation). On standard tasks, TMNRE is shown to perform on par with state-of-the-art methods. Furthermore, the sequential truncation of the prior allows for local amortisation and thus makes it easy to perform (local) consistency tests.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "### Originality\n\nThe novelty of the method consists in the sequential constraining of the prior distribution by truncation. Previous work is for the most part appropriately cited and discussed, and it is clear how this work differs from previous contributions. However, I believe it would be important to also discuss the paper by Blum and François (2010 - Non-linear regression models for Approximate Bayesian Computation), as this proposes an ABC method also with prior truncation.\n\n\n### Quality\n\nThe paper is technically sound, and claims are mostly well supported by empirical evaluation. However, I have a couple of comments I would appreciate the authors to address:\n\n-the authors write \"Targeting marginals directly, by estimating only the marginal posterior for the parameters of interest, is simpler and sufficient for most purposes.\" As a first minor point, I believe \"most purposes\" to be quite an overstatement, and would recommend the authors to instead write \"many\" or something similar (see also a related point under section \"Limitations\");\n\n-throughout the manuscript, the authors put quite some emphasis on the fact that TMNRE is more efficient, given its sequential prior truncation feature. However, while this efficiency is convincingly shown against MNRE (no truncation), NRE and some other variants (Figures 2,3,5), it would have been important to also show this against other methods (e.g. SNLE, SNPE), and for several simulation budgets.\n\n\n### Clarity\n\nThe paper is clearly written, and provides enough information for an expert reader to understand all the steps to reproduce the results. I would just like to point out a few typos:\n\n-lines 161-162, rather than \"r(x|theta)\", \"r(x,theta)\";\n\n-Figure 2, right panel, the x-axis is in units of the common logarithm, and that should be specified;\n\n-the authors write \"That means simulations from within this constrained region are likely informative while simulations from outside the region are likely uninformative\". I would suggest the authors to rephrase this sentence, as the wording \"(un)informative\", without stating about what, is a bit imprecise and potentially misleading.\n\n\n\n### Significance\n\nThis new method will be of interest to the simulation-based inference community and inspire further method development.\n\n\n### After rebuttal\n\nThank you for the careful answer to my review. After reading all the reviews and authors' rebuttal, I believe that this is an interesting paper and that the authors made an effort to address all the reviews, and in my opinion, to satisfaction. While TMNRE is a method for the exploration of the posterior (or more concretely, its marginals) rather than a method for full Bayesian inference, I think this will be a valuable/useful addition to the SBI toolkit. I increased my score accordingly.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have for the most part adequately addressed the limitations of their work. However, I would like to raise a point regarding the limitations of estimating marginals rather than the full posterior: arguably, in many cases the practitioner of simulation-based inference would indeed be interested in having good posterior predictives. These posterior predictives are not possible with TMNRE (as the authors acknowledge in line 345), and I would recommend the authors to more explicitly acknowledge this potentially impactful limitation.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "ZOICbbGnDDM", "reviewer": "Reviewer_8ziR", "summary": "The authors suggest the use of neural ratio estimation to target the marginal posteriors directly, thereby circumventing the more difficult problem of obtaining the full Bayesian posterior.\nThe method comes with other benefits such as local amortization as well as a sequential truncation technique for targeted inference. \nThe method is validated on several benchmark examples. \n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "__Originality:__ There is something to the idea to circumvent the trouble of full Bayesian inference and target the marginal posteriors directly. The authors also consider modifications that help to improve the performance by truncation. With its aims this paper is a different take on simulation-based inference than previous approaches. \n\n_Some minor comments:_\n- References to related work have a recency bias. For example, when writing “...the traditional approach is Approximate Bayesian Computation (ABC) [7]...” the reference [7] is a survey from 2018. However, ABC techniques already emerged in the 1990s. If the authors don’t want to add any more references, then I suggest something like [see e.g. 7 for a reference], in latex `\\cite[see e.g.][for a reference]{bibkey}`.\n\n- Inference for simulators goes back at least to the 1980s: \nDiggle, Peter J., and Richard J. Gratton. \"Monte Carlo methods of inference for implicit statistical models.\" Journal of the Royal Statistical Society: Series B (Methodological) 46.2 (1984): 193-212.\n\n__Quality:__ \nWhile it might be true that many practitioners mostly consider marginal posteriors (be it just because of visualization) I think the paper’s claims are too strong in parts. \n\nFor example, Table 1 is very suggestive, effectively showing the properties of the method but indicating that those are _the_ desirable properties of an algorithm. In particular, the “Simulator efficient direct marginals“ are not by necessity a desirable property as I would assume that most Bayesians would prefer to have access to the joint distribution if cost was not an inhibiting factor.\n\n__Clarity:__ The paper is generally clearly written and well organized. I did find some minor points the authors might want to change in a revision to additionally improve the clarity of the paper. \n\n_Some minor comments:_\n- May I suggest the authors use `\\usepackage[sort]{natbib}`\n- I am not a fan of the formula\n$$\n\tf(x | \\theta) = \\int \\delta(x - g(z, \\theta)) f(z | \\theta) dz\n$$\nand I feel it adds more confusion than clarity, not least because of its double abuse of notation. First, it abuses notation by using the same symbol for the distribution of $X$, $f( \\cdot \\mid \\theta)$ and for $Z$, also $f( \\cdot \\mid \\theta)$. They would be better references as $f_X$ and $f_Z$. Second, the use of the Dirac delta as a density requires some mathematics that are absent from the rest of the paper and might especially be confusing to non-physicists.  Third, I fail to see any additional content by writing the simulator likelihood in this way, effectively denoting an unknown quantity $f(x \\mid \\theta)$ as integral over another unknown density $f(z \\mid \\theta)$ and an additional unknown function $g$. \n“The full joint posterior can be unnecessarily informative.” I don’t think that makes sense as a criticism. The additional information of the posterior might not be worth the additional cost afforded, but “unnecessarily informative” is too blunt.\nThe notation $\\{g(\\theta, z) : \\theta \\in \\Omega, \\forall z\\}$ is confusing. The quantifier “for all” ($\\forall$) is usually followed by a predicate, but I don’t see what predicate should be satisfied by all $z$.\n\n\n__Significance:__ \nApproaches that help reduce the computational burden of simulation-based inference are an important part of research. The current draft of the paper needs some minor improvement before it is ready to be published. However, but I am happy to increase my score if the author can address the points raised here. \n\n__Edit after rebuttal:__\nThe authors have addressed all my concerns on clarity and have suggested changes that in my opinion improve the quality of the manuscript. I have therefore decided to increase my score towards acceptance. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Societal Impact has been adequately addressed.", "review_text": "", "extra_scores": {"ethical_concerns": "- ", "needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "9"}, "extra_text": {}}, {"review_id": "JuXWEwYoGdh", "reviewer": "Reviewer_BG8g", "summary": "This paper presents a method for performing likelihood-free inference in parametric simulators.  The method presented is thematically similar to, and builds upon, a recent tranche of work using neural function approximators to ameliorate the inherent difficulties in likelihood-free settings.  The method takes the interesting and unique approach of modelling an arbitrary set of many smaller marginals of the full posterior, instead of directly targeting the full posterior.  These marginals are estimated by using neural ratio estimation between the target marginal posterior and the full marginal.  The prior is then truncated (to a hyperrectangle smaller than the support of the original prior in each dimension) to concentrate simulations into regions of significant posterior mass.  These estimators (the ratio estimators and the truncated prior) can then be used to establish an estimate of the posterior distribution.  The method is evaluated on reasonably standard tasks, and an interesting cosmology example (although this was inexplicably demoted to the supplementary materials).  ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "### Summary+\n\nThe authors posit that modelling many lower-dimensional marginal posterior distributions is a fundamentally easier task than modelling a high-dimensional posterior distribution.  This seems like an interesting and intuitive design choice.  It also seems to be a relatively novel formulation in this context (see comments below).  I can believe that learning multiple cheap classifiers/ratio estimators is computationally streamlined, and defines a fundamentally easier inference task.  \n\nThe prior art is well surveyed (although the nuts and bolts of the prior art are not explored and hence some familiarity is required).  The problem formulation paragraphs (S2.P1,2) are compact and neatly introduces the problem domain, and indicates the solution path that the authors will take.  \n\nThe algorithm block is beautifully presented and makes the method very easy to understand.  I believe that I could implement the method using pretty much just the algorithm block.  Chapeau.  \n\nThe authors benchmark their method on the LFI benchmark dataset, as well as applying their method to two further examples.  The authors TMNRE method performs slightly worse on the benchmark tasks compared to the baseline methods.  A major point in the SBI baselines paper is that the choice of metric is significant, and so I would have liked to have seen these methods compared using a slightly broader range of metrics than a single metric.  \n\nI broadly like the submission, although I am not entirely convinced by it and its significance  (see comments below).  I have erred on the side of weak acceptance because I believe the work is sound, the paper is well put together and executed, and it has been well evaluated and the results thoroughly analysed.  The paper is generally well written, and is actually quite enjoyable to read.  The authors also release code, for which they are to be commended for.  However, there is no theoretical innovation or particularly inspiring or transferrable methodological innovation.  The performance on the standard benchmarks is not outstanding, and the baselines were not fully tested on the other experiments, making meaningful comparison difficult.  Finally, the extent to which this method advances our understanding of LFI more broadly is not especially clear to me.  For me, these points limit the reach and impact of this paper.  I would, however, consider upgrading my score if my concerns were successfully allayed.\n\n\n### Major Comments:\n\n1 - Can the authors comment on the fidelity of modelling a posterior distribution as a collection of marginals?  I am not convinced that this would transfer well to distributions with complex and richly structured posteriors.  I also do not agree with the author's assertion that a joint distribution can be “unnecessarily informative”.  I would rather “pay more” to recover the full joint, and then marginalize post-hoc, as opposed to recovering a simplified representation from which I cannot recover the full representation.  When targeting the full joint, I believe this method just reduces to NRE+truncation.  NRE is known to perform relatively poorly compared to more sophisticated methods, although the truncation _may_ help somewhat.\n\n2 - There are two methods presented, MNRE and TMNRE, correct?  Why wasn’t MNRE tested in Figure 1?  It also strikes me that the torus model and the eggbox model were well-chosen experiments for this method -- the posterior does basically factorize as a product of marginals with well-concentrated mass, and hence I am not especially surprised that these methods perform well.  I would be interested to see how well something like SNL performs on the torus model.  I suspect it would perform exceptionally well, and provides tractable marginals as stipulated by the authors, and hence I do not see a major contribution from this method there.  \n\n3 - Truncating the prior is a reasonably low-overhead method for increasing sample efficiency.  However, for the eggbox example, surely this truncation method leads to the hypervolume retained having a much larger (hyper)volume than one would desire?  This would drive down sample efficiency, especially as regions with posterior mass are driven towards the edge of this volume.  This will _dramatically_ reduce sample efficiency in high dimensions, in a way that more sophisticated sequential methods are not affected.  Furthermore, if the axis were rotated through 45 degrees (in all dimensions), then the volume of the hyper rectangle increases by a factor of 2^5=32, and hence the truncation will be a factor of 32x less efficient.  This seems to be a major drawback to me.\n\n4 - It seems to me that truncation may be a step back from the SNL-esque methods.  I would prefer to see something along the lines of {SMC, MCMC}-MNRE, where instead of truncating the prior, a discrete/atomic representation is used.  This would allow computation to be concentrated more efficiently, would relax the dependence on axis-aligned truncations, and prevent regions of parameter space from being aggressively deleted early in training.  I realise this is not the paper that has been written, but it is a sticking point in my head that a set of strictly contracting axis-aligned truncations is just about the least flexible approach that could have been taken.  You allude to this in the discussion (Lines 349-352).  Do you have any further comments on this?  This seems like the biggest weakness of the method.  (I suppose you want to write Canonical TMNRE next…!)\n\n5 - I do not understand these “consistency checks through local amortization”.  I figure you are suggesting that through the use of synthetic data we can test the inference result by recovering the ground truth.  If this is the case, this seems trivial/generally true to me and does not seem to be a real plus-point of your method in particular.  The real issues with methods such as this arise from model-mismatch in the generative model, and hence there are no real tests for this.  I think this claim is overstated, and unless I’ve gotten something wrong, I would rather see this removed.  (It could be referenced in passing as a sanity check or for debugging -- but nothing more).  I could not follow Section 3.3 as a result.\n\n6 - According to Algorithm 1, the data points aren’t drawn from the prior, they are drawn from the truncated prior, and so is the ratio estimator targeting the correct quantity?  This may just be a typo, i.e. Line 5 and Line 7 are identical (modulo a dash), whereas I believe Line 7 should drop the indicator term.  \n\n7 - Using the C2ST score seems a little biased to me.  Since the other methods are computing full posteriors, and yours is only computing a single marginal,  performing a two-sample test comparing yourself to the full posterior actually seems a little harsh on your own method -- even _incorrect_ as a metric!  I suppose this shows your method is working admirably, but also shows that maybe the solutions just happen to admit a factorized posterior (and hence are amenable to your method).  Either way, I find using this as the only performance metric a bit of a head-scratcher.  As mentioned above, I would like to see a more diverse suite of performance metrics examined.  \n\n8 - You recover N^2 marginal pairs, and N marginals.  The N^2 + N marginals that you recover will all be dramatically different as a result of marginalizing over N-2 other variables.  Practically speaking, how do you select which distribution to draw from for a particular distribution?  Moreover, how do you actually go about generating a sample from the joint...? \n\n\n### Other Comments:\n\nA - Are the authors able to comment on the difference between their method and that of Jeffrey & Wandelt?  I believe that the work is sufficiently different to be easily classified as novel throughout, but I think a more substantiated explanation of any similarities and differences would be useful.\n\nB - Does truncating the prior lead to sections of parameter space being permanently deleted?  I don’t think the algorithm could recover if a section with mass were not covered.  I realise this is true for most sampling algorithms, and Fig 4 suggests that the distribution starts broad and then narrows.  But it strikes me as though it could be particularly dangerous in this setting if there is categorically no way to expand the distribution.  Maybe consider adding some discussion of entropy regularization to the prior to the conclusion.\n\nC - There is a relatively substantial limitations section in the supplementary materials.  I would prefer it if some of these limitations were brought up and explored in the main text.  \n\nD - The cosmology experiment is by far the most interesting of the experiments.  I do not understand why it was demoted to the supplementary materials.  \n\nE - There also seems to be a missed opportunity using the marginals recovered to establish independencies or conditional independencies in the posterior, where this method explicitly enforces the efficiency gains that these independences can provide. \n\nF - An interesting point that is not strongly emphasised is that the same samples can be (re-)used to learn all of the ratio estimators.  \n\nG - I am also a little confused about the “local amortization”.  This method is inherently sequential and conditioned on a single $x_o$, and so there is no real amortization going on.  Around Line 190, you say that you train “locally amortized posteriors that are valid for parameters $\\theta \\in \\Gamma^{(m)}$”.  Are suggesting that the function approximators are going to be someway accurate/informative over the entire domain of $\\Gamma^{(m)}$?  This should be explained more clearly if this is the case.\n\n\n### Minor/Typographical Comments:\n\n- Inline or wrapped figures should be avoided.  They are bad for readability and flow.  Figures should really be floated to the top (or bottom at a push) of the page.  \n- I don’t understand the sentence beginning Line 30: “With existing…”.\n- Line 129: I don’t understand how a likelihood ratio estimator can be overconfident...! \n- Line 147: Incorrectly rendered characters (I only see boxes).\n- Some of the font sizes in the figures are TINY.  \n- I don’t really understand what Figures 4(b-d) are showing.  \n- Figure 6 does a much better job of showing the truncation compared to Figure 4a. \n- The discussion is very long and doesn’t add much (partly because the rest of the paper is pretty well written).  This could be cut down and used for the cosmology example instead, or, for Figure 6.\n- The societal impact statement is too long.  I think it is safe to say here: “There are no societal impacts.” (as is explicitly permitted in the submission instructions). \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "No societal impacts of note.", "review_text": "", "extra_scores": {"ethical_concerns": "No ethical concerns of note.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "1vdIVN86czL", "reviewer": "Reviewer_6wmD", "summary": "This work proposes a novel algorithm for estimating the posterior distribution in simulation-based inference (SBI) settings, where the likelihood is not known but it is implicitly defined via a stochastic simulator. The work proposes truncated marginal neural ratio estimation (TMNRE), which is able to approximate the posterior distribution only for a small number of parameters of interest and making the sampling more efficient by sequentially truncating the prior for a given observed data. In addition, the posterior distribution is amortized and can be used to check the goodness of fit of the posterior distribution, albeit only in the areas which have not been truncated during training. The paper offers extensive experiments for different parameter dimensionality, as well as a comparison with the state of the art in SBI and theoretical analysis of how much error is introduced with the prior truncating process.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I found the paper clear and easy to read, with the main algorithm being presented clearly (the comments in the algorithm box are much appreciated), as well as the relationship with related work. The experimentation is extensive and tackles different settings that could be challenging: an asymmetric low-dimensional distribution and a high-dimensional multimodal example, in addition to a series of standard tasks in SBI which have been recently established by [1] and a physics-motivated example in Appendix E.\nAdditionally, the authors elaborate on how much the \\epsilon impacts the approximation error in a couple of settings, which helps the reader get an idea of the tradeoff in posterior inference; having said that, the fact that the \\epsilon is set to be very low likes make a negligible difference in inference (at least in the example shown in the paper).\n\nOverall, I feel this is a solid paper and would be a great addition to the SBI community.\nI am including some (minor) comments below:\n\n- The amortization for every data and parameter value is a nice property, but it seems like it is restricted to the constrained region of the prior. Given the truncation in this paper seems to be pretty mild (in terms of tail trimming), I assume that checking the posterior fit for different observed data would still not be affected by the truncation. Can the author comment on how empirical check would be affected by a more aggressive trimming?\n\n- The truncation idea is quite interesting and potentially beneficial to other methods. As the author observed, creating a hyper-rectangle is not the most efficient way of doing truncation, but I assume it's probably computationally faster as one does not need to consider the correlation between parameters. Would it be possible for the truncation mechanism to just consider the correlation between the parameters of interest and doing 1D truncation among the nuisance parameters as a tradeoff? In addition, can the authors comment on how many initial samples one should use in order to avoid missing a potentially relevant area of the parameter space and how much is this number affected by the dimensionality of the parameter space? \n\n\n[1] Lueckmann et. al, \"Benchmarking Simulation-Based Inference\", AISTATS 2021\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes, limitations and societal impact have been addressed adequately.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3.5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a method for simulation-based inference that is based on neural ratio estimation and estimates marginals of the posterior distribution.

The reviewers are generally positive about the submission, citing the usefulness of the proposed method, the clarity of the presentation, and the good variety of experiments as strengths. There were some initial concerns, but the reviewers appreciated the extensive and thorough responses provided by the authors and most concerns were allayed. Overall, I'm happy to recommend acceptance.
Decision: Accept (Poster)