Active Sequential Posterior Estimation for Sample-Efficient Simulation-Based Inference
Authors: Sam Griesemer
Defu Cao
Zijun Cui
Carolina Osorio
Yan Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: simulation-based inference
likelihood-free inference
active learning
Bayesian methods
Primary area: probabilistic_methods

Abstract: Computer simulations have long presented the exciting possibility of scientific insight into complex real-world processes. Despite the power of modern computing, however, it remains challenging to systematically perform inference under simulation models. This has led to the rise of simulation-based inference (SBI), a class of machine learning-enabled techniques for approaching inverse problems with stochastic simulators. Many such methods, however, require large numbers of simulation samples and face difficulty scaling to high-dimensional settings, often making inference prohibitive under resource-intensive simulators. To mitigate these drawbacks, we introduce active sequential neural posterior estimation (ASNPE). ASNPE brings an active learning scheme into the inference loop to estimate the utility of simulation parameter candidates to the underlying probabilistic model. The proposed acquisition scheme is easily integrated into existing posterior estimation pipelines, allowing for improved sample efficiency with low computational overhead. We further demonstrate the effectiveness of the proposed method in the travel demand calibration setting, a high-dimensional inverse problem commonly requiring computationally expensive traffic simulators. Our method outperforms well-tuned benchmarks and state-of-the-art posterior estimation methods on a large-scale real-world traffic network, as well as demonstrates a performance advantage over non-active counterparts on a suite of SBI benchmark environments.

Active Sequential Posterior Estimation for Sample-Efficient Simulation-Based Inference
Sam Griesemer'
Defu Cao
Zijun Cui
Carolina Osorio "HEC Montréal
Yan Liu'
USC
MSU
Google Research
(samgriesemer, defucao, yanllannllaa.... calaill..s@eueeeee cusc. edu
cuiziju @msu. edu osorioc@google.com com
Abstract
Computer simulations have long presented the exciting possibility scientific insight into complex real-world processes. Despite the power of modern comput- ing, however, it remains challenging to systematically perform inference under simulation models. This has led to the rise of simulation- based inference (SBI), class of machine lcarning-emabled enabled techniques for approaching inverse problems with stochastic simulators. Many such methods, however, require large numbers of simulation samples and face difficulty scaling to high-dimensional settings, often making inference prohibitive under resource-intensive simulators. To mitigate these drawbacks, we introduce active sequential neural posterior estimation (ASNPE). ASNPE brings an active learning scheme into the inference loop to estimate the utility of simulation parameter candidates to the underlying probabilistic model. The proposed acquisition scheme is easily integrated into existing posterior estima- tion pipelines, allowing for improved sample efficiency with low computational overhead. We further demonstrate the effectiveness of the proposed method in the travel demand calibration setting, high- gigh--imensional inverse problem commonly requiring computationally expensive traffic simulators. Our method outperforms well-tuned benchmarks and state-of-the-an posterior estimation methods on large- scale real- world traffic network, as well as demonstrates performance advantage over non-active counterparts on suite of SBI benchmark environments.
Introduction
figh-fidelity computer simulations have been embraced across countless scientific domains, fur- thering the ability to understand and predict behaviour in complex real-world systems. Modern computing architectures and flexible programming paradigms have further lowered the barrier to capturing approximate models for scientific study in silico, enabling wide-spread use of computational experiments across disciplines. However, despite the relative ease of capturing real-world gener- ative processes programmatically, the resulting black-box programs are often difficult to leverage for inverse problems. This is a common challenge practical applications: the simulator often computationally expensive to evaluate, its implicit likelihood function is generally intractable, and the dimensionality of high-fidelity outputs typically prohibitive. To address these issues, likelihood- free inference methods have been introduced, operating under the broadly applicable assumption that no tractable likelihood function is available. Early success along this direction was achieved through easy-to-use methods like Approximate Bayesian Computation (ABC) [1] [ or extensions
of kernel density estimation. The scale real-world applications demands more flexible and scalable approaches, which has lead to the integration of aptly suited deep learning methods in likelihood-free
Work completed while USC.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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settings. Neural network- based methods [3 29 17 have since been proposed, introducing greater flexibility when approximating probabilistic components (e.g.. the posterior, likelihood ratio, etc) in the inference pipeline. The use of the term "simulation-baseed inference" (SBI) has since been when referring to this emerging class of techniques.
colloquially embraced
SBI methods primarily leverage deep learning through their use of neural density estimators (NDE), neural network-based parametrizations of probability density functions. Common choices of NDE in practice include mixture density networks [8] and normalizing flows 40 34 along with popular
extensions (e.g,, Real NVP [2]. MAE [6 MAF 35 etc). Methods also vary in the probabilistic
form they elect to approximate; the posterior 3S the likelihood [36], and the likelihood ratio [99 13
are all common choices for estalsstabli methods.
While many SBI techniques leverage basic principles from active learning (AL), they are mostly established helpful heuristic for increased sample efficiency, rather than explicit optimization over a defined acquisition function. For example, methods like Sequential Neural Posterior Estimation (SNPE) 333 boost sample efficiency (over non-sequential methods) by iteratively updating a proposal prior p (). steering simulator parameters to values expected to be more useful for learning the posterior under observations T, of interest.
While sequential proposal updates are an effective first-order step to more informative simulation runs, there are many key factors that remain overlooked. For example, the updated proposal does not take into account the current parameters of the NDE itself, and parameter samples the batch drawn from the current proposal independently. This fails fully utilize myopic AL strategies and batch optimization, leading to large amounts of simulation runs with high expected information overlap and wasted computation. To address this issue, we formulate active learning scheme that (I) selects samples expected to target epistemic uncertainty in the underlying probabilistic model, and (2) makes acquisition evaluation simple and efficient when using any Bayesian NDE,
We demonstrate the effectiveness of our method on the origin-destination (OD) calibration task. OD calibration aims to identify OD matrices that yield simulated traffic metrics that accurately reflect fbeld-obeerved traffic conditions. can be seen as parameter tuning process, akin to model fitting in machine learning. From the machine learning perspective, OD calibration presents challenges due to the requirement of calibrating specific unique samples from observed traffic information, such as link flows, trip speeds, etc.
Our contributions are summarized follows:
Active Sequential Neural Posterior Estimation (ASNPE), an SNPE variant that incorporates active acquisition of informative simulation parameters @ to the underlying (direct) posterior estimation model, without the use of additional surrogate models. This helps to drive down uncertainty in parameters of the utilized NDE and improve sample efficiency, both of which are particularly important when interfacing with computationally costly simulation based models.
An efficient approximation the proposed acquisition function above, along with means of training Bayesian flow-based generative models for density estimation during posterior approximation (both with open source implementation Leveraging this class of models enables direct uncertainty quantification in the acquisition function, and is more flexible, efficient to train, and scalable to digh-dinnssional data than many traditional Bayesian model choices (e.g., Gaussian processes).
A Bayesian formulation of the OD calibration problem and coupled statistical framework for performing sequential likelihood-free inference with neural posterior estimation methods. We show ASNPE outperforms baseline methods across a wide variety of simulation scenarios on a large-scale traffic network. We also evaluate ASNPE on three broader SBI benchmark environments and find it acheives performance advantage over non-active counterparts.

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2 Background
2.1 Neural posterior estimation
Given observational data of interest 2% and prior p(0), we want to carry out statistical inference to approximate the posterior p(0|æ = zo) under the model p(r d). We assume p(r () is defined implicitly via a simulation-based model, where direct evaluation p(z(0) is not possible but samples ()) can be drawn. Conventional Bayesian inference is thus not accessible in this setting, and we instead look approximate the posterior using N generated pairs {(O,, x,))
Neural Posterior Estimation (NPE) methods attempt to approximate the posterior directly with a neural density estimator 4s (f]zz) trained on samples (00, =4)];== where 8 p(f) and 2; p(z]0,),
by minimizing the loss
(()) B |- log (0)x2)]
for learnable parameters 6. Provided qo is sufficiently expressive, qo (0|z) will converge to the true posterior p(0[z) as N + 0o.
Sequential Neural Posterior Estimation (SNPE) methods break up the NPE process across several iterations, and can improve sample efficiency by leveraging the fact that p(ß]æ xo) is often far more narrow than p()). While accurately representing p(0)z) for any E X' is ideal (where X is the simulation output space), doing so can require prohibitively large simulation samples, including outputs from parameters with low posterior density under T... To combat this, SNPE methods draw / expected be more informative about p(0) xr)) by using a successively updated proposal distribution p(() which approximates p(0].r = To)- Training the NDE qo on samples 0 -(0) when p not the true prior, however, will cause it converge instead to the proposal
posterior p(0 z)) = p(0[z) ((0)p(2)
F(a)p(o) rather than the true posterior (as shown in 33 Existing SNPE
methods correct for this in different ways: SNPE-A 3 trains go (f]z:) to approximate p(f) during each round and employs importance reweighting afterward, SNPE-B [29] directly minimizes an importance weighted loss, and SNPE-C [[7] (also known as Automatic Posterior Transformation,
APT) maximizes an estimated proposal posterior that easily transforms to the true posterior.
2.2 Bayesian active learning
Bayesian active learning is a selective data-labeling technique commonly employed in data-scarce learning environments. Active learning assesses the strength of candidate data points using a so-called acquisition function, often capturing some notion of expected utility to the underlying model given the currently available data. Given an acquisition function a, computing the next best point to label includes optimization of over a domain of as yet unlabeled points U: x" (r (() D)). where x are input data points and p(0) D) is the posterior of the Bayesian model parameters 0 given the current training dataset, i.e., the distribution over parameters after training the model. In modern Bayesian deep learning pipelines, this model is often a Bayesian neural network [15] 21]. Many acquisition functions used in practice extensions approximations of expected information gain (EIG) 20
argmax, () E By-poola,D) HO ((,,)))
where [H] is conditional entropy, and y are input labels.
Several existing works explore the use of Bayesian optimization (BO) the likelihood-free inference setting. [18 [2) employ Gaussian processes (GPs) as surrogate models for the discrepancy
function of f, and select parameter candidates by optimizing this surrogate with BO. GPs are also used as surrogate by [30] to represent the proposal distribution in MCMC ABC. [4] further extends these principles to deep Gaussian processes and leverage these models as surrogate likelihoods. In this work, we explicitly avoid the use of likelihood surrogates and aim to leverage only the approximate posterior NDE model, with the express intent of subverting additional computational overhead and enabling the use of powerful NDEs (c.g., flow- generative models).
3

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Acquisition
Simulator
3
Neural density estimator
(0: 1:8
Zj p(ar
$ (
(oox-hoed)
(op-1)
Intractable laneut trajectovies
1 p1pts,00p(=)00dd
truin
(1)
(2)
(())
(0:)
drow
p(())
p(0)2,)) 20(0(20))
(0(0)z))
pool
Round y (l....t))
Proposul
Pusterior
Figure 1: Depiction of the proposed active iaanineg-ntegraaed method. Demonstrates the high-
level ASNPE pipeline. Samples 9g are drawn from sequentially updated proposal distributions p(0), filtered according to the acquisition function a(f1.N. C.l...pp66ll p(o and run through the simulator p(æ ((z))) to generate B pairs (0,, x;) for training the approximate posterior qa- The learned posterior is then conditioned by the target observation Ta, producing the next round's proposal f(() = Qo (0)x)))
2.3
Origin-destination calibration
OD calibration an important task for transportation agencies and practitioners who develop traffic simulation models of road networks and use them to inform a variety of planning and operational decisions. Calibrating the input parameters of these simulators an important offline optimization problem that agencies must face on a regular basis (e.g., when new traffic data are made available, when changes the road network have occurred, etc). These simulators are often computationally expensive to evaluate, however, and highlight the practical importance of developing sample efficient calibration methods.
Existing works primarily approach OD calibration using general-purpose simulation-based opti- mization (SO) algorithms, such Simultaneous Perturbation Stochastic Approximation (SPSA) methods [5] 47. 22, [7). genetic algorithms [24] 43 47). and neural interaction models [48]. Many general-purpose SO algorithms tend require large numbers of simulation evaluations, which can be computationally costly. To address this issue, recent extensions of SPSA have been pro- posed [TO 27 45 Analytic metamodels have also been considered and shown reduce the need
for large numbers of function evaluations 31 32 49 3
3 Methodology
3.1 Active learning for SNPE
SNPE methods produce iteratively updated proposal priors ((r+1)(0) = p(f]z = x.) from (f] D(r)) at each round r in the inference process, where ( is the accumulated dataset {(0,, )r 1 by round r and the number of newly collected pairs per round. These sequentially updated proposals provide a means drawing parameter values g(r) ~ (0) that are increasingly useful (i.e., high likelihood) the posterior estimate of interest p(f)xx). As such, SNPE offers clear possible benefit for improved sample efficiency over non-sequential NPE, which cannot explicitly sample new values at expected high likelihood regions under x.. Despite this, it's difficult to quantify theh value of any particular 0 ~ ((++)(()) and whether it's worth the computational cost to obtain p(x(0) (0) with respect its utility to the underlying NDE qa (0|z, D(r)).
In high-cost simulation environments, we want to take every measure to sample only at highly informative regions of the parameter space to improve our estimate qs(0)zz of p(f]2 = x.). This entails a more principled analysis candidate simulation parameters 0 before investing in simulation run p0z In an attempt to quantify the prospective impact of any particular on our posterior estimate, we look to Bayesian active learning, and acquisition functions such as FIG. EIG considers the reduction in uncertainty of model parameters under the inclusion of new data in

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expectation over the predictive posterior. Adapting Eq. T to our NPE context gives
argmax,HiolDD -Ez-mato.D
H D U {(0, ((0,))]1}
which seeks to drive down uncertainty in NDE parameters o by optimizing for 0 with simulation out- puts z ~ p(=10, (0, D) expected to be most informative. Unfortunately, EIG and related approximations require p(=/0, D), which we cannot evaluate nor do directly approximate in the NPE setting, This makes it considerably more difficult to quantify the utility of candidate values, as we have no direct means of sampling likely simulation outputs.
3.2
Characterizing posterior uncertainty
Instead of relying on the marginal distribution p(= D), we seek instead to capture uncertainty across different parameterizations of the NDE. Broadly speaking, want simulate 0 expected to be informative to our NDE model, reducing epistemic uncertainty as measured by p((o) D) and further elucidating parameter sets likely explain the probabilistic mapping from simulation outputs r to parameter inputs 0. Given that do models this relationship as the conditional distribution Is (0)z)) we consider the uncertainty over distributional estimates induced by D)) This allows for the targeting of epistemic uncertainty in the NDE model according to how that uncertainty appears across feasible target posterior forms ( z²). More precisely, for some divergence measure D(-II.), we represent this distributional uncertainty as
(x)x) Esp (p(0 D)llp(0(a, 6)),
where p(0).z [z, 6) is the corrected posterior produced by qa (f|x: (see Section 2.1). and p(0)2, D) is the NDE's marginal posterior (under model parameters @) for simulation posterior estimates (parameters 0):
p(0 D)
p(0)z, D)d(o)
Intuitively, HDD captures notion of dissimilarity between the posterior estimates from different draws of é ~ p(@)D) D). Put another way. HDD indicates how certain the NDE is in assigning likelihood values across
θ under a chosen x;; relatively low value would indicate that likely parameter draws & ~ D) D) produce posterior estimates p(0) æz, Φ) that tend to agree with the "marginal" posterior p(0 zr D). for instance. Note that when we let the divergence measure be the KL divergence D DKL, we have HDu. = L[Ø;; 0 x.. D] (see proof II. Computing this exactly is difficult in practice, however, and we therefore seek a practically appropriate approximation below.
3.3 Acquisition of informative parameters
Eq. provides a basis for evaluating uncertainty in NDE without relying access to or approximations of the likelihood px=(0). Given the posterior estimation problem at hand, we'r we're particularly interested in how to select simulation parameters expected to reduce Hp at around T. Here we take inspiration from [23), who seek to drive down uncertainty at 0 with noisy estimates of the log joint probability (albeit in context where likelihoods p(x]()) are available). While H.D provides a measure of distributional uncertainty, we can target specific whose assigned likelihood is widely disagreed upon across draws of & p(o) D):
0 argmax B&I (p(0) D) p(0/x.: 0))2
While optimization of Eq. 3 over all 0 in the prior support may be ideal, this is computationally infeasible given the posterior estimates from all parameters & 2lD required in expectation. Additionally, note that Eq. 3 does not explicitly account for relative likelihoods of 0 under the posterior or available approximations, possibly leading to 0* with high uncertainty under but with low-likelihood under p(f]x..) expectation. We account for this explicitly during integration with specific SNPE approaches, seen in the section below.
5

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Algorithm Active Sequential Neural Posterior Estimation (ASNPE)
Input: Prior p(0), target observation To, round-wise selection size B, round-wise sample size N,
total rounds R
Output: Approximate posterior qp(0) x.)
Let [ {
Let p(0) = p(0)
for E , R] do
Draw N samples {0ihaN ~ p(0)
Sort I0: 1:N by expected divergence (Eq (4) select top-B {0:)1:
for b E (1.... : do
Simulate 24 p( (6) Set D(r-1) D(r-1) {(0g,
end for
Set D(r) D(r-1)
Train NDE 96 on D 6- argmin - E log (6,12))
Let p(0) (0(xz)
end for
3.4 Integration with APT
In order tractably approximate Eq. acquisition into the SNPE loop:
3 we impose additional restrictions to bring the parameter
1. Require the NDE be updated according to APT [7], i.e., trained via maximum likelihood
L(Ø) = E log d (0: where
(0)x) (0 ((0) 1
p(0) Z(x, 6)'
and by Proposition / of 33 ensures qa (f|z) - p(0]z) as N - oo without requiring post- hoc updates to the NDE's distributional estimate. This allows the model parameter posterior p( D) to be used directly in the contexts Eq. (2) and Eq. (3) whereas otherwise the corrective terms involved would need to be accounted for explicitly.
2. To account for the likelihood
under the posterior estimate as captured by the proposal prior ((0) = p(0 xo) in a given round of SNPE, adjust Eq. (
(0,0, (() p(0) ES-alD [I0(0]E,, D) p(0) 2))²
(4)
Further, in practice we approximate this by optimizing Eq. 4 over samples (() straightforwardly integrating the acquisition mechanism into the standard SNPE pipeline. Round-wind wise proposals (fr)) (0) ( (x) are set after N = rB samples are collected (for round r with B samples collected per round), and qo ()) > p(0[z))) as N + se. All round- wise proposal distributions share the support of the prior p()). which itself is established as having support over the entire parameter domain of interest O
Thus, optimizing a over sample of size N drawn from a proposal distribution (f) at any round r' recovers the true optimum of N + oo; each proposal's support connects back to the prior's support, which covers O. As result, at each round fixed sample size N drawn from the proposal can be used to approximate the acquisition maxima, while additionally adhering to the round-wise proposal sampling required to ensure ga (0) p(0]ar). Refer to Section for additional discussion on the functional form of the acquisition function.
3. (Optional, depending model) To approximate the Bayesian model parameter posterior p( D). neural setwork-based NDEs (such as flow- bowed generative models or mixture density networks) can be trained via MC-dropout [4 See additional details regarding consistent sampling and log probability evaluation MAFs under MC dropout in Appendix
Altogether, this constitutes the ASNPE method, which is more succinetly described in Algorithm See Figure for visual depiction of this process.

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3.5 Bayesian origin-destination calibration
We now position OD calibration as Bayesian inference problem, with a posterior density of interest to be approximated by SNPE methods. During a time interval of interest [t, te] on a traffic network G, we consider a single OD matrix d {d= {d=] where dı represents the expected travel demand for the origin-destination pair 2. 2 is the set of OD pair indices, i.e., 2 = (1, 2, ... 7.), for all pairs of interest on G. OD pairs are typically defined between elements in a fixed set of Traffic Assignment Zones (TAZs) whose size may not be uniform due to variable demand density; see Figure f zones drawn on two candidate networks. Figure 2 loosely depicts the acquisition pipeline for traffic data, corresponding the collection process shown in Figure
Conventionally, OD calibration is formulated as simulation- based optimization problem over a traf- fic simulator S-: U1, U2), where U1, U2 are vectors of endogenous simulation variables and exogenous simulation parameters, respectively. The goal is to obtain an OD matrix d" that yields simulation results x." = S(d"; ui, U2) that are sufficiently close to avail- able observational data To.-
(OD matrices)
Acquistiio
10
ett plo
While many pre-existing methods adopt a traditional optimization scheme and iteratively produce point es- timates for d", we formulate the calibration problem under the Bayesian paradigm and instead seek pos-
Simulation
terior p( (d d)
p(d[=; d) = p(xld)p(ddd) a)
p(xld)p(d:d) d)
X, pUr0N
p(a;d) [ didd
(Network flows)
(5)
Simple depiction of the data ac- quisition and simulation process for the OD calibration application. The acquisition step selects parameter candidates (OD matrices) to then be simulated (via SUMO) and pro- duce outputs (network flow observations) that are used to update the approximate pos- terior model.
Figure 2:
where p(d; d) represents the prior distribution over OD matrices, often defined around noisy historical esti- mate d. The posterior estimate under our observation p(d|z = z d) can then be used to compute different point estimates for d", used in other downstream tasks as an informative prior, and can represent intrinsic un- certainty in the calibration problem. This formulation achieves parity with existing approaches, where d is otherwise used as noisy starting point. Additionally, the traffic simulator S is treated as black-box that implicitly defines the likelihood p(z|d):
p(=ld)= / ps(s, - (2 u1, u2 u2ld)durdug
(6)
i..., marginalizing over all possible latent trajectories x. As is typical in simulation- based inference settings, this integral is intractable for simulators of sufficient complexity.
4 Experimental results
We explore the performance of the proposed ASNPE method in the context of OD calibration on a challenging real-world traffic network. Our goal here is to compare the general purpose utility of our approach in complex settings against tuned benchmark methods, and 2) verify ASNPE's candidacy as a sample efficient posterior estimation tool in high-dimensional, data-scarce settings. These objectives are directly in line with the needs of practitioners, both in the urban mobility community and broadly across scientific disciplines.
4.1 Experimental setup
We conducted case study on the large-scale regional Munich network seen in (37. The Munich network includes 5329 configurable origin-destination pairs (constituting simulation input), as
7

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Hours 5:00-6:00
Hours 8:00-9:00
Cong. level A
Cong. level B
Cong. level
Cong. level B 0,150
Prior OD
0.178
0,165
0.181
Setting prior
0.396
0.488
0.539
0.387
Prior
r 0.6
SPSA
0.563 + 0.089
0.521 + 0.052
0.453 + 0.078
0.384 + 0.049
q= = 0.3
PC-SPSA
0.193 + 0.063
0.185 + 0.097
0.159 + 0.036
0.159 + 0.046
MC-ABC
0.275 = 0.047
0.295 + 0.066
0.343 + 0.036
0.305 = 0.036
SNPE
0.201 0.085
0.167 + 0.092
0.187 = 0.059
0.314 = 0.025
(ours) ASNPE
0.147 + 0.011
0.157 + 0.097
0.165 + 0.064
0.161 + 0.079
Setting prior
0.340
0.311
0.245
0.277
SPSA
0.316 + 0.074
0.342 + 0.045
0.258 + 0.061
0.189 + 0.025
Prior II r 0.75
PC-SPSA
0.180 + 0.029
0.189 + 0.055
0.163 + 0.032
0.155 + 0.031
q = 0.45
MC-ABC
0.143 + 0.034
0.190 + 0.036
0.169 + 0.023
0.140 + 0.010
SNPE
0. 0.137 = 0.025
0.157 + 0.032
0.142 0.024
0.135 + 0.016
(ours) ASNPE
0.130 = 0.024
0.148 + 0.034
0.138 + 0.025
0.132 + 0.016
Table I: RMSNE scores on the Munich traffic network, described in Section 4.1
Note that methods like SPSA (poor convergence aside) can produce RMSNE scores larger than the reported setting prior due to noise in the starting sample. The "setting prior" value an average RMSNE score over many I draws from the shifted prior. Reported errors are empirical standard deviations computed over the five trial runs.
Prior (r- 0.6, q 0.3)
Prior (r- 0.75, 0.45)
0.7
0.6
0.5
MC-ABC
RMSN
SPSA
0.4
PC-SPSA
0.3
SNPE
ASNPE
0.1
60
100
120
80
100
120
Simulator samples
Simulator samples
(a)
Prior (r= 0.6, 9
Prior 0.75, 0.45)
0.4
RMMSN
0.3
SPSA
PC-SPSA
0.2
ASNPE
Wallslock
(b)
Figure 3: Plots of the (averaged) calibration horizons for each of the evaluated methods the Prior I, Hours 5:00-6:00, Congestion level A scenario. (a) RMSN(E) scores reached throughout the 128 sample simulation horizon for each evaluated method, averaged over five repeated trials (mean line plotted) and with error bars calculated as bootstrapped 95% confidence intervals. (b) The same scores shown in (a), but instead plotted against the wallclock time passed before the score reached (for each method's i's single best run). Note that the full 128-sample method trajectories are included, and the variability in line lengths demonstrates both 1) the impact of NPE-base methods' ability to run simulations in parallel, and 2) noisiness in simulation runtimes due to the variable inputs explored by each method. See Appendix for all scenario plots.

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Method
C2ST
MMD
MED-DIST
MEAN-ERR
Bernoulli
SNPE-C
0.749
0.017
0,210 + 0.024
11.454 + 0.255
0.188 + 0.117
GLM
ASNPE
0.725
0.146
11.993 0.172
0.150 + 0.085
SLCP
SNPE-C
0.987 + 0.001
0.172 0.0001
16.716 + 1.014
0.899 + 0.065
distractors
ASNPE
0.985 + 0.002
0.148 0.022
16.547 + 0.499
0.906 + 0.220
Gaussian
SNPE-C
0.773 + 0.009
0.167 0.006
1.051 0.037
0.532 + 0.074
mixture
ASNPE
0.771 + 0.006
0.150 + 0.025
1.010 + 0.066
0.440 = 0.164
Table 2: Results comparing SNPE-C and ASNPE for various metrics on the Bernoulli GLM, SLCP distractors, and Gaussian mixture tasks from (28) Experimentation details, metrics, and associated plots can be found in Appendix B
well as 507 detector locations (positions of reported output traffic flows), resulting in a highly underdetermined system.
We build and evaluate a number of synthetic demand scenarios, following an established framework for fair evaluation of urban demand calibration methods (2 9).. Each of the test demand scenarios are constructed from combinations of the following factors:
Time interval: time interval of interest is specified through which to simulate traffic flows on the network. Prior ODs are chosen to reflect real-world traffic patterns for the affiliated times. We evaluate peak morning demand for hour-long intervals at 5:00am-6:00am and 8:00am-9::0aam
Congestion level: within given time interval, can further control the level of traffic congestion exposed during the hour. The distribution of frequencies present in the starting ODs plays a critical role in determining route time across the traffic network. Here we define two congestion levels. "A" and "B," to reflect different average frequencies assigned to OD pairs. Here we use a truncated normal distribution (lower bound at 0) to sample OD counts with varying means and variance: (1) (μ = 5, 25) for hours 5:00-6:00, congestion level A, (2) (u 10, 50) for hours 5:00-6:00, congestion level B, (3) (μ 25, o = 50) for hours 8:00-9:00, congestion level A, (4) (p 50,a a 100) for hours :00-9:00, congestion level B.
Prior bias and noise: under each time interval and congestion level, we further perturb the generated OD matrices to represent realistic variance found in real-world sampling of traffic observations. Here we use the following noise model, mirroring that of [3] I = (r + q x 6) x d, where ó ~ (0, }). We then formulate two perturbed settings: 1) Prior r 0.6, q = 0.3, and 2) Prior I1: r 0.75. 0.45. Prior constitutes heavily under-congested estimate with relatively little added noise, while Prior II is less biased from the true OD but noisier. Both priors represent underestimations of the true demand, reflecting the fact that most prior ODs from real-world settings are constructed from historic travel demand observations.
The eight synthetic combinations constitute starting ODs/priors that span a variety of different settings important for real-world urban demand calibration tasks. Each synthetic setting yields particular prior OD estimate d, which is then used to construct a prior p(d: d). fixed sample is drawn from p(d; d) and passed through the open-source traffic simulator Simulation of Urban MObility (SUMO) 26 to generate associated "true" network flow Fo-
4.2
Comparison to SOTA Calibration Methods
We evaluate the effectiveness of the proposed solution by comparing against available SOTA bench- marks commonly employed in the OD calibration space: Simultancous Perturbation Stochastic Approximation (SPSA) [42] and principal component (PC)-based SPSA. or PC-SPSA 37]. SPSA is a widely employed algorithm for travel demand calibration, and PC-SPSA is an effective exten- sion that optimizes over parameters in lower-dimensional subspace, as defined by the principal components of computed travel demand history matrix. Both of these methods are conventional aetimization-based methods, and do not leverage neural networks. Additionally, these methods in their canonical form cannot be parallelized, requiring serial simulation evaluations across each iteration.
For NPE-based approaches, we evaluate our proposed method ASNPE alongside SNPE-C (or APT [7 and Approximate Bayesian Computation (ABC)
41 6). ABC serves primarily as a less

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As alluded above, ASNPE updates its internal model only after batch of simulation samples While generating sophisticated baseline that reflects early approaches to likelihood-free inference, and it typically faces difficulty scaling and is far less flexible compared its NPE counterparts.
For all of the eight scenario priors p(d; d), each calibration method is ran for a maximum of 128 SUMO simulations, seeking to recover To-- The root mean squared normalized error (RMSNE) is recorded for each method's simulation horizon, as used in 39 (see also Appendix A..) To account for the stochasticity across evaluations, we report RMSNE averaged of five repeated simulation runs. See Table for reported values for each method across each of the eight synthetic scenarios, as well as paired prior plots in Figure 3
4.3 Analyzing calibration performance
ASNPE outperforms all other methods across most explored settings: as can be seen in Table] [ our method outperforms both the well-tuned PC-SPSA method commonly employed by the urban mobility community, as well as popular simulation-base almost all of the explored settings. In general, PC-SPSA tends quickly converge but demonstrates a limited ability to further improve beyond the first 10-20 encountered simulations. Both SBI methods tend to make steady improvements throughout the entire trial, however, albeit often doing so more slowly in the first 20-40 simulations than PC-SPSA, This is primarily due to the limited feedback SNPE/ASNPE receive comparatively, only incorporating new simulation data batches (in this case, every 32 simulation draws).
inference (SBI) methods like SNPE, across
Additionally, ASNPE reliably reaches better RMSNE scores than SNPE with fewer simulations, as well as Approximate Bayesian Computation (MC-ABC). This can be seen as early as the first NDE update, before which the two methods encounter the same (seeded) simulation samples. This also empirically supports our central methodological contribution, i.c.. optimization over informative simulation parameters can more efficiently improve the accuracy of the inferred posterior estimate.
ASNPE is outperformed some cases: ASNPE is outperformed by PC-SPSA in two of our explored settings (under Prior Hours 8:00-9:00). While we wouldn't expect single method to be the best choice for all variations in such high-dimensional setting, this particular scenario serves as an opportunity better understand possible failure modes of the proposed method.
generated, whereas PC-SPSA adjusts its parameters after each simulation samples in batches can be beneficial (and is often necessary) for early stability of ASNPE, it can mean informative simulation data is incorporated later in the trial. This explains the occasional gap that opens up between SBI methods and PC-SPSA A in the first 30 simulations, only after which is ASNPE/SNPE able incorporate the samples to improve its posterior estimate. Note, however, that most of the early advantage PC-SPSA may have over ASNPE is dwarfed by the ability to obtain its simulation draws in parallel, whereas PC-SPSA must run simulations serially. This allows for larger, more stable improvements in less time, which can be seen in subplot (b) of Figure]
4.4 Performance on common SBI benchmarks
We additionally report results on several common SBI benchmark environments, and compare against the performance of (non-active) SNPE. Numerical results can be found in Table _ along with plots and more details in Appendix [ These additional results demonstrate the wider applicability of our method beyond the travel demand calibration task.
Conclusion
In this paper, we introduced Active Sequential Neural Posterior Estimation (ASNPE), an SNPE variant that actively incorporates informative simulation parameters 0 to drive down epistemic uncertainty in the neural density estimator and improve sample efficiency for high-quality estimates. We evaluate this method on complex. nigh-dimensional problem in urban demand calibration, and show it reliably outperforms available benchmark methods across variety of scenarios with variable bias and noise. We additionally provide results on several common SBI benchmark environments, and find ASNPE capable of outperforming state-of-the-an SNPE methods on key posterior approximation metrics.
10

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Acknowledgements
This work was supported in part by the National Science Foundation under awards #2226087 and # #1837131.
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A Appendix
A.1 Connection between mutual information and DKL
Expanding the definition of mutual information for L[ø;; 0 02 D}, we have the following (note that Zo and D are fixed);
[ 0 F..... D) = / p(t, 0oz0. log
p(0, 6 D)
d0do
e
p(0 co.
()p(olzor D)
1 p(olo, 2o. D)p Do
p(0 p)p(p)ppppp) ())( D)
p(0|z D)p(0 Io. D) d0ds
pio|zo. D) // p( の) log
p(0 Φ)
dB de
p(0 D)
[ p(olzo. D) [DEL (p(0 no. @) ro. D))] dø
D) [DRr. (p(0) F.. (p0 D))]
This aligns with the expected divergence term introduced in Eq 2 when we let the measure D = DKL. The connection mutual information helps to position our motivation for the acquisition function ultimately introduced in Eq. That is, by seeking to drive down disagreement between "marginal" and "component" posteriors p ( Jx, D) and p(0) rr o), respectively, we are attempting to realize the information we expect 0 to tell us about our NDE parameters ø, thereby minimizing the remaining mutual information between the two.
A.2 Regarding the functional form of the acquisition function
The family of functions a00, D)) = (E.) under parameter À constitutes valid choices for the acquisition function for any facilitating different levels of emphasis on uncertainties at values of 0 relative to their likelihoods under P. While several values of X may be justifiable, the choice to use À 1, implicit in Eq. 4 intuitively captures a desirable balance in the relationship between uncertainties and likelihoods of 6.
In particular, under level sets a(-, P))
(where p(ø) D) is held constant), likelihoods ((0) decrease by factor of n, the average deviation between p(0 a. D) and p(0[x, +) need only increase by factor of /n, i.e., changes in uncertainty
sub-linear in the likelihood ratio. With the introduction of variable X, this factor generalizes to n (22A), and may require additional measures to balance the resulting sensitivity between the terms. We find that = is a natural choice that reasonably captures the desire explore potentially unlikely parameters with high uncertainties without ignoring them (e.g., X + 0) or relying too heavily on them (e.g.. X + oo).
A.3 Additional Definitions
The root mean squared normalized error (RMSNE) for simulated output è with respect to an observational reference T., as used [39], is defined as
Σ 1L (+(1)
xo
(
RMSNE
Σ
To
where there n the number of observed segment flows, and both No and are n-dimensional vectors.
B Additional SBI benchmarks
In order to appeal to the general utility of our proposed method, we provide additional experimental results between ASNPE and SNPF-C [17] on three common SBI benchmark tasks: SI.PP distructors, Bernoulli GLM, and Gaussian Mixture. Each of these settings corresponds to reproducible task
14

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Metric: C2ST
Metric: MMD
0.755
SNIPEC mean SNIE 95% a
SNIE-C meaa
SAPE C 95%
ASNIE memn
ASNIE mean
ASNIE 95% C
0.30
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WND
0.775
0.20
3.770
0.765
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15
2.0
3.5
Metric: Posterior Median Distance
Metric: Posterior Mean Error
2.2
95%
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14
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Proteiiee Ermm
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Round br)
Figure 4: Results on various metries between ASNPE and SNPE-C across four rounds of sequential inference for the Gaussian mixture task.
environment from (28), and the corresponding implementation the sbibm Python package used to run our experiments.
We additionally evaluated method these tasks using metrics beyond RMSNE, the primary metric used for the travel demand calibration case study. These include classifier 2-sample tests (C2ST), maximum mean discrepancy (MMD), the posterior median distance (median L norm between simulated samples 0; ~ p(f)z,) + 24 ~ p(z and the observation v,), and the posterior mean error (normalized absolute mean). Full details for each of these tasks and metrics can be found in (88
between the true posterior
and the approximate posterior
We evaluate both methods for a total of 1024 simulation samples. Note that this is eight times larger than the sample sizes collected for the trials the travel demand task. For reference, 256 samples in (non-parallelized) SUMO environment takes hours, whereas 256 samples from the SLCP simulator takes ~10 seconds hardware.
medium-size sample horizons: rounds with 256 samples per round,
Trials were repeated five times for each method, and the average score and standard deviation for each metric over these trials are shown in Figures 4 5 6 Note that smaller values are better for each metric (C2ST ranges between 0.5 1.0). While these simulation horizons are relatively small, we find that, by the final round, ASNPE tends to outperform SNPE across most settings and most metrics. In particular, ASNPE outperforms SNPE on C2ST and MMD across all settings, along with the distance-based metrics on all but the median distance for Bernoulli GLM and mean error for SLCP Distractors.
While SNPE-C is state-of-the-an benchmark method, comparing against it also constitutes an ablation test for ASNPE's acquisition component. Although parameter sets are chosen differently and the underlying NDE varies (minimally to accommodate the need to approximate p(o across methods, the sequential inference procedures are otherwise identical. These results help to isolate and identify the contribution of the active learning scheme across wider range of tasks and metrics for the overarching goal of producing accurate posterior approximations holistically (i.e., not just well-calibrated point estimates).
15

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SNPE-C
0.40
SAPE-E
3.00
0.35
0.30
0.15
0.75
0.10
0.05
1.0
13
2.0
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SNPE mear SNPE-C
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0.15
12
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95%
11
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Rourd 1o
Figure 5: Results on various metrics between ASNPE and SNPE-C across four rounds of sequential inference for the Bernoulli GL.M task.
Metric: C25T
Metric: MMD
SNIE-C
0.990
0.988
0.18
WN0
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SNPE-Cmean SNPE 095%
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ASNPE 95%
0.10
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13
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Bound in)
Round br)
Metric:
SNPE-Cmean mean
SNPE-C 95%
1.1-
Medan Dizancce
17.0
Mewn
0.9
16.5
26.0
0.7
ASNPE mean ASNPE 95% CI
0.5
1.0
1's
2.0
3/0
3/s
1.5
2'0
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3.5
40
Round in
Round in
Figure 6: Results on various metrics between ASNPE and SNPE-C across four rounds of sequential inference for the SLCP Distractors task.
16

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C Code reproducibility
In the spirit of reproducibility and in the hopes our code may be of use for follow-up works, we provide the following Python packages:
1. seqinf package: this package includes full implementation of ASNPE and convenient abstractions around the popular sbi [44] and sbibm Python packages for general se- quential inference pipelines. This package also includes implementation of the masked autoregressive flow (MAF) with consistent MC-dropout that was used as an NDE for all experiments. Available at uthtpps: : http:/gggbtb/////// com/ Saaggriseeer//eeee /seinff
2. sumo_cal package: short for SUMO calibration, this package includes many general programmatic utilities for calibrating traffic models using results from the SUMO traffic simulator [26] with high-level Pythonic interface, including collecting results from several runs, running in medti- contexts, employing standardized configuration, etc. See Figure [1] for snapshot of the SUMO interaction scheme. Available at htttp:///gthh: com/ htt////aairreeeeeeeee sumo sumo__call _cal
D Additional demand calibration experimentation details
D.1 Descriptions of SOTA Calibration Methods
SPSA: SPSA (Simultaneous Perturbation Stochastic approximation) is optimization algorithm for systems with multiple unknown parameters, which can be used for large-scale models and various applications. It can find global minima, like simulated annealing [46). SPSA works by approximating the gradient using only two measurements of the objective function with gradients, making it scalable for high-dimensional problems 42
PC-SPSA: PC-SPSA proposed to address fundamental scalability issues with SPSA. This is because SPSA searches for the optimal solution in a high-dimensional space without considering the structural relationships among the variables. PC-SPSA combines SPSA with principal components analysis (PCA) to reduce the problem dimensionality and limit the search noise. PCA captures the structural patterns from historical estimates and projects them onto a lower-dimensionn space, where SPSA can perform more efficiently and effectively 39
Specifically, we implement the SPSA and PC-SPSA algorithms according to 38 and associated open-source implementations"] In addition, we employ the so-called Method-6, titled "Spatial, Temporal, and Day-to-Day Correlation," as detailed (38). This provides means of systematically generating needed historical data for PCA, and according to the original work constitutes the most robust and optimal solution among the proposed variants.
D.1.1 Hyperparameter details & computing resources
Here we include a brief discussion on the implications of the hyperparameters found in ASNPE, as well as the settings used in our experiments. Keeping the total number of simulation samples constant,
1. The number of rounds R dictates how many times we update the proposal distribution over the course of the simulation horizon. Increasing this value can enable quicker feedback to the NDE, requiring fewer simulation samples before re-training the model. When the prior is well-calibrated and simulation samples are representative of the observational data, this can have a positive compounding effect that boosts the rate convergence to the desired posterior. However, for larger R the resulting batch sizes are smaller and the NDE receives noisier updates, which can have the opposite effect and hurt early performance when the prior is poor.
2. The number of selected samples B per round is directly determined by R when the total number of simulations is held constant, and thus the above effects apply here.
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(b)
Figure 7: Depiction of traffic networks in the SUMO simulator. (a) depicts relatively small synthetic network for reference, with approximately 500 configurable OD pairs. (b) shows the larger Munich network used in reported experimental settings, with order of magnitude more configurable OD pairs than the synthetic network.
3. The number of proposal samples N per round governs the size of the parameter candidate pool over which the acquisition function is evaluated. Increasing this value allows to consider more potentially relevant candidates under p(f). and can thus increase the quality of the resulting B-sized batch. Given the acquísition function pool very efficiently (i.e., as a batched inference step through the NDE model), one can practically scale this up arbitrarily to increase the sample coverage the proposal support (but with decreasing marginal utility).
be evaluated over this
The following are additional hyperparameter details for the evaluated methods:
the SNPE loop: total number of rounds reported experiments), round wise sample size N (between 256-512), round-wise selection size B (32 in reported experiments).
2. Neural Density Estimator (NDE) model: our model architecture (used for both SNPE and ASNPE) is masked autoregressive flow with 5 transform layers, each with masked feedforward blocks containing 50 hidden units, and trained with a (consistent) MC-dropout setting of 0.25, When collecting distributional estimates as described in Eq 4 we used 100
weight samples @ ~ D) (as generally recommended in (21).
3. PC-SPSA: this method uses PCA to optimize OD estimates in Nower-dimensione subspace the 229-dimensional parameter space. The number the principal components is chosen such that 95% of the variance is recovered in the provided historical OD estimate (which is further dictated by the choice of prior distribution). The number of PCs used by this method across the many explored settings presented in Section 4.1 varies from 99-117.
All experimentation code is written in Python 3. To run experiments, employed our own hardward locally, which an linux-based machine running an Intel(R) Core(TM) 19-10900X CPU @ 3.70GHz 64GB memory, and NVIDIA GeForce RTX 2080 Ti.
E
Additional demand calibration plots
Figures and are plots of calibration horizons for the remaining settings the demand calibration task not shown in the main paper (which highlighted the first scenario, Prior I, Hours
5:00-6:00, Congestion level A). The associated RMSNE scores can all be found in Table T in the
main paper body.
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Prior 1 (==
0.6, 9.3)
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6000
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6000
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Wallclock time
(b)
Figure 8: Plots of the (averaged) calibration horizons for each of the evaluated methods on the Prior II, Hours 5:00-6:00, Congestion level scenario.
Prior 1 (/= 0.6, q 0.3)
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Figure 9; Plots of the (averaged) calibration horizons for each of the evaluated methods on the Prior I. Hours 8:00-9:00, Congestion level scenario.
19

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Prior 1 (== 0= 0.33)
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Figure 10: Plots of the (averaged) calibration horizons for each of the evaluated methods on the Prior II, Hours 8:00-9:00, Congestion level scenario.
Additional distribution plots and schematics for demand calibration
F.
Simulation schematic
Figure I provides a more detailed look at the programmatic interaction with the SUMO simulator.
F22 Additional posterior plots
Figure 12 and figure 1.3 provide additional plots of the posterior approximation for the primary travel calibration task.
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S0NT IEECEE SOUTTE
DRUECTIV
Figure 11: This diagram provides a more detailed look at some of the internal details behind the preparation of input to and the transformation of output from the simulator. An input OD vector 0, drawn from some proposal distribution in the outer method context, 1) "enters" the diagram at the left, 2) transformed into suitable representation for SUMO, 3) combined with additional configuration and network files, and 4) run through the SUMO simulator, after which the output is parsed to produce the resulting segment flows z under demand 6.
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Figure 12: Pairwise density plots of 20-dimensional slice of the final approximate posterior
(()) = ()(0(x.) produced by ASNPE on the Prior I, Hours 5:00-6:00, Congestion level A scenario.
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0.05
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Figure 13: Pairwise density plots of 20-dimensional slice of the "empirical likelihood" under the final ASNPE posterior on the Prior I, Hours 5:00-6:00, Congestion level A scenario. Figure 12 shows the approximate posterior qo (0)x))) whereas here we draw samples 0, ~ (0)xz) and feed them back through the simulator (0, 1:N - p(=:(f) to visualize the resulting data space. The target observational data point x, is shown on top these pair plots as a red "plus", which provides a visual anchor for how well calibrated the posterior is around the observational data point of interest.
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Reviews:
[{"review_id": "utmPKy3sYw", "reviewer": "Reviewer_garc", "summary": "The paper introduces ASNPE, a modification of sequential neural posterior estimation (SNPE) that uses active learning to determine the set of most informative simulation parameters. The method is benchmarked in a synthetic scenarios based on a real-world traffic network and outperforms domain-specific optimization schemes as well as simulation-based inference methods.", "questions": "- In the training setup of SNPE and ASNPE, the number of simulations is not clear to me. The authors state that they perform R=3 round with selection size B=32, which should result in 96 simulations seen by the neural nets. How does this match the 128 sample simulation horizon?\n- I (and maybe also future readers) would be interested in the authors opinion on the reasons for the clear gap between SNPE and ASNPE in the prior 1 but not prior 2 setting.", "limitations": "The explicit analysis of failure modes (which of course, are to be expected in any method comparison since no method always performs the best) is a big plus. As stated above, this does not incorporate a comparison between SNPE and ASNPE, which would be interesting for the wider SBI community.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The clear and concise communication throughout the paper makes it an enjoyable read. \n- It features a simple and elegant approach to target a gap in current SNPE methods. \n- Considering the empirical evaluations, a high-dimensional setting with real-world importance is chosen. ASNPE demonstrates a substantial improvement over both domain-specific optimization schemes and, at least for prior 2, SNPE. Again, the results are analyzed in a clear and informative way. \n- Lastly, I appreciate the detailed description of the planned open source Python packages in Appendix D.", "weaknesses": "Major:\n- While the reported experimental setup is exciting, the proposed method is tested only in a single setting. Since ASNPE seems to be a promising general improvement of SNPE-C, it would have substantially benefitted the work to demonstrate its usefulness in a wider range of experimental settings (e.g., by adapting tasks from [1] as a compact first experiment). I acknowledge that this is likely not feasible in the limited rebuttal period, but I believe it would make the paper much more informative for the broader SBI audience.\n\nMinor:\n- The paper directly jumps from theory to experimental setup and the results without mentioning the implementation of ASNPE, but this would be quite informative for the reader.\n- Section 4.2 does not feature all benchmark comparisons - MC-ABC and SNPE suddenly appear in the results and I expected to find more information concerning MC-ABC at least in Appendix C. As a related minor comment, I suppose the benchmark uses APT / SNPE-C for SNPE, but it would be helpful to state this explicitly.\n- Lastly, it would have been informative to see ablation studies (potentially in a computationally less demanding setting) or at least a theoretical discussion regarding the impact of the hyperparameters R, N, and B on the performance of ASNPE.\n\n[1] Lueckmann, J. M., Boelts, J., Greenberg, D., Goncalves, P., & Macke, J. (2021). Benchmarking simulation-based inference. AISTATS.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work. Please find our responses to your questions and comments below.\n\nNote: Where applicable, we prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively, numbered by the order in which they were mentioned.\n\n**[W-1]** Please see our global rebuttal reply for details on additional empirical evaluation.\n\n**[W-2]** We agree this transition could be smoother. Section 3.4, however, sets up several practical considerations for how ASNPE can be implemented (e.g., efficiently approximating Eq. 4, training neural network-based NDEs with MC-Dropout, etc), and specific model details and hyperparameter settings are relegated to Appendix C.1.1 (primarily due to space constraints). In our final manuscript, we will incorporate a discussion on concrete implementation choices prior to the experimental setup, and direct readers to the Appendix for additional details.\n\n**[W-3]** The intended scope of Section 4.2 was to introduce methods specifically used in OD calibration literature (i.e., just SPSA and PC-SPSA), while we relied on prior context for the likelihood-free inference methods (i.e., SNPE and MC-ABC). However, as you point out, MC-ABC is not properly introduced, and the exact version of SNPE remains unclear (although it is indeed APT). In our final manuscript, we’ll broaden the scope of Section 4.2 to include all reported methods and properly introduce them.\n\n**[W-4]** This is a good point, and something we will incorporate alongside the smaller scale experiments run on benchmarks mentioned in [W-1]. As a quick theoretical discussion, however, the following reflects our observations on the large-scale setting we explore in the paper. Keeping the total number of simulation samples constant, \n\n- The number of rounds $R$ dictates how many times we update the proposal distribution over the course of the simulation horizon. Increasing this value can enable quicker feedback to the NDE, requiring fewer simulation samples before re-training the model. When the prior is well-calibrated and simulation samples are representative of the observational data, this can have a positive compounding effect that boosts the rate of convergence to the desired posterior. However, for larger R the resulting batch sizes are smaller and the NDE receives noisier updates, which can have the opposite effect and hurt early performance when the prior is poor.\n- The number of selected samples B per round is directly determined by R when the total number of simulations is held constant, and thus the above effects apply here.\n- The number of proposal samples N per round governs the size of the parameter candidate pool over which the acquisition function is evaluated. Increasing this value allows us to consider more potentially relevant candidates under $\\tilde{p}(\\theta)$, and can thus increase the quality of the resulting B-sized batch. Given the acquisition function can be evaluated over this pool very efficiently (i.e., as a batched inference step through the NDE model), one can practically scale this up arbitrarily to increase the sample coverage over the proposal support (but with decreasing marginal utility).\n\n**[Q-1]** Thank you for pointing this out. It should instead say R=4 (selection size B=32 is correct),  and will be corrected in our final version.\n\n**[Q-2, L-1]** Generally, prior I captures relatively high bias/low noise around the true OD parameter, while prior II is comparatively low bias/high noise. Our opinion as to why gaps between SNPE and ASNPE relies on a few observations:\n\n1. Empirically, all methods under prior II benefit from its lower bias (seen very clearly in Figure 2) without suffering too greatly from the higher noise, indicating a particular sensitivity to the prior bias for this congestion setting and time of day. \n2. ASNPE can identify useful simulation parameters more quickly than SNPE due its explicit acquisition mechanism, aligned with uncertainty in the NDE model. \n\nGiven the sensitivity around the prior in this setting, ASNPE can have an edge over SNPE by making early strides to correct for the bias through exploration, and this can have a significant compounding effect for multi-round inference on such short simulation horizons. Put another way, ASNPE attempts to counter uncertainty in the NDE model by employing a more principled exploration of the parameter space than SNPE, and the effects of doing so are magnified under the higher bias under prior I (and less prominent under the less biased prior II).", "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": "", "extra_scores": {}}, {"review_id": "YVvVkukDCA", "reviewer": "Reviewer_GNuF", "summary": "A recent class of methods that have shown to perform well at simulation-based inference are based on modeling the posterior density as a neural network, using mixture density networks, normalizing flows, or other popular architectures. With enough data, these methods tend to produce salient estimates of the posterior density on parameters conditional on data (or summary statistics). With any simulation-based inference method, proposing parameters efficiently is a relevant concern because the posterior can have most of its mass in a tiny region of the support of the prior. Standard SNPE constructs sequential proposal distributions from the current posterior approximation and uses importance reweighting, which is a simple heuristic but not optimal. The main contribution in this paper is an active learning method which filters potential parameter values by ones that are sufficiently large in terms of an acquisition function, such as expected information gain. The authors come up with a reasonable acquisition function that can be approximated via Monte Carlo. The ASNPE algorithm is then applied to an origin-destination calibration problem, and performs well compared to competing approaches under a budget of simulator evaluations.", "questions": "1.) I am curious how much the computational cost increases when this methodology is used compared to the simple sequential updating in SNPE. Of course, when the simulator is extremely expensive, the cost of evaluating the acquisition function will be small in comparison, but would be it still be recommended if the simulator is not too onerous.\n\n2.) On a similar note, I would be interested for the authors to carve out and add a little exposition describing exactly the class of SBI problems that they posit would have the largest boon from this active learning approach when compared to the standard sequential updating in SNPE. What types of problems are most rewarded by the acquisition function filtering step, and on which class of problems is it less rewarding?\n\n3.) I may have missed it, but is it obvious that q_\\phi still converges to the true posterior in the presence of the filtering step? In the presence of the filtering step, the proposal is no longer just \\tilde{p}(\\theta), but instead it is a transformed distribution essentially weighting \\tilde{p}(\\theta) by the acquisition function. Shouldn't this need to be considered when updating the NDE in order to ensure q_\\phi really converges to the posterior?", "limitations": "The authors provided an honest assessment of the performance of their method in comparison with other methods in the experiments section. I believe that potential negative societal impact is not a concern here. As I mentioned earlier, I would be interested to know if the authors believe there are limitations in the context of which SBI problems their method is not as well-suited for.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is clear about the contributions, and contextualizes them well relative to the previous work. Active learning is a desired traits in simulation-based inference methods, where simulators can be costly. The active learning component that is introduced in this work seems like a clear improvement over the standard heuristic employed by SNPE.", "weaknesses": "I would be interested to see more experiments/applications than just the Bayesian O-D calibration example. Particularly, I'd like to see how this method compares to competing approaches on a standard \"benchmark\" in likelihood-free/simulation-based inference, i.e. Lotka-Volterra, a queue model, Heston model, a Gaussian toy model as in SNL, etc. This particular simulator model seems to have an extremely high-dimensional parameter space relative to the budget of simulations used for the experiments, and I am curious if it is this particular setting in which this method shines compared to others. I.e. the benefit is clear when the simulation budget is 128, but if it is on the order of 10^4 or 10^5, does it vanish?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work. Please find our responses to your questions and comments below.\n\nNote: Where applicable, we prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively, numbered by the order in which they were mentioned.\n\n**[W-1]** Please see our global rebuttal reply for details on additional empirical evaluation.\n\n***Regarding vanishing benefit***: While we haven’t been able to experiment with large sample sizes (e.g., of the order 10^4-10^5) in our high-dimensional traffic setting due to computational constraints in our academic environment, we expect ASNPE’s advantage over SNPE to diminish as we collect increasingly large amounts of simulation samples. This is primarily due to the fact ASNPE and SNPE converge to the true posterior in the limit under samples from the simulation model, and both take similar steps to improve the accuracy of the posterior approximation under the observational data of interest. Precisely at what simulation budget we might expect the difference between methods to become negligible, however, is difficult to anticipate, and would require a more principled empirical study.\n\nTo this end, we will explore this question alongside the experimental results on the common SBI benchmarks mentioned above in our final manuscript. Given the smaller scale simulation models, we can better characterize the relationship between the dimensionality of the parameter space and the behavior of ASNPE and SNPE over longer simulation horizons.\n\n**[Q-1, Q-2, L-1]** You raise a valid concern: the cost required for evaluating the acquisition function may not always be worthwhile when the simulation model is very cheap to evaluate. In these cases, more data may be more valuable than a principled exploration of the parameter space, and the time ASNPE spends on active learning may be better spent collecting additional simulation samples.\n\nPrecisely characterizing the kinds of problems where we expect the cost for acquisition to be worthwhile is difficult, but in general we posit our method will provide the most benefit in complex settings (e.g., stochastic, non-trivial, large parameter/observation spaces) where simulation samples are limited (due either to slow simulation models, limited computational resources, or both). Our primary traffic case study is a problem very much characterized by these qualities, which is at least in part why we elected to focus most of our efforts on this single problem. In these cases, ASNPE may exhibit large benefits over traditional methods given its more principled exploration of the parameter space, and with sequential inference, early improvements can compound heavily over time (i.e., informative samples produce better proposal distributions, from which the next round’s samples are drawn, and so on).\n\nRegarding ASNPE’s relative computational overhead, note that acquisition evaluation can be performed very efficiently as a batched inference step under the NDE model, and typically yields a negligible difference in terms of raw runtimes between the two methods. For our primary traffic task, the reported wallclock times (found in subfigures (b) for each of the calibration plots) provide the raw runtimes for each method when obtaining the 128 simulation samples for each setting. Empirically, this allows us to compare the total time spent by our algorithm (including acquisition evaluation), and we observe a negligible difference compared to SNPE over our explored horizons. \n\n**[Q-3]** The convergence of $q_\\phi$ to the true posterior is discussed in Section 3.4. In particular, we leverage a result from APT [1] (also known as SNPE-C), which states that training our NDE model parameters via maximum likelihood $\\min_\\phi \\tilde{\\mathcal{L}}$, where\n\n$$\\tilde{\\mathcal{L}}(\\phi) = -\\sum_{i=1}^N \\log \\tilde{q}_{x,\\phi}(\\theta_i)$$\n\nand\n\n$$ \\tilde{q}_{x,\\phi}(\\theta) = q_{\\phi}(\\theta)\\frac{\\tilde{p}(\\theta)}{p(\\theta)}\\frac{1}{Z(x,\\phi)} $$ (line 178)\n\nimplies $q_\\phi(\\theta|x) \\rightarrow p(\\theta|x)$ as $N \\rightarrow \\infty$ (along with the proposal posterior $\\tilde{q}_{x,\\phi}(\\theta)\\rightarrow \\tilde{p}(\\theta|x)$). This setup allows us to incorporate the true proposal prior $\\tilde{p}(\\theta)$ in the training loss without additional explicit corrective terms. Further, this limiting behavior holds even when optimizing $\\tilde{\\mathcal{L}}$ over samples drawn from the transformed distribution under the acquisition filter, as it shares its support with the proposal prior (and the same can be said for any distribution with this quality, in the limit as $N \\rightarrow \\infty$).\n\n[1] D. S. Greenberg, M. Nonnenmacher, and J. H. Macke. Automatic posterior transformation for likelihood-free inference, 2019.", "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": "", "extra_scores": {}}, {"review_id": "0KdqwEVdQe", "reviewer": "Reviewer_JwDn", "summary": "This paper proposes an approach to performing neural simulation-based inference – specifically, sequential neural posterior estimation – in a simulation-efficient manner for complex and expensive simulation models. The idea is to use active sampling to sequentially generate datapoints from the simulator and to train the neural density estimator using batches of datapoints generated this way. The authors test their approach experimentally on an origin-destination estimation problem in an urban mobility simulator, comparing against other common simulation-based Bayesian inference methods (\"vanilla\" SNPE and ABC) and against two alternatives that are commonly used in the urban mobility modelling literature ((PC-)SPSA).", "questions": "See above ^ and thanks in advance for your responses!", "limitations": "I do not see that this work especially threatens any negative societal impact. The main limitations of the work that I see are already addressed in the Weaknesses section above.", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "_Originality_\n\nAs far as I am aware, active sampling via an explicit acquisition function as described in this paper has not been explored yet in the neural SBI literature, although some less formal active sampling already occurs in sequential/round-based training of neural SBI methods.\n\n_Quality_\n\nThe method looks overall reasonably sound and some comparison against multiple alternatives is presented for the case of a complex urban mobility model.\n\n_Clarity_\n\nThe paper was mostly clear in my reading, although there were one or two points that were a little unclear to me that I will detail below.\n\n_Significance_\n\nOverall I think this paper has (and approaches to making simulation-based inference procedures as simulation-efficient as possible, more generally, have) the potential to be significant and will assist practitioners in real-world settings to make use of simulation models effectively.", "weaknesses": "The two main weaknesses in my reading are:\n\n- Sorry if I missed something, but clarity was lacking in one key respect for me, namely what exactly the relationship was between $q_{\\phi}(\\theta \\mid x)$, $p(\\theta \\mid x, \\phi)$, $\\tilde{q}_{x,\\phi}(\\theta)$ on Line 178, and `q_{\\phi,x}(\\theta)` (I had to do this last one `in this format` rather than in $math\\ mode$, sorry, because it wasn't rendering properly in $math\\ mode$ and I couldn't figure out why) on Line 178. Is `q_{\\phi}(\\theta \\mid x)` (now $math\\ mode$ isn't working here either...) the same as `q_{\\phi,x}(\\theta)`? And `\\tilde{q}_{x,\\phi}(\\theta)` the same as `p(\\theta \\mid x, \\phi)`? If so can you use consistent notation throughout for clarity? And if not could you please clarify what the difference between all of these guys are?\n\n- The empirical evaluation was quite weak in my opinion. The authors present a new approach to training SNPE models and it's great that they've applied it to a complex simulator to get some idea of whether it works on real practical simulators of interest, but I think it's also important to test a new method on some simpler benchmark models for which a good (even if approximate) ground-truth posterior can be obtained. This will allow the authors to properly test whether their method produces good posteriors, which isn't really tested at the moment by looking purely at RMSNE values. In general I would like to see something that demonstrates how well the proposed Bayesian pipeline can actually estimate the full posterior distribution, since this is ultimately what it's trying to do. I think the paper needs revision to include such experiments, even if the detail on this is mostly relegated to the appendix.\n\nA couple of less substantial weaknesses are:\n\n-  Perhaps the literature review could be a bit more extensive/comprehensive. For example, reference [1] below is a(n admittedly very) recent paper on origin-destination matrix estimation in mobility models, and while it might not be necessary or possible to compare against, it would perhaps be worthwhile including in the literature review as a SOTA method for estimating OD matrices when evaluating the likelihood is extremely expensive. Further, reference [2] below also considers the problem of efficiently performing SBI for complex and expensive simulators, albeit with a focus on obviating the task of learning summary statistics so a slightly different focus.\n\n- Some proof-reading and fixing of formatting issues is needed. For example: Line 268 is missing a word (\"...PC-SPSA is an effective extension _that over_ parameters in a lower-dimensional subspace...\") as is Line 294 (\"...ASNPE is outperformed by _PC-SPSA two_ of our explored settings...\"). Also the lines following Lines 141 and 156 have no numbers, not sure what's happened there (not a big deal currently but might cause problems down the line).\n\n[1] _Zachos, I., Damoulas, T., & Girolami, M. (2024). Table inference for combinatorial origin‐destination choices in agent‐based population synthesis. Stat, 13(1), e656._\n\n[2] _Dyer, J., Cannon, P. W., & Schmon, S. M. (2022, May). Amortised likelihood-free inference for expensive time-series simulators with signatured ratio estimation. In International Conference on Artificial Intelligence and Statistics (pp. 11131-11144). PMLR._", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work. Please find our responses to your questions and comments below.\n\nNote: Where applicable, we prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively, numbered by the order in which they were mentioned.\n\n**[W-1]** There is indeed some overlap in notation, and the differences between these forms are mostly context-dependent: \n\n- $q_{\\phi}(\\theta | x)$: the approximate posterior NDE model, with parameters $\\phi$\n- $p(\\theta | x, \\phi)$: a concrete posterior density produced by a particular setting of NDE weights $\\phi$\n- $\\tilde{q}_{x, \\phi}(\\theta)$: the approximate proposal prior\n- $q_{\\phi, x}(\\theta)$: same as $q_{\\phi}(\\theta | x)$, but with syntax adjusted to mirror the key result being used from [1] at line 178.\n\nIn particular, the form used on line 178 was constructed to resemble the setup of a key result we leverage from [1]. However, we agree that this may ultimately have made the connection to the notation elsewhere in our methodology unclear, and will revise this in our final manuscript (by both introducing each term if using a different form, and simplifying where applicable).\n\n[1] D. S. Greenberg, M. Nonnenmacher, and J. H. Macke. Automatic posterior transformation for likelihood-free inference, 2019.\n\n**[W-2]** Please see our global rebuttal reply for details on additional empirical evaluation, including discussion around new simulation benchmarks and more representative metrics.\n\n**[W-3]** _Regarding paper [1]_: We distinguish between the following two problems in OD estimation: \n\n1. General OD estimation problems (also known as travel demand estimation problems), where the outputs are stand-alone OD matrices that can be used for a variety of planning and operational network analysis, and\n2. Model calibration problems, where the goal is to calibrate (or estimate) the inputs (such as the demand inputs specified as OD matrices) of a specific traffic simulation model and the output is a calibrated traffic simulator that can itself be used for analysis. \n\nPaper [1] addresses Problem (i), while our work addresses Problem (ii). We would like to stress that this distinction is an extremely important one. In particular, the main challenges of Problem (ii) are due to using an intricate (e.g., stochastic, non-differentiable, high compute cost) traffic simulator. This calls for likelihood-free methods, sample-efficient methods, and methods robust to simulator stochasticity. In contrast to this, Problem (i) can be formulated as a differentiable problem, which can be tackled with a likelihood-based approach, has little-to-no compute cost challenges, and no need for sample efficiency. This distinction is also discussed in [A].\n\n[A]: C. Osorio (2019) High-dimensional offline origin-destination (OD) demand calibration for stochastic traffic simulators of large-scale road networks. Transportation Research Part B: Methodological, Volume 124, Pages 18-43\n\n_Regarding paper [2]_: This work addresses the similar problem of performing likelihood-free inference under expensive simulators/low budgets. However, it focuses almost entirely on formulations suited for time-series data, and aims principally to jointly learn both summary statistics and a classifier for performing density ratio estimation. These are no doubt important settings in this space, but this particular work ultimately differs in several significant ways form our paper (e.g., we use a single observational data point, work exclusively with direct posterior approximations, require a means of inducing uncertainty over NDE parameters, etc).\n\n**[W-4]** Thank you for pointing these out, lines 268 and 294 are indeed missing words and will be corrected in our final manuscript. We also noticed the missing line numbers in several places (appears to be an odd formatting bug) and will ensure these are fixed.", "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": "", "extra_scores": {}}, {"review_id": "enMq9JL4f6", "reviewer": "Reviewer_S6Jd", "summary": "The paper addresses the problem of computational cost, or alternatively, of sample efficiency in simulation-based inference (SBI). By employing an active learning scheme in sequential neural posterior estimators (SNPE), the proposed method achieves improved sample efficiency, which is paramount when dealing with expensive to evaluate scientific simulator-based models.", "questions": "* I did not understand the reasoning behind adjusting equation 3. Why multiplying with the prior makes sense?\n* Line 140-141: not clear what 'cost' is referring to.\n* Line 172-173: I did not understand what is being conveyed.\n* Line 191-192 talk about recovering the true optimum of  $\\alpha$ as $N$ tends to infinity. But in practice $N$ is much smaller. Does this statement hold even in the finite case?\n* What is the computational overhead of the proposed method compared to SNPE?\n\nTypos/grammatical errors:\n* Line 96\n* Line 127 (genetic instead of generic)\n* Line 121-122 doesn't parse well", "limitations": "Please include a paragraph on the limitations of the proposed approach.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "The paper has the following strengths:\n* It addresses a relevant problem in SBI.\n* The idea of using active learning to guide the sequential steps is interesting.\n* The proposed method is applied to real-world traffic simulation problems.", "weaknesses": "* The methodology is not motivated properly, and at times seems quite ad hoc. The authors first mention the expected information gain (EIG), which is a principled criteria for active learning, but then they propose equation 2 as the criteria they use. It was not clear to me why this is a good choice, and how it related to EIG. The distance $\\mathcal{D}$ in equation 2 became the quadratic loss in equation 3 without any justification/reasoning. I did not understand how the equation above line 157 was estimated. This seems like a critical quantity in the proposed method, and it would be nice if there is a discussion about it.\n\n* Existing methods for SBI using neural density estimators do not typically provide uncertainty quantification for the weights and biases ($\\phi$) of the neural network. There is very little emphasis placed on discussing this aspect. The authors mention Bayesian neural networks once in Section 2.2, and talk about MC-dropout for approximating the posterior of $\\phi$. The paper lacks an in depth discussion and experiments about using them in the context of SBI (which has so far not been done to the best of my knowledge), and how they affect the proposed method (for instance, how to set the prior for the Bayesian neural network, sensitivity to the dropout rate, etc.).\n\n* The paper includes experiments only on the traffic simulators. This wouldn't be an issue if this was an applied paper, or if the method was motivated by certain aspects of this problem which would generalise to other simulators as well. However, the method is presented in general, which is why I would expect a varied set of benchmark experiments. Given that a major portion of the references are from transportation research literature, perhaps this work, in its current state, would be more relevant for audience of that community.\n\n* The clarity of writing can certainly be improved. \n\n* The paper lacks discussion of the hyperparameters of the proposed method, its sensitivity to the choice of hyperparameters, and how to set them in practice.\n\n* The literature on active learning for simulator-based models is not cited and discussed, see for instance the following papers. This makes it difficult to judge the technical novelty of the proposed method. \n    * Gutmann and Corander (2016): https://arxiv.org/abs/1501.03291\n    * Kleinegesse et al (2019): https://proceedings.mlr.press/v89/kleinegesse19a.html\n    * Kleinegesse et al (2020): https://arxiv.org/abs/2003.09379", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work.\n\nNote: We prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively.\n\n**[W-1]** The discussion around EIG was principally intended to help set the stage for desirable qualities of an acquisition function, despite the fact it is itself not directly possible to optimize in LFI settings due to its reliance on the likelihood $p(x|\\theta,D)$. This drawback motivates an alternative formulation that captures these qualities, which begins with our introduction of a notion of distributional uncertainty (up to a divergence measure $\\mathcal{D}$) in Equation 2. This is further connected to Equation 3 via the discussion in Section 3.3, which makes concrete an approach inspired by the general form in Eq. 2 and that seen in [1].\n\nLine 157 is an expansion of the marginalization involved over NDE model parameters $\\phi$ in defining $p(\\theta|x,D)$, and is included primarily to highlight the relationship between $p(\\theta|x,D)$ and $p(\\theta|x, \\phi)$. More exposition will be added to our final manuscript to better contextualize this form.\n\n[1]: K. Kandasamy, J. Schneider, and B. Poczos. Bayesian active learning for posterior estimation. (IJCAI ’15)\n\n**[W-3]** CI-1 Please see our global rebuttal reply for details on additional empirical evaluation, including discussion around new simulation benchmarks and more representative metrics.\n\n**[W-4]** Appendix C.1.1 provides the hyperparameters used by our algorithm across explored experimental settings, as well as our NDE model’s structure, dropout rate, etc. We agree that a more principled analysis of hyperparameter sensitivity of heuristics for setting them in practice would be valuable. While performing such a study is prohibitive on our primary task due to computational constraints, the effects of the algorithm hyperparameters (e.g., $R$, $N$, $B$) and NDE hyperparameters (e.g., network structure, dropout rate, etc) will be explored on the smaller scale SBI benchmarks discussed above and reported in our final manuscript. \n\n**[W-5]** These are valuable resources and indeed provide useful context for leveraging implicit models in Bayesian optimization contexts. While we believe mentioning these works in our literature review is warranted, there are a few reasons why they might otherwise have been seen as indirectly relevant:\n\n- They primarily leverage GP surrogate models rather than models seen in more recent SBI literature (e.g., flow-based generative models) and those used in our work.\n- Our efforts are focused on formulations that only require a direct posterior approximation, rather than assuming surrogate likelihoods or likelihood ratios are also available.\n- They primarily focus on experimental design contexts, which embrace a slightly different set of assumptions, formulations, and intended applications. \n\n**[Q-1]** The adjustment to Eq. 3 is motivated in Section 3.3 (lines 172-175), which describes how the $\\theta^*$ that maximizes this notion of distributional uncertainty is not necessarily a likely parameter under the observational data $x_o$. Multiplying by the proposal prior, which captures the most up-to-date estimate of $p(\\theta | x_o)$ at each round of the algorithm, allows us to re-weight this uncertainty by the approximate likelihoods of each parameter under $x_o$ and prioritize more likely $\\theta$ during acquisition. See also [Q-3] below. \n\n**[Q-2]** “Cost” here is referring to the time or compute resources spent evaluating the simulator under the mentioned parameter. We agree this should be more clearly stated. \n\n**[Q-3]** Eq. 3 captures a means of quantifying uncertainty over the likelihood values assigned to $\\theta$ under the NDE and its various realizations under $\\phi\\sim p(\\phi|D)$. This does not capture the desire for $\\theta$ to also be likely under the posterior $p(\\theta|x_o)$, however. Our approach, along with many sequential SBI methods, is primarily concerned with learning an accurate view of the $p(\\tilde{\\theta}|x_o)$ under observational data $x_o$. Lines 172-173 are simply connecting this larger goal to the implications of optimizing Eq. 3, suggesting $\\theta^*$ may be of low value if it’s unlikely under the observational data $x_o$. This motivates the adjustment to Eq. 3 that appears in Eq. 4 (addressed in [Q-1]).\n\n**[Q-4]** Lines 191-192 convey that the optimum of $\\alpha$ (call it $\\alpha^*$) is captured in a sample $X_n$ of size $n$ (drawn over $\\alpha$'s domain) in the limit $n \\rightarrow \\infty$. This could be expanded to include the relevant implications, namely,\n\n$$p(\\alpha^* \\in X_n) \\rightarrow 1, n \\rightarrow \\infty$$\n\nor  \n\n$$\\text{argmax}_{\\theta\\in X_n} \\alpha(\\theta, p(\\phi | D)) \\rightarrow \\alpha^*, n \\rightarrow \\infty$$\n\nThis does not suggest that there exists a finite sample of size $n$ within which $\\alpha^*$ is present, and instead only captures the limiting behavior for increasingly large sample sizes. In practice, we leverage this consistency with $\\alpha^*$ given this limiting behavior.\n\n**[Q-5]** The primary difference between ASNPE and SNPE in terms of computational overhead is the optimization of the acquisition function in Eq. 4. As is mentioned in Section 3.4, however, the acquisition function is computed over a fixed size parameter batch and NDE realizations $\\phi\\sim p(\\phi|D)$ (with specific hyperparameters reported in Appendix C.1.1). This can be performed efficiently as a batched inference step under the NDE model, and typically yields a negligible difference in terms of raw runtimes between the two methods. For the traffic case study we explore in the paper, the raw runtimes of both methods (found in subfigures (b) for each of the calibration plots) allow us to compare the total time spent by our algorithm (including acquisition evaluation), and we observe a negligible difference compared to SNPE over our explored horizons.", "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": "", "extra_scores": {}}, {"review_id": "4PHpY3LvNJ", "reviewer": "Reviewer_s4rF", "summary": "The paper considers the problem of efficient neural density estimation for simulation-based inference, in settings where we want to estimate the parameters of a model from which we can draw samples but cannot define a likelihood function. Essentially, the main contribution of the paper is to integrate an active learning approach for exploring the parameter space efficiently, for which they define a measure by which they decide which parameter values should be \"explored\" next, i.e. used to generate more data to train the posterior. The approach is used for the Origin-Destination Matrix estimation problem in traffic simulation.", "questions": "1) How to choose a good prior for the OD matrix? Can we use some insights based on additional knowledge about the locations?\n2) What are the other endogenous and exogenous parameters used for this simulation? \n3) The technique proposed is quite generic, but only one application is shown: OD matrix estimation for traffic simulation. Can you discuss other applications, maybe within the traffic simulation domain itself?\n4) Is there any other way of dealing with the parameters \\phi without defining a distribution over them? In other words, if we do not choose 'q' to be a Bayesian Neural Network, can we still use the active learning approach to choose candidate '\\theta'?\n5) Is Equation 3 basically the variance of \\theta? Does the active learning basically choose those values at which \\theta has maximum variance according its current posterior distribution?\n6) When we are sampling candidate parameter values before applying the selection criteria, should we consider a distribution different from the current posterior? Choosing the current posterior essentially means we are \"exploiting\" the region of the parameter space where we already have observed some values - maybe we should \"explore\" the other parts of the parameter space using a different distribution?", "limitations": ".....", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The main contributions of this paper are:\n\n1) In case of simulation-based inference, an important question is how to explore the parameter space efficiently. This paper uses the concept of \"active learning\" for this purpose, by choosing the next few samples to \"label\" (by running simulations) using a measure\n2) The paper brings on the concept of model uncertainty through Bayesian Neural Network, and marries it off with simulation-based inference\n3) The paper considers elaborate experiments of origin-destination matrix calibration based on SUMO traffic simulations.", "weaknesses": "My general comment is that the paper has several useful and interesting ideas, but they have not been sufficiently explored or developed.\n\n1) The experiments show improved sample complexity due to the \"active learning\". But if the true posterior distribution p(\\theta|x_0) over parameter space is multimodal, then we may risk finding a suboptimal solution, especially if the prior is not suitable. \n2) A measure is proposed to choose the next few samples to be \"labelled\", but we don't really get to understand why that measure should be used.\n3) Although the general approach is quite generic, experiments are shown for only one task (OD calibration)\n\nMinor comments:\nAlgo 1: I think you should initialize D(r) with D(r-1) outside the \"for\" loop for variable b, and inside the loop, it should be D(r)=D(r) U (\\theta_b,x_b)\nFig 1 part b: internal text is illegible, should be expanded\nEq 4: p(\\theta|x_0, \\phi) should be p(\\theta|x_0, \\phi ')", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work. Please find our responses to your questions and comments below.\n\nNote: Where applicable, we prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively, numbered by the order in which they were mentioned.\n\n**[W-1]** This is true, although SBI/Bayesian methods in general are sensitive to a choice of prior, and this is not a limitation exclusive to our method. Further, the active learning mechanism does not change the limiting behavior of the method insofar as its relation to APT [1], as discussed in Section 3.4 (i.e., we still have $q_\\phi(\\theta |x)\\rightarrow p(\\theta|x)$ as $N \\rightarrow \\infty$).\n\n[1]: D. S. Greenberg, M. Nonnenmacher, and J. H. Macke. Automatic posterior transformation for likelihood-free inference, 2019.\n\n**[W-2]** The acquisition function proposed for selecting simulation samples is motivated across Sections 3.1, 3.2, and 3.3, with the principal motivation being to improve on traditional SBI methods by exploring the simulation parameter space in a more principled manner over the course of the multi-round inference procedure. In particular, we want to do so in a way that is maximally informative to our model of the posterior, which can yield compounding benefits in posterior accuracy under $x_o$ that are particularly important in settings with low simulation budgets and/or high simulation costs.\n\n**[W-3]** Please see our global rebuttal reply for details on additional empirical evaluation, including discussion around new simulation benchmarks and more representative metrics.\n\n**[W-4]** _Regarding syntax issues and small figure text_ Thank you for pointing these out, they will be corrected in our final manuscript.\n\n**[Q-2]** Many of these parameters are intrinsic to the SUMO simulator or the traffic network. For example, exogenous parameters include the route choice set and link attributes (e.g., free flow speeds, number of lanes). Endogenous variables include link and path traffic statistics (travel times, flows, speeds), and departure times. \n\n**[Q-3]** Our proposed approach falls into the class of simulation-based inference methods, and is indeed intended to be generally applicable in settings where statistical inference under an arbitrary mechanistic model is needed. There are many scientific domains that define problems of this nature (e.g., inferring parameters of biological processes, calibrating physics models to real-world observations, etc). In the transportation space, SBI methods like the one proposed can be used not only for inferring network demand (OD matrices), but also for many other dynamics in urban settings, such as human mobility and public transportation. Urban designers and city planners can reason about effects of various proposed changes using simulation models, as well as leverage inverse models learned with SBI to better characterize parameters required to achieve desired outcomes.\n\n**[Q-4]** Evaluating the acquisition function as-is requires some means of inducing a distributional estimate over the parameters of the chosen model. BNNs and MC-dropout are flexible ways to achieve this in practice, but without this form our acquisition function cannot be used as intended. We are not otherwise aware of a clear way to sidestep this requirement without considering an entirely different acquisition function (and no such formulation exists in the SBI/BO literature, to the best of our knowledge). \n\n**[Q-5]** Equation 3 captures average differences in the assigned likelihoods between the posterior model ensemble ($p(\\theta | x_o, D)$, see also the marginalization above line 157) and any particular model instantiation under the weight posterior. It could perhaps be likened to a notion of \"distributional variance\" around $\\theta$ under the model weight posterior $p(\\phi |D)$, but it is distinctly different from the variance of $\\theta$ under the posterior $p(\\theta | x_o)$.\n\n**[Q-6]** This is a great question, and very similar to the one that motivated this work. It is worth first noting that sequential SBI methods in general are focused on learning a particularly accurate picture of $p(\\theta | x_o)$, i.e., the posterior under known observational data of interest. These methods explicitly condition on $x_o$ to produce each round’s proposal distribution, effectively “refocusing” the next round’s samples to reflect the model’s current understanding of $\\theta$ that explain $x_o$. As a result, the posterior at a given round does not strictly correspond to regions previously explored in the parameter space (and so “exploitation” may be a bit misleading); rather, it captures the parameters expected to explain $x_o$ under the simulation model, given the simulation samples that have since been observed.\n\nIn any case, there are a few challenges when thinking about using the alternative distribution you mention in this case: \n\n1. How can we ensure convergence to the true posterior if sampling from a modified distribution?\n2. (similarly) How can we ensure the resulting exploration is consistent with the prior?\n\nOur approach effectively implements your proposition by allowing the underlying model to \"explore\" regions of the parameter space it doesn't currently \"understand\" well, while remaining consistent with prior specifications and converging to the true posterior in the limit. Put another way, our method effectively produces the mentioned alternative \"exploration distribution,\" but through a combination of components (the posterior model and a selection mechanism) that can be tractably mapped onto properties required for consistent Bayesian inference.", "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": "", "extra_scores": {}}, {"review_id": "Qhmp4ie3eU", "reviewer": "Reviewer_jL6p", "summary": "This work introduces a new way to do active learning in simulation-based inference. This algorithm allows for the sampling of data points not only in regions of interest but also in regions that lead to high information gain. They then apply this algorithm to the problem of urban travel demand calibration, which consists of estimating the expected travel demand between different points of interest.", "questions": "Do you have results with a metric other than RMNSE that would not suffer from the same issues?", "limitations": "The limitations regarding the computational cost of the acquisition function are not mentioned.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "### Originality\n* I'm not aware of other pieces of work introducing active learning in sbi the same way as done in this paper. \n* I'm unfamiliar with the urban travel demand calibration literature, but I am also unaware of a similar piece of work. \n\n### Quality\n* The idea of targeting samples of high epistemic uncertainty in SBI is sound. \n* The different approximations made give rise to a practical algorithm.\n* The experiments suggest that the active learning scheme allows the reduction of the number of samples required from the simulation for the task of urban travel demand calibration.\n\n### Clarity\n* Overall, the paper is clearly written. The method and motivation are well described, step by step and the urban travel demand calibration task is also well described.\n\n### Significance\n* Introducing active learning in sbi is of high significance.\n* I'm unfamiliar with the urban travel demand calibration literature but I would trust the authors on the fact that this is an important topic.", "weaknesses": "### Originality\n* I have no concerns regarding originality.\n\n### Quality\n* I find that Equation 4 is not well motivated. Why are the terms coming from equation 3 and the proposal prior simply multiplied ? I would have expected an hyperparameter controlling the weight assigned to both those terms but there is not, and this choice of simply multiplying the terms seems arbitrary. \n* In my understanding, optimizing equation 4 can be very computationally costly as it requires evaluating the Bayesian neural network on many samples and each evaluation of Bayesian neural network consist itself in the evaluation of several neural networks. I find that this limitation is not well stated in the paper. \n* RMNSE is a strange metric to use for posterior evaluation, in my opinion. This metric is designed to evaluate the quality of point estimation but not distributions. In particular, it falls short when the posterior is multi-modal; the metric would prefer an unimodal approximation in the middle to the correct multimodal distribution. \n\n### Clarity\n* The paper is a mix of methodological and applicative paper and does not have a clear scope. While the title suggests an applicative paper, the authors introduce a new algorithm that is valuable in all generality for the field of simulation-based inference. The experiments are, however, limited to urban travel demand calibration and hence do not demonstrate the this method is effective for other problems. \n\n### Significance\n* I have no concerns regarding significance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort providing detailed feedback on our work. Please find our responses to your questions and comments below.\n\nNote: Where applicable, we prefix sections with `W-<x>`, `Q-<y>`, or `L-<z>` to reference itemized comments in Weaknesses, Questions, and Limitations, respectively, numbered by the order in which they were mentioned.\n\n**[W-1]** The motivation for Equation 4 is primarily discussed in Section 3.3 (lines 172-175), which describes how the $\\theta^*$ that maximizes this notion of distributional uncertainty is not necessarily a likely parameter under the observational data $x_o$. Multiplying by the proposal prior, which captures the most up-to-date estimate of $p(\\theta | x_o)$ at each round of the algorithm, allows us to re-weight this uncertainty by the approximate likelihoods of each parameter under $x_o$ and prioritize more likely $\\theta$ during acquisition. This multiplication is primarily motivated by the desire to more closely align the effective likelihoods of parameters $\\theta$ with the true proposal prior, which is ultimately present and corrected for in the NDE loss (see line 178).\n\n**[W-2, L-1]** Regarding computational cost of acquisition: Section 3.4 describes the mechanism by which Equation 4 can be optimized over a fixed set of proposal samples during each round. While we do indeed need to evaluate the NDE for several candidate parameters and many network realizations, this acquisition evaluation can be performed very efficiently as a batched inference step under the NDE model. \n\nFurther, the additional computational overhead can be compared against SNPE (i.e., no acquisition function) for our primary traffic task, as the reported wallclock times (found in subfigures (b) for each of the calibration plots) provide the raw runtimes for each method when obtaining the 128 simulation samples for each setting. Empirically, this allows us to compare the total time spent by our algorithm (including the cost of optimizing the acquisition function), and we observe a negligible difference compared to SNPE over our explored horizons. \n\n**[W-3, Q-1]** You raise a valid point: RMSNE is indeed a non-standard metric for evaluating SBI methods in the literature, and is employed in this work almost entirely due to its prevalence in the OD calibration community. In our traffic calibration setting, there are few why it’s used:\n\n1. We are ultimately interested in producing good point estimates of the true OD matrices,\n2. RMSNE is a standard in the OD calibration space and allows us to compare with other baseline methods like SPSA and PC-SPSA, and\n3. While we may benefit downstream from having a good distributional estimate under $x_o$ (as SBI methods generally attempt to provide), for our task, we are mostly interested in how approximating the posterior can improve our method's intermediate exploration of the parameter space in service of producing good point estimates.\n\nNevertheless, we agree a systematic exploration of the final posterior accuracy under metrics better suited to capture distributional differences (not just point estimates) would be warranted in general. Please see our global rebuttal reply for more details here.\n\n**[W-4]** Please see our global rebuttal reply for details on additional empirical evaluation, including discussion around new simulation benchmarks and more representative metrics.", "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": "", "extra_scores": {}}]

Author rebuttal:
We’d like to thank all reviewers for their time and effort in providing insightful feedback on our work. Please find our responses to your individual questions and comments in the rebuttal replies to each review.

Due to space constraints, we would like to address a common critique raised by many reviewers here in our global rebuttal regarding limited empirical evaluation. In summary, we agree that our work would benefit greatly from additional benchmark results, particularly for common simulation models in simulation-based inference (SBI) literature. To this end, we’ve evaluated our method on three tasks under three new metrics. Our hope is that these initial results provide additional depth when it comes to judging the empirical value of our contributions. These settings will be scaled up and explored in a more systematic fashion for presentation in our final manuscript.

### New benchmark tasks

We evaluated our method (ASNPE) and SNPE-C [1] on three common SBI tasks found in the literature (namely [2], which provides a suite of benchmarks across several SBI methods): **SLCP distractors**, **Bernoulli GLM**, and **Gaussian Mixture** (note that each corresponds to a reproducible task environment from [2]).

We further evaluated our method on these tasks using metrics beyond RMSNE that are better suited for capturing the accuracy of the approximate posterior, including _classifier 2-sample tests (C2ST)_, _maximum mean discrepancy (MMD)_, and _Kernelized Stein Discrepancy (KSD)_.

**Note**: Full details for each of these tasks and metrics can be found in [2].

We evaluated both methods over small sample horizons: 4 rounds at 125 samples per round, for a total of 500 simulation samples. Note that this is nearly four times larger than the sample sizes collected for the primary traffic setting explored in paper. For reference, 125 samples in our (non-parallelized) SUMO environment takes ~3 hours, whereas 125 samples from SLCP takes ~5 seconds on our hardware (see also Appendix C.1.1). 

Note that these trials were repeated five times for each method, and the average score and standard deviation for each metric over these trials is reported in the tables below.

### Results

|       | MMD            | C2ST          | KSD           | L2            |
| :---  | :---           | :---          | :---          | :---          |
| SNPE  | 11.3820 ± 1.1043 | 0.9938 ± 0.0003 | 0.2901 ± 0.1217 | 5.9121 ± 0.3323 |
| ASNPE | **11.2673** ± 0.5796 | 0.9938 ± 0.0002 | **0.2558** ± 0.1331 | **5.6731** ± 0.1988 |
**Table 1: SLCP Distractors**

|       | MMD            | C2ST          | KSD           | L2            |
| :---  | :---           | :---          | :---          | :---          |
| SNPE  | 18.7130 ± 1.8315 | **0.9952** ± 0.0028 | 0.2496 ± 0.0579 | **37.2472** ± 3.4843 |
| ASNPE | **16.1893** ± 0.5625 | 0.9980 ± 0.0007 | **0.2340** ± 0.0467 | 39.4204 ± 0.2686 |
**Table 2: Bernoulli GLM**

|       | MMD            | C2ST          | KSD           | L2            |
| :---  | :---           | :---          | :---          | :---          |
| SNPE  | 1.2655 ± 0.0412 | 0.9950 ± 0.0000 | 0.0886 ± 0.0292 | 6.6242 ± 0.9629 |
| ASNPE | **1.1216** ± 0.0780 | 0.9950 ± 0.0000 | **0.0633** ± 0.0256 | **3.7100** ± 0.3214 |
**Table 3: Gaussian Mixture**

Note that _L2_ refers to the average $L_2$ distance between $x_o$ (observational data point) and samples $x\sim p(x|\theta), \theta\sim p(\theta|x_o)$. This provides some insight into how well the posterior is calibrated around our data point $x_o$, and whether samples drawn from the posterior $\theta\sim p(\theta|x_o)$ ultimately yield synthetic data close to $x_o$ under the mechanistic model.

While these simulation horizons are relatively small, we find that ASNPE tends to outperform SNPE across most settings and on most metrics. In particular, ASNPE outperforms SNPE on MMD and KSD metrics across all settings. Outside of _Bernoulli GLM_, ASNPE either matches or exceeds SNPE in terms of C2ST and reported $L_2$, whereas on _Bernoulli GLM_ SNPE is better on these metrics.

### Additional studies for our final manuscript

Due to the difficulty of evaluating our transportation simulator for large simulation samples, in-depth ablation studies or hyperparameter sensitivity analysis has been difficult to carry out at scale. Given the efficiency of running our method on the much faster, lower-dimensional simulations explored above, we will use these scenarios to provide 1) hyperparameter ablations, 2) comparison between ASNPE and SNPE over long simulation horizons, and 3) diminishing effects/failure modes of the acquisition function in the SBI loop.

### In defense of our single traffic task

We’d like to note that we intentionally focused our efforts on a single problem that captures many facets of challenging real-world settings, including high-dimensionality, realistic (traffic) dynamics, and slow simulation times. Further, the availability of real-world, metropolitan-scale traffic simulations is very limited, and most high-impact works in the urban mobility space limit their empirical evaluation to synthetic networks or small-scale road networks. This is due to the difficulty of accessing and/or developing realistic road network models for major metropolitan areas. While investigating additional simulation models is undoubtedly valuable and can improve the general appeal of our contribution, we ultimately believe the transportation problem focused on in this paper best captures the realistic qualities of slow, real-world, high-dimensional simulation models.

[1] D. S. Greenberg, M. Nonnenmacher, and J. H. Macke. Automatic posterior transformation for likelihood-free inference, 2019.

[2] Lueckmann, J. M., Boelts, J., Greenberg, D., Goncalves, P., & Macke, J. (2021). Benchmarking simulation-based inference. AISTATS.

Meta-review: The reviewers recommend acceptance (8-6-4-5-6-7). The paper introduces ASNPE, a new method for simulation-based inference based on active learning. The authors-reviewers discussion has been very constructive and has led to a number of improvements to the paper.  In particular, the reviewers noted that ASNPE was a generic improvement over SNPE and, as such, should be evaluated on common SBI benchmarks instead of focusing only on the task of urban demand calibration. The authors have provided convincing results SBI benchmarks during the rebuttal, which has been largely appreciated by the reviewers. The borderline rejection from Reviewer JwDn is not an obstacle to acceptance, as most the major concerns have been addressed, despite the lack of acknowledgment from the reviewer.

Given the scope of the paper and the potential of the proposed approach beyond the traffic task, the authors should consider the opportunity to reframe the paper and its title to better reflect the generic contribution of ASNPE. Finally, the authors are asked to implement the changes and clarifications discussed with the reviewers in the final version of the paper.
Decision: Accept (poster)