Reverse-Annealed Sequential Monte Carlo for Efficient Bayesian Optimal Experiment Design
Authors: Jake Callahan
Andrew Chin
Jason Pacheco
Tommie Catanach
Venue: NeurIPS 2025 poster ()
Keywords: Bayesian optimal experiment design
expected information gain
mutual information
model evidence
sequential Monte Carlo
Markov chain Monte Carlo
Bayesian methods
Primary area: probabilistic_methods
TL;DR: We propose a sequential Monte Carlo estimator for expected information gain that reduces computation in Bayesian optimal experiment design using backward tempering.

Abstract: Expected information gain (EIG) is a crucial quantity in Bayesian optimal experimental design (BOED), quantifying how useful an experiment is by the amount we expect the posterior to differ from the prior.
However, evaluating the EIG can be computationally expensive since it generally requires estimating the posterior normalizing constant.
In this work, we leverage two idiosyncrasies of BOED to improve efficiency of EIG estimation via sequential Monte Carlo (SMC).
First, in BOED we simulate the data and thus know the true underlying parameters.
Second, we ultimately care about the EIG, not the individual normalizing constants.
Often we observe that the Monte Carlo variance of standard SMC estimators for the normalizing constant of a single dataset are significantly lower than the variance of the normalizing constants across datasets; the latter thus contributes the majority of the variance for EIG estimates.
This suggests the potential to slightly increase variance while drastically decreasing computation time by reducing the SMC population size, which leads us to an EIG-specific SMC estimator that starts with a only a single sample from the posterior and tempers \textit{backwards} towards the prior. Using this single-sample estimator, which we call reverse-annealed SMC (RA-SMC), we show that it is possible to estimate EIG with orders of magnitude fewer likelihood evaluations in three models: a four-dimensional spring-mass, a six-dimensional Johnson-Cook model and a four-dimensional source-finding problem.

Reverse-Anneled Sequential Monte Carlo for Efficient Bayesian Optimal Experiment Design
Jake Callahan'
Andrew Chin" Department of Biostatistics Johns Hopkins University
Program in Applied Mathematics The University of Arizona
Jason Pacheco
Tommie Catanach
Department of Computer Science The University Arizona
Computational Data Science Sandia National Laboratories
Abstract
Expected information gain (EIG) a crucial quantity in Bayesian optimal experi- mental design (BOED), quantifying how useful an experiment is by the amount we expect the posterior to differ from the prior. However, evaluating the EIG can be computationally expensive since it generally requires estimating the posterior normalizing constant. In this work, we leverage two idiosyncrasies of BOED to improve efficiency of EIG estimation via sequential Monte Carlo (SMC). First, in BOED we simulate the data and thus know the true underlying parameters. Second, ultimately care about the EIG, not the individual normalizing constants. Often we observe that the Monte Carlo variance standard SMC estimators for the nor- malizing constant of single dataset significantly lower than the variance of the normalizing constants across datasets; the latter thus contributes the majority of the variance for EIG estimates. This suggests the potential slightly increase variance while drastically decreasing computation time by reducing the SMC population size, which leads to EIG-specific SMC estimator that starts with only single sample from the posterior and tempers backwards towards the prior. Using this single-sample estimator, which we call reverse-ane reverse-annealed SMC (RA-SMC), we show that it is possible to estimate EIG with orders of magnitude fewer likelihood evaluations in three models: a four-dimensional spring-man -man,, a dix-dimensional Johnson-Cook model and four-dimensional source-finding problem.
1 Introduction
Optimal experimental design (OED) is a powerful method for selecting design parameters for experiments that update model uncertainty using observational data. By quantifying the utility U of design d, one can maximize the utility over all the designs as: d' arg max, (d). In Bayesian optimal experimental design (BOED) are interested in the information gain (IG) from an experiment for an unknown parameter given the dataset y (Lindley, 1956). This is quantified by the Kullback-Leibler (KL) divergence from the prior to the posterior (Rainforth et al., 2024):
IG d) Dr (p(( d,d) |l p(0)) p
p(0 y. d) log p(0 y, d) d0.
(1)
p(0)
DDenotes equal contribution. These authors are listed in alphabetical order.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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As we do not have access to y before an experiment is run, our utility function is the expected information gain (EIG):
( Eejaa v, [[(0) - -
log p(u 0.d) ply d)
EIG =
do dy
(2)
This expectation has no analytical solution outside of the most basic examples, and thus one typically resorts Monte Carlo integration. This proceeds by drawing dataset, estimating the IG, and repeating this many times to obtain an EIG. However, the IG itself is also generally intractable due to the presence of the posterior normalizing constant p(y d), also known as the model evidence or marginal likelihood. Thus, considerable effort is often placed on estimating p(y d) or its logarithm (Ryan, 2003), and not all methods are guaranteed to give accurate results. In low dimensions, the simple nested Monte Carlo (NMC) is a common choice (Rainforth et al., 2018; Zheng et al., 2018)
Higher dimensions or more informative data require more stable, but expensive, Monte Carlo estimators. Principal among these is a class of estimators based on sequential Monte Carlo (SMC) (Del Moral et al., 2006; Chopin al., 2020). SMC starts with many samples from known distribution and evolves those samples through a sequence of intermediate distributions toward the distribution of interest. Likelihood values computed during this evolutionary process can then be used to compute the model evidence (Xie et al., 201 I). Often, hundreds of millions of costly likelihood evaluations are required to compute the EIG for a single design, making SMC infeasible for many models.
Our work adapts and extends the bidirectional Monte Carlo approach introduced by Grosse et al. (2015) for the specific context of BOED by leveraging two unique aspects of the problem. First, observe that in BOED we require Ellog p(y)), and the variance in log p(y) across different y often dominates the variance in estimating log p(y) for a single y when using SMC. We find that we can significantly reduce the number of particles to achieve similar Monte Carlo error but with far less r
computational cost. The second observation is that generating y requires drawing @ ~ p(0) and then y pyy 0), resulting in the joint sample (0, y) ~ p(0, y). Instead of discarding 6, we can treat this joint sample as if we drew y p(y) and I p(0 y), giving us free posterior sample. Together, these two facts allow us to modify the framework proposed in Grosse et al. (2015), which starts with a single posterior sample and draws from a sequence of distributions tempered backwards towards the prior, and then estimates an upper bound on p(y) accordingly.
This approach yields reverse-anmealed sequential Monte Carlo (RA-SMC), an algorithm that starts with a single posterior sample and draws from a sequence of distributions tempered backward towards the prior. Unlike Grosse et al. (2015), which sandwiches the true value of pog(( RA-SMC directly provides practical EIG estimates and remains robust to overestimating under poor MCMC mixing. We demonstrate this estimator on a coupled spring-mas mass system, Johnson-Cook model of plastic deformation, and a sequential source location problem-al) all with multimodal posteriors. Not only do we show that traditional SMC estimators can be used with an order of magnitude fewer particles, but also that our reverse estimator provides further fourfold improvement in computational cost.
2 Related Work
EIG estimation in BOED has been approached with various methods. Nested Monte Carlo (NMC) uses inner-outer loops (Ryan, 2003: Beck et al., 2018) but scales pooorly and has slow-decaying
bias (Rainforth et al., 2018: Zheng et al., 2018). Variational approaches optimize approximate
posterior (Dahlke et al., 2023; Foster et al., 2019, 2020); they scale better but can be costly and miss complex posterior structure (Pacheco & Fisher, 2019). Likelihood methods bypass liekelihood via density-ratio estimation, as in LFIRE (Kleinegesse et al., 2021), or by directly learning the objective, as in MINEBED (Kleinegesse & Gutmann, 2020). For sequential design, reinforcement learning can learn a design policy (Huan & Marzouk, 2016; Foster et al., 2021; Ivanova et al., 2021; Blau et al., 2022).
Asymptotically consistent alternatives include tempering methods like annealed importance sampling (AíS) (Neal, 1996). Sequential Monte Carlo (SMC) extends this by resampling and rejuvenating particles along the temperature sequence (Chopin et al., 2020; Del Moral et al., 2006), which has been used for evidence estimation (Xie et al., 2011 1) and BOED (Ryan et al., 2016). We note here that in this work, "sequential" refers to the progression of particles through a sequence of tempered distributions, as distinct from the temporal sequence in state- and filtering applications. Our

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estimator is a equally weighted each temperature and rejuvenated via MCMC, making conceptually closer to MCMC.
zero-ressan seromresample instance of SMC tempering, Unlike AIS, our particle is
Bidirectional Monte Carlo (BDMC) (Grosse et al., 2015, 2016) runs forward and reverse annealing to produce stochastic upper and lower bounds on ply)a A reverse pass starts from an exact posterior sample and forward pass from the prior, "sandwiching" the marginal likelihood for model comparison. We adopt BDMC's use of a free posterior draw and reverse anneal but omit the forward pass and bounding machinery.
Our estimator can be regarded as a modification of the BDMC framework for the BOED context. While BDMC "sandwiches" the marginal likelihood, repurpose the everse-annnallln pass the create a practical EIG estimator. Crucially, we use the thermodynamic integral instead BDMC's stepping stone algorithm to mitigate the bias of reverse estimators. Furthermore, while BDMC requires many partieles for tight bounds, our analysis of EIG's variance structure shows a single particle provides sufficient accuracy at a fraction the compuational cost. To our knowledge, this is a novel application of reverse-annealed SMC to BOED and constitutes a new EIG estimator.
The methods described above present diverse set of trade-off's between computational cost, scal- ability, and asymptotic guarantees. This work specifically focuses on improving the efficiency of tempering-baie based Monte Carlo estimators, like AIS and SMC, which provide asymptotically-ext estimates crucial for high-fidelity applications, and we show that the RA-SMC estimator makes this powerful but expensive class of methods more practical. To that end, we first review the standard SMC framework, which forms the basis of our proposed approach.
Preliminaries: Annealed Sequential Monte Carlo
We briefly review the SMC framework (Chopin et al., 2020) and the SMC-based stepping stone algorithm (Xie et al., 2011) 1) that is commonly used for computing model evidence p(y). SMC evolves population of N partieles from simple initial distribution (e.g., the prior) to the posterior. It does so by progressing through a sequence of intermediate tempered distributions using importance resampling and Markov chain Monte Carlo (MCMC) steps. One commonly-chose sequence of tempered distributions is the power posterior sequence (Friel & Pettitt, 2008):
Pt. (0 | v) ply p())
/. p(y (0) do
where, ze. (1)
(3)
Zt, (3)
and 0 = lg .... ... S Iy = 1 an increasing sequence of temperatures. The lowest temperature lo recovers the prior, and at the highest temperature iv. the posterior. At each level,
compute:
p(u 0) ply 0) p(0) do
So p(v) 0) p(0) do
24,+1 (3)
Elp(y | 00)
21.
24.
21, (v)
where At; = 1441 1i2 Taking the product up to the N level yields the marginal likelihood p(y):
N-1
(y) 2t, (v) Ztg (y)
2t, (v)
ZzN (v)
p(v)
II
(5)
tto (v) 21 (v)
(v)
Zeo (v)
ZNN
==0
This is known as the stepping-stone algorithm (Xie et al., 2).1). and in SMC the number of levels and their temperatures can be chosen adaptively (Catanach & Beck, 2018). The tempering gradually
guides the samples to areas of high posterior density, and leads to some of the most accurate estimates of ply) (Fourment et al., 2020) via the following estimator:
N-
p(v) = II
Σ
ply (gg) Ar,
where (0), 1 ~ Pt.
(6)
4 Reverse Annealed Sequential Monte Carlo
4.1 Motivation: Balancing Variances
In BOED we are not interested in solely estimating p(y): we are interested in EIG(d), for which the individual IG(y) will depend on their own p(y). Hence, for estimate of the EIG there are two
3

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sources of variance. The first is variance in (v()) across the datasets y, and the second is variance from its estimate (y) for given y, which denote Var(IG(y) y). If we assume Var(IG(y) v) constant across y and IG is unbiased, we can formalize this through the law of total variance:
Var(IG(y)) = E. ([aa((IG())
[b] + Var(E[IG(y)) v)) = Var(IG(y) (v) + Var(IG(v))
(7)
Our experiments find that SMC can be overly precise in the sense that Var(IG(y) y) < (Vr()
Significant cost is spent on achieving excellent information gain estimates for each dataset, but this precision is unwarranted in light of the overall variance across datasets. As an example, we demonstrate this the coupled spring-mass system of Sec. 5.1. For fixed design,
draw 100 different y using different 0, and for each we compute 30 estimates IG()) using an SMC algorithm (Catanach & Beck, 2018) with 250 particles, The results are summarized in Table la, where we find that the variance of IG is two orders of magnitude less than Var(IG(y)) = 22.4.
Dataset
(1:)
Var (w:) W))
# Particles
(g
Var(IG(y;) 16)
1/1
20.71
0.059
250
20.210
0.081
Ys1
11.40
0.039
150
20.377
0.133
yeo
19.24
0.093
50
20.513
0,570
yr1
23.39
0.110
20
20.399
1.384
y100
11.57
0.021
10
24.802
25.123
(a)
(b)
Table 1: (a) Selected variances of thirty SMC estimates of IG (250 particles). The variance of IG across all 100 datasets is 22.4. Full results in Figure 10 (appendix). (b) Comparison of Var(IG(ys) y1 for different numbers of SMC particles.
We modulate the variance of the forward SMC estimator by reducing the number of particles. Shown in Table 1b, we find that reducing particles from 250 to 20 still yields estimators with an order of magnitude less variance than Var(IG(y)). Only when we get to 10 particles does Var(IG (y) s Var(IG(y)). These few-particle SMC estimators are no longer useful for estimating individual IGy)) due their variance and thus could not be applied in other contexts-their utility unique to calculating EIGs.
Single particle estimates We now consider the extreme case of using only single particle. This seems all-advised, however failure a consequence of poor MCMC mixing, not failure of estimation using few samples. To show this, we consider whether we use single sample from well-mixed Markov chain at each temperature. To ensure proper mixing we run forward SMC using 1000 particles, but estimate IG using only single random particle from each temperature. This results in variance of 14.903, which is roughly half Var(IG(y)). This demonstrates that a single particle IG estimate, from a well-mixed chain, is feasible for experimental design. To obtain such a result we temper backwards, from the posterior to the prior distribution, which we discuss next.
4.2 Reverse-annealed SMC Algorithm
Now make use of a second unique feature of BOED: datasets are simulated when computing the EIG. This means that we have access to the true underlying parameter 6*, Crucially, we can treat each joint sample (y, (*) being generated first by drawing y from its marginal, and then 0* from the posterior. This gives us one free draw from the posterior, and so we can start SMC sampling from d* and consider sampling a sequence of tempered distributions beginning from the posterior and going to the prior, taking a single sample at each level. This simplifies sampling as we start with a sample from the most difficult distribution and decrease temperature over time, preventing degeneracy issues where our samples end up stuck in low likelihood regions.
We call this scheme reverse-annedie SMC (RA-SMC). Since we will be increasing the variance in our estimates, we use the thermodynamic integral (Gelman & Meng, 1998) to help reduce bias by directly targeting log p(y):
logp(y)= = - log
ply ())p(()) z(y)
log p(y 0)
do dt.
(8)
A proof of this identity is provided in Appendix A.I.

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Algorithm Reverse annealed sequential Monte Carlo
Require: Dataset y0", Temperatures / {la..... ................................ LN
Ensure: Output IG(y)
1: Initialize 8 <- 8", < N 11,1 Array|logp(y (0)]
2: while 10 do
3: f MCMC
Run MCMC on pt, starting at 0)
4: i-i-1
5: Lappendd(ogg p(u 0))
6:
end while
7: return loyply 0*) Simpson(t,
{ Use Simpson's 's rule for thermodynamic integral.) )
The temperature sequence provides discretization of & on [0. 1]. and we then use Simpson's rule (Young & Gregory, 2012) to approximate the outer integral (Calderhead & Girolami, 2009). At each tempering level, the particle is advanced via MCMC runs targeting Pt, (0 y), which is then used in evaluating the thermodynamic integral. By interchanging the order of integration, we can also view our estimator as first doing numerical integration with Simpson's 's rule using single draw at each temperature, and then averaging over all the integrals. This enables us to estimate Monte Carlo standard errors by computing the variance across the integrals. The full RA-SMC estimator is thus defined as follows.
Definition 4.1. For a fixed y and design d, define
(y, d) logp(y 0!M,,a) ,d)
and
log Sx( (y. d), (1 (y, d)................ ..... 1000 (y,
Here, 0f denotes a sample drawn from an M M-step MCMC sampler targeting Pt, as its stationary distribution, and SN is the composite 1/3 Simpson's 's rule estimator of the input integral evaluated at N temperature levels t; (i/N), i, 0, I.. Then the RA-SMC estimator for design d is:
EIG(d)
pog 0x) log PN (8k. d), Σ
where ((yk dx)) iid ply.0) 0)
We now establish main property of RA-SMC-it an asymptotically unbiased estimator of EIG: Lemma 4.2. Define the RA-SMC estimator as in Definition 4.1. If the MCMC sampler used to draw each M.M) defines an ergodic Markov chain with stationary distribution pt. and Ep, [log ply 0)] is four times continuously- diffruussstddfffetttabbllll for all E [0. 1], then the RA-SMC estimator is asymptotically unbiased.
The proof of Lemma 4.2, which can be found in full in Appendix A.1, relies on the asymptotic convergence of the composite Simpson's 's rule estimator as well as the convergence to the correct. stationary distribution of a valid MCMC sampler.
The thermodynamic integral approach is also valid for forward SMC, but the number of tempering levels needed for accurate integration tends be higher than the number of tempering levels needed for good SMC estimates, and the bias is less of an issue when the estimate of p(y) is precise. The full algorithm is presented in Algorithm 1, and despite only using a single sample each iteration, we will see in Section 5 that its accuracy is comparable to forward SMC with far more particles.
Experimental Results
We compare our algorithm to the forward SMC algorithm of Catanach Beck (2018), which adap- tively chooses the tempering levels and MCMC iterations based on effective sample size calculations. Different numbers of particles are also considered illustrate the discussion in Section 4.1, with a 250 particle run considered the gold standard against which we compare. For our backward estimator, we use fixed tempering sequence based on Calderhead Girolami (2009), with = 100 levels and temperatures t (i/N)3 (see Corollary A.I). For the MCMC kernel use simple random walk
5

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Metropolis, where the proposal standard deviation is adapted based on the previous temperature 's acceptance rate using the feedback controller of Catanach (2017) to target an ideal acceptance rate of 0.234 (Gelman et al., 1997). Based on an analysis shown in Appendix 1. we fix the number of tempering levels at 100 and only adjust the number of MCMC iterations per temperature for simplicity. We implement an early stopping criterion for MCMC at each level once the Spearman correlation between the starting log likelihoods and the current log likelihoods drops below 0. I. We also halve the number of steps taken once the proposal standard deviation equals the prior standard deviation, indicating that the power posterior diffuse enough such that more iterations are not eritical.
We observe better performance with fewer MCMC iterations across almost all configurations, though overestimation is now more likely. For tuning, we fix design and run multiple MCMC iterations, stopping roughly when our estimates stabilize while accounting for Monte Carlo standard error, which ended up being 60 iterations. Any initial SMC runs used to determine whether this sampler is viable as mentioned in Section 4.1 can also be used inform tuning.
5.1 Coupled spring-mass system
We first explore a coupled spring-mast mass system depicted in Fig. 1. Two masses, mi and m2, are surface with respective friction coefficients b and by and joined by a spring with spring constant k2. The first mass then joined to a fixed point by a second spring with spring constant kip assumed to have length 0 when forces are applied to them, and the starting positions and veloc- ities of the masses are set O. Each of the two masses, their friction coefficients, and the two spring constants are
ki
k-
000000
000000
770
1/1
2/2
The springs
Figure 1: Coupled spring-mass manng system. Fach of the masses also has an friction coefficient bn and b2. considered unknown parameters, so the posterior is 6D: (ma1 m2, b1, b2, k1, kg). All parameters have log-normal priors with mean 0 and SD 1. The experiment imparts damped oscillating force on the system representing a
vibration, given by:
11 U1, 4/2 U2, 1/1 -b b1u1 (k1 + 12) + k2x22
b202 + 12(21 - x2) + f(x,t)
m2
where f(r,t) = 5 sin())) exp(- is the forcing function, z; is the position of the ith mass, and is is the velocity of the ith mass. We observe noisy position 2 at 100 equally spaced time points. The noise is Gaussian with mean 0 and SD 0.025, with observed data y {yI.... .,, y1o00 and,
Er snd (O, 0.025 t= I, 100
V/ = Z1t + Et.
where Tn is the position of the first mass at time t. Fig. 2 shows three randomly generated datasets. We consider ten equally spaced designs y between 0 and of the form 0.2j.J 1, I..., 10. By
1.0
Spring postion
0.5
Observed value
0.55
300
40
10
50
10
20
30
40
Time (t)
Figure 2: Three examples of observed data y in orange, along with the true signal x in blue for the spring-mass model. The true signal corresponds to the position of the first mass my between time = and = 50, and observed over 100 equally-spaced time points. observing only one mass, induce multimodality and curvature in the posterior distribution. Fig. shows four dimensions of an example posterior. 1000 datasets are drawn for each method, with performance measured by the number of likelihood evaluations required and how well the final EIG values correspond to those of an expensive forward SMC run. Likelihood evaluations are the dominant cost and are therefore used as a hardware agnostic metric for computational efficiency.

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Magnetic reverse-annealed SMC steps To aid MCMC mixing we use knowledge of the target distribution (the prior), when tempering backwards, to inform the initial- ization at each level. 'he idea that the next temperature level should place more mass towards the prior. Hence, instead of starting at the last position of the previous level, we nudge the starting point of the MCMC towards the prior mean. Between levels, we take up to a tenth of the planned MCMC iterations toward the prior mean using the proposal standard deviation as the step size, stopping if the log likelihood at any step is less than tenth of the starting log likelihond. We note that this is an optional enhancement, rather than core component of the method.
Figure 3: Example spring-mass posterior for four of the parameters based on kemel density estimation using samples from 250- particle forward SMC, with the standard nor- mal priors shown in grey. The posterior demonstrates multimodality and curvature.
22
Likelihood evaluations
10a
RA-SMC, 60 MCMC steps
20
Forward SMC 250 particles
Forward SMC particles
10a
Forward SMC particles
107
0.2 0.4 0.6 0.8 1.0 1.2 14 1.6 1.8 2.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4
1.8 2.0
Design (y)
Design (y)
10 designs of the spring-mass model, with vertical bars representing Monte Carlo standard errors above and below estimate. Right: Number likelihood evaluations each estimate. The adaptive nature forward SMC results a varying number of evaluations per design.
Figure 4: Left: EIG estimates
Results A comparison of the EIG curves resulting from traditional forward SMC estimators with varying numbers of particles to the backward SMC esti- mator's EIG curve is shown in Figure By decreasing the number of forward SMC particles by an order of magnitude to 20, we gain an order of magnitude of efficiency with little loss of performance compared to the gold standard 250 particle runs.
At 10 particles, significant upward bias and noise oc- curs. Notably, the failure is catastrophic opposed to gradual; below some threshold, mixing fails and the estimates poor, while above that threshold the estimates are still stable. Our reverse estimator performs well with another fourfold decrease in likeli- hood evaluations compared to the 20 particle forward SMC.
Estimated EIG
10a
SMC NMC PCE
10a
10a
1.4 1.6
2.0
Design
Other Monte Carlo Baselines As a baseline, we also evaluated Nested Monte Carlo (NMC) and Prior Contrastive Estimation (PCE) on the spring-mass problem. allocating the same computational budget as the 20-particle forward SMC run. As shown in Figure 5, both estimators performed poorly. NMC produced inaccurate estimates with large standard errors, while PCE exhibited strong downward bias and failed to locate the optimal design. Since the data are informative, the resulting posterior sharply peaked. The prior-based sampling of these methods inadeuuate to explore this region, which leads to unstable and biased evidence estimation.
Figure 5: Estimated expected information gain (EIG) for the springmass design problem across design values from 0.2 2.0. Each curve cor- responds to different estimator: Forward SMC, Nested Monte Carlo (NMC), and Prior Contrastive Estimation (PCE). Error hars indicate estimator standard errors, computed across replicates. SMC estimates use 100 outer samples and 20 particles per stage; NMC and PCE estimates use 100 outer samples and 300,000 inner samples. Error bars for the SMC and PCE curves are present but too small be visible.

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Stress (o)
100
50
True stress
Observed valuey
0.1
0.2
0.4
0.5
0.0
0.1
0.3 Strain percentage
0.2
0.4
0.5
0.0
0.1
0.2
0.3
0.5
Figure 6: Three examples of observed data in orange, along with the true signal a in blue for the Johnson-Cook model. The true signal corresponds to the output stress of material for strain percentage between O and 0.5, observed 100 equally spaced values.
5.2 Johnson-Cook model
We rum a Johnson-Cook model of stress-strain curve for hypothetical material under plastic deformation. The data follows the following model, where the experiment involves observing the stress at 100 equally spaced strain percentages ε from 0 to 0.5, with varying measurement noise dependent on whether the material is in the elastic or plastic phase:
EE + 5c
if &E < A
A ))) x (1 Colo (E)) x (1- T 293 A E E
+ 6p
otherwise,
775 293
where 5g
NO., 1) and 6p sid N(O,, 10). We consider the strain rate and temperature T as experimental variables. Other parameters are unknown material constants with the following priors:
E~ (73000,, 10000"), A ~ N3500, 100"), B ~ N6600, 200"),
C (eta(2, 10),
(2ta 5),
Beta(2, 5).
Figure 6 shows examples of three randomly generated datasets for = 0.1 and T = 300. For simplicity, evaluate five strain rates {0.001, 0.005, 0.01, 0.05, 0.1} for the same temperature 300, and then multiple temperatures (300, 400, 500, 600, 700} for the same strain rate 0.1. For the MCMC kernel, use proposal standard deviation equal to the prior standard deviations scaled to target the 0.234 acceptance rate, though no adaptation not multivariate Gaussian. Similarly to the coupled spring mass model in Section 5.1, we use 1000 datasets with 60 MCMC steps as well as the magnetic SMC steps.
the proposal to the prior is done since the prior is
Results Figure 7 displays the EIG curves resulting from forward SMC estimators with different population sizes compared to the EIG curve generated by the reverse-anneale SMC estimator with varying MCMC iterations when using both strain rate and temperature the design variable. Similarly to the spring mass model, we see that we can decrease the number of particles to without significant impact
the EIG curve in both cases. We also find that the reverse-annealed SMC estimator with 60 MCMC steps achieves good results. Moreover, the computation budget for either configuration is roughly equivalent to forward SMC with 10 particles, while yielding substantially more accurate EIG estimates.
Unlike with the spring-mast model, the reverse-annealed SMC estimator with 10 MCMC steps is not as heavily biased downward, providing better estimator than the 10-sample forward SMC estimator and requiring order of magnitude fewer likelihood evaluations. When using temperature the design variable, the EIG curve from the reverse-ad reverse-annealed SMC estimator matches the "true" EIG curve even better than when using strain rate the design. Unsurprisingly, standard error than reverse-annealed SMC with 60 MCMC steps, although the standard error lower in the temperature design setting than in the strain-rate design setting.
still higher
5.3 Source Finding
Finally, we apply our reverse-annealed SMC estimator to the source-localization problem described in Foster et al. (2021), in which two hidden sources emit a signal whose intensity decays according the inverse-square law. each step, sensor placed at a design location ç, records noisy measurement of the aggregate signal intensity. The objective is to sequentially choose sensor placements {E, JT

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Strain rate
Temperature
13
12
10
Likellhood eval evaluations
10a
10a
10a
10a
107
10
10
10
300
460
500
600
Design
RA-SMC, 10 MCMC steps RA-SMC, steps
Forward SMC, 250 particies
Forward SMC, 20 particies
Forward SMC,
particles
Figure 7: EIG and computational cost results different under two different choices design variable the Johnson-Cooe model. Left column: EIG and computational cost for five choices of strain rate at a fixed temperature 300. Reverse-annealed SMC yields accurate EIG curves with orders of magnitude fewer likelihod evaluation compared to forward SMC baselines. Right column: EIG and computational cost results for five choices of temperature fixed strain O.I. Here, forward SMC able effectively adapt the number of itcrations required at the higher temperatures. Error bars indicate two Monte Carlo standard errors.
efficiently infer the unknown source coordinates from the noisy observations {ye) The noiseless observation model is:
m( b+
m7 + 1 + E1I
where b, m are constants that control background and maximum signal, respectively, We observe the log intensity, and use the following prior and likelihood models:
Og Si
NOO, 12), log 0, (log ()).
Each observation is chosen by selecting: & = arg max; EIG(S 91---1, 11.--1). Once & is chosen, the observer records Vt, the posterior updated to p(0 V1.1s EI.e), and the design-measure-update cycle is repeated until terminal time T is reached.
In our experiments we choose m 10-4 and b 10 matching the values used in Foster et al. (2021), and we restrict observation locations to 20 x 20 grid covering the square domain [-3,3] [-3,3] 3). We perform T steps of sequential optimization using RA-SMC and forward SMC with 50 and 100 particles.
Results Figure shows the results of the fource-finding problem using reverse-annealed SMC and forward SMC with 50 and 100 particles. We find that reverse-annealed SMC not only produces tightly clustered sensor placements around the true sources, but does so with substantially less computation. Over 15 measurements, reverse-anneales SMC required 8.4 x 10 likelihood evaluations, whereas forward SMC consumed about 3.7 x 10t0 and 4.8 1000 evaluations for 50 and 100 particles, respectively.
Despite this reduction in evaluation cost, reverse-annealed SMC yields better convergence than 50-particle forward SMC and similar performance to 100-particle forward SMC. Figure 9 shows the KL divergence between the prior and the posterior at each iteration of the sequential design process. We see that RA-SMC performs similarly to 100-particle forward SMC, while 50-particle forward SMC gives erratic results.

---

RA-SMC
Forward SMC (50 particles)
Forward SMC (100 particles)
15
Intensity
6-
2
Ooservaation order
11
0
60
22
Figure 8: Side-by-side comparison of sequential sensor placements for source localization using reverse-annealed SMC (left), forward SMC with 50 particles (middle) and forward SMC with 100 particles (right). The background heatmap indicates the log-signal intensity over the domain, and each marker shows sensor location colored by its observation order (dark to light). Reverse annealed SMC and 100-particle SMC rapidly concentrate measurements near the true source positions, whereas 50-particle forward SMC exhibits more dispersed sampling pattem. The likelihood evaluations were 8,4 x 10 3.7 x 10 and 4.8 10 respectively.
5.4 Discussion and Limitations
reverse trajectory of RA-SMC has key limitations. It complicates adaptive MCMC proposals and tempering schedules, which often require population statis- tics. Poor MCMC mixing when annealing from peaked posterior toward the flatter prior can yield conservative IG estimates. Determining its suitability for problem is currently ad hoc, requiring preliminary runs to find effi- ciency gains. A robust diagnostic for this would therefore be valuable.
The
Da (p(0 y) || ((0))
5.5
5.0
KL Divergence
4.0
3.5
3.0
On the other hand, because RA-SMC initializes from true posterior sample, may be less susceptible to the particle degeneracy issues that hinder traditional multi- particle SMC methods in high dimensions. Future research could explore developing diagnostics to assess estimator suitability, incorporating more sophisticated MCMC ker- nels (e.g., Hamiltonian Monte Carlo (Neal, 2012) or the No-U-Turn sampler (Hoffman et al., 2014)), studying the method's scalability in high dimensions, devising adaptive strategies using multiple particles to improve both MCMC mixing and the selection of tempering levels, extending the framework to real-worke data by initializing the re- verse trajectory with approximate posterior draws from likelihood-free methods, and conducting more extensive comparison other estimators for BOED.
Forward SMC, 50 particles
2.5
Forward SMC, 100 particles RA-SMC, 60 MCMC steps
2.0
3 101112111415
Observation iteration
Figure 9: Evolution of the KL divergence over 15 sequential observations for forward SMC with 50 particles (red), forward SMC with 100 particles (orange), and reverse- annealed SMC with MCMC steps (blue). RA-SMC converges as smoothly and rapidly 100-particle forward SMC, while 50- particle forward SMC yields more erratic KL divergence estimates.
Acknowledgments and Disclosure of Funduud Funding
Sandia National Laboratories (SNL) is multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's 's National Nuclear Security Administration under contract DE-NA0003525. SAND#2025- 13387C. Initial research was supported by SNL Laboratory Directed Research and Development. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper not necessarily represent the views of the U.S. Department of Energy the United States Govermment.
10

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References
Beck, J., Dia, B. M., Espath, L. F., Long, Q. and Tempone, R. Fast bayesian experimental design: Laplace-based importance sampling for the expected information gain. Computer Methods in Applied Mechanics and Engineering, 334:523-553, 2018. ISSN 0045-7825, doi: https://doi.oo hrg.o/ooorr////////////
1016/j.cma.2018.01.055 URL hhttps://wwttt hwtp://wwwwncccccccctccccm com.:ettttcc.coeeeeeeeeeeeeeeeee wtt.c/ooo.cceeeeeeeeeccceceeeeee attpc///aricleepii.... $0045782518300616.
Blau, T., Bonilla, E. V., Chades, I., and Dezfouli, A. Optimizing sequential experimental design with deep reinforcement learning. In Chaudhuri, K., Jegelka, S., Song, L., Szepesvari, C., Niu, G., and Sabato, S. (eds.), Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pp. 2107-2128. PMLR, 17-23 Jul 2022. URL html.
https: http://pocceeen.....// mlr.
Calderhead, B. and Girolami, M. Estimating Bayes factors via thermodynamic integration and population MCMC. Computational Statistics & Data Analysis, 53(12):4028-44 4042, 2009.
Catanach, T. A. Computational methods for Bayesian inference in complex systems. California
Institute Technology, 2017.
Catanach, T. A. and Beck, J. L.. Bayesian updating and uncertainty quantification using sequential tem- pered MCMC with the rank-one modified Metropolis algorithm. arXiv preprint arXiv: 1804.88735 2018.
Chopin, N., Papaspiliopouloss O., et al. An introduction to sequential Monte Carlo, volume Springer, 2020.
Dahlke, C., Zheng. S., and Pacheco, J. Fast variational estimation of mutual information for implicit and explicit likelihood models. In International Conference on Artificial Intelligence and Statistics, pp. 10262-10278. PMLR, 2023.
Del Moral, P.. Doucet, A., and Jasra, A. Sequential monte carlo samplers. Journal of the Royal Statistical Society Series B: Statistical Methodology, 68(3):411- 1-436, 2006.
Foster, A., Jankowiak, M., Bingham, E., Horsfall, P., Tch, Y. W., Rainforth, T., and Goodman, N. Variational bayesian optimal experimental design. Advances in Neural Information Processing Systems, 32, 2019.
Foster, A., Jankowiak, M., O'Meara M., Teh, Y. W. and Rainforth, T. A unified stochastic gradient approach to designing bayesian-optimal experiments. In Chiappa, S. and Calandra, R. (eds.), Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine Learning Research, pp. 2959-2969, PMLR, 26-28 Aug 2020. URL https: http///poeeeennnee///eeeeeeeestt mml. press/ /aaess/vv00e v108/foster20e html.
Foster, A., Ivanova, D. R., Malik, I., and Rainforth, T. Deep adaptive design: Amortizing sequential bayesian experimental design. In Meila, M. and Zhang, T. (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 3384-3395. PMLR, 18-24 Jul 2021. URL https: httpt:tt/////ccediii v139/foster21a. html.
mlr.press.
Fourment, M., Magee, A. F., Whidden, C., Bilge, A., Matsen IV, F. A., and Minin, V. 19 dubious ways to compute the marginal likelihood phylogenetic tree topology. Systematic biology, 69 (2):209-220, 2020.
Friel, and Pettitt, A. N. Marginal likelihood estimation via power posteriors. Journal of the Royal Statistical Society Series B: Statistical Methodology, 70(3):589 -07,, 2008.
Gelman, and Meng, X.-I.. Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statistical science, pp. 163-185, 1998.
Gelman, A., Gilks, W. R., and Roberts, G. O. Weak convergence and optimal scaling of random walk metropolis algorithms. The annals of applied probability, (01):100-100, 1997.

---

Grosse, R. B., Ghahramani, Z., and Adams, R. P. Sandwiching the marginal likelihood us- ing bidirectional Monte Carlo, November 2015. URL http://arxxi prggabs/15111.11 02543. arXiv: 11.022543 [stat].
Grosse, R. B., Ancha, S., and Roy, D. M. Measuring the reliability of MCMC inference with bidirectional Monte Carlo. In Lee, D., Sugiyama, M., Luxburg, U., Guyon, and Garnett, R. (eds.). Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016. URL https:
pttppppp__iiieeepppp////////1 /pppe//011////1ll neurips. cttptippp.ccpppppeeee http/appr_fffeee pdf.
Hoffman, M. D., Gelman, A., et al. The No-U-Tum sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1):1593-1625 2014.
Huan, X. and Marzouk, Y. M. Sequential bayesian optimal experimental design via approximate dynamic programming, 2016. URL https: / arxiv. org/ /1044 1604.08320.
Ivanova, D. R., Foster, A., Kleinegesse, S., Gutmann, M. U., and Rainforth, T. Implicit deep adaptive design: Policy-based experimental design without likelihoods. In Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Advances in
Neural Information Processing Systems, volume 34, pp. 25785-25798. Curran Associates, Inc., 2021. URL https: htttps://proceedinggs www.nninnnnnnuuus...rr..mrr neurips. ctciiis...ppppppee cc/paper files/llllpaee wptpappepp//////////ee 0511000116654334447777711111112- Pdf...ddf.
Kleinegesse, S. and Gutmann, M. U. Bayesian experimental design for implicit models by mutual information neural estimation, 2020. URL. https:tp//// htwph:ttt///areee http://aaavvvvie/i
08129.
Kleinegesse, S., Drovandi, C., and Gutmann, M. U. Sequential Bayesian Experimental Design for Implicit Models via Mutual Information. Bayesian Analysis, 16(3):773 802, 2021. doi: 10.1214/20- 11//20-BA120 1255. URL https::/doi..rgg
1214/20-BA122
Lindley, D. V. On a measure of the information provided by an experiment. The Annals of Mathemat- ical Statistics, 27(4)9886- 1005, 1956.
Neal, R. M. Sampling from multimodal distributions using tempered transitions. Statistics and Com- puting, 6(4):353-366, December 1996. ISSN 0960-3174, 1573-1375. doi: 10.1007/BF00143555. URL http: //linkk httpp//s.isssssiiieger.... www/co.mocccccc 1007 BF0044355.
Neal, R. M. MCMC using Hamiltonian dynamics. arXiv preprint urXiv:1206.100 2012.
Pacheco, J. and Fisher, J. Variational information planning for sequential decision making. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 2028-2036 PMLR, 2019.
Rainforth, T., Cornish, R., Yang, H., Warrington, A.. and Wood, F. On nesting monte carlo estimators. In International Conference on Machine Learning, pp. 4267-4276. PMLR, 2018.
Rainforth, T., Foster, A., Ivanova, D. R., and Bickford Smith, F. Modern Bayesian experimental design. Statistical Science, 39(1):100-114, 2024.
Ryan, E. G., Drovandi, C. C., McGree, M., and Pettitt, A. N. A review of modern computational algorithms for bayesian optimal design. International Statistical Review, 84(1):12--154, 2016.
Ryan, K. J. Estimating expected information gains for experimental designs with application to the random fatigue-limit model. Journal of Computational and Graphical Statistics, 12(3):585-603 2003.
Xie, W., Lewis, P. O., Fan, Y., Kuo, L., and Chen, M.-H. Improving marginal likelihood estimation for Bayesian phylogenetic model selection. Systematic biology, 60(2):150 150-160, 2011.
Young, D. and Gregory, R. A Survey of Numerical Mathematics, Volume I. Dover Books on Mathematics Series. Dover Publications, Incorporated, 2012. ISBN 9780486145051. URL https://booke. google. e com comm/books?iddgosAEYreee
Zheng. S., Pacheco, J., and Fisher, J. A robust approach to sequential information theoretie planning. In International Conference on Machine Learning, pp. 5941-5949, PMLR, 2018.
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A Appendix
A.1 Proofs and additional theory
Proof of Lemma 4.2.
Proof. By the linearity of expectations,
EllogNN.M d)] (x( (y. d), 1 (y.d)...... ((M) (
(v: d))
At
M( (d) // (d)
(c)
Σ (d) (d)
E
evvnn
N
N-
At
E
( (d) + (d) + 4 0
E ( (d) + 2 Σ
ECC m0 (d))
2=
odd
even
For each i, we have
B (d) = Ep,, [logp(y 0,d] = E(t.. d).
lim
M +00
so
Al
N-1
N-2
Jim M M0000
EV d))
((lo. E(IN, W. d) +
Σ
(l1,y,d) d) + Σ (1
3
odd
ivven
= QQ.
When the fourth derivative is bounded, the composite Simpson rule satisfies the remainder bound:
QN f. (e((,u,)) di O)N
so
lim QN ['" ((t, (tt,,a d) d) [' Eonp. logp(y d.d dt log)(y).
Hence,
lim PN,M( log p(y d). N.M
Finally, observe that
lim N.M.K
E E.GGG,,MK(()
lim E K0000
logp(uk Oz. d) lim (NM((and) N.M
lim ((. logpíy (0,d) k0000
lim (Vp
N,M++
(.. [logply 0, log p(y d)] EIG(d).
Thus, the RA-SMC estimator of EIG(d) is asymptotically unbiased.
Corollary A.1. Under the same conditions Lemma 4.2, the RA-SMC estimator is asymptotically unbiased for geometrically spaced temperature levels t, = (uy),, where = j/N are uniformly spaced points in [0, 1] and is constant.
Proof. We use a standard change of variables. The integral for log p(y d) is transformed:
[. A(u y, d)
S/'' e(t, d)dt = log p(y d).
13

---

The estimator for log p(y d) is then constructed by applying the composite Simpson's 's rule (-) (with uniform step Au 1/N) to MCMC-based estimates 9) g) (y, d) of the transformed integrand g(usy, d). Specifically,
of c( (v.d).
where () (y. d) is the MCMC estimate of E((u,) ( y.d).
The asymptotic unbiasedness argument then applies directly to this estimation of So g(, y, didu because of the following two facts. First, the transformed integrand g(u, y, d) is four-times continu- ously differentiable on [0, 1]. Second, the MCMC estimates 9) () (y. d) are asymptotically unbiased in expectation for g d) as M > oc: observe that
lim Elg (y. M-000
lim Bieu 100 (v. M -0000
Elcu--1 e(t,, y, d)]
= Elg(us y. d)}.
Under these conditions on the transformed problem, the same O convergence for the Simpson's 's rule discretization error (for g(u)) and the vanishing MCMC error ensure the asymptotic unbiasedness for estimating log logp(y d) with geometrically spaced temperatures.
Proof of Equation 8.
Proof. Observe that
logp(y) = log(zz (v)) log(=o(v))
S/' d
lo)( di
dt
z((y) dt (y) dt
1
p(y 0)' 0))p(0) do dt
()(
di
1
L. d
| ply p))p(())))))))))))))))))))))(p)) do dt
1 20(1) / logg(y 0)p(y (0) do
[- / logr(y 0) Ply (())(()) do dt. (x0
as desired.
A.2
Additional Results
A.2.1 Inner-loop variance outer-loop variance
To validate that Var(IG(y) vv) Var()G(y)), draw 100 datasets Vi, i = 1,..., 100 from the spring mass observation model in Sec 5.1. For each these datasets, we repeatedly compute IG(y;) thirty times using forward SMC with 250 particles. Then, we compute Var(IG(y) y y:) across the thirty replicates of IG();) and compare the estimator variance for given dataset that dataset's value of yy) Figure 10 displays this comparison for all 100 datasets. It is clear that the inner-loop
estimator variance (I) v) is much smaller than I IG().
14

---

Within-dataset variance vs IG
10a
VarlIGly) y)
10
IG(y)
10-1
0
20
40
60
80
100
Dataset index
Figure 10: A comparison between Var(TG(y) y), the estimator variance from computing IG(y) for a single dataset, and the value of IG). We see that Var(IG[y) y) is on average two orders of magnitude smaller than IGv)).
A.2.2 Choosing the tempering levels
There is balance between the number of temperature levels and number of MCMC iterations because more temperatures means that the difference between subsequent distributions is smaller; fewer MCMC iterations should then be required achieve convergence. Assuming the number of levels is sufficiently large to calculate the thermodynamic integral, it is unclear whether, for example, 100 levels 100 iterations is generally better than 200 levels of 50 iterations.
Figure 11 demonstrates how the absolute error changes for the model Section 5.1 as we vary these parameters when compared to gold standard forward SMC run, and it appears that there is significant difference past a certain threshold.
6.66
3.65
2.16
1.36
1.21
0.89
-0.65
-..30
0.32
4.79
2.01
-0.93
0.51 0.47 0.62 -0.8 0.22 0.17
60
-4,80
2.54
-1.27
-0.99
0.81
-0.79
-0.66
-0.30 -0.14
-2.84
..63
-0066
0.06
-0.13
0.21
0.19 0.06 -002
80
-3.92
2.10
-0.96
-0.86
0071
-0.35
-0.40
-0.29 0.02
-1.75
-0.06
0.30
0.07
0.24
-0.00 0.03 0.02 0.05
tempering
1.00
1.777
-1.06
0.46
0.10
120
-3.02
-1.76
-1.04
-0.73
-045
-0.41
-0.29
-0.08 -0.02
-0.24
1.13
0.61
0.35
0.50
0.07
0.06 0.09
0.04
140
-2.63
-1.48
-0.79
-0.88
0.37
0.35 0.07
-0.11
0.01
0.26
1.26
0.79 0.57 0.43 0.32
0.29 0.22 0.05
10
20
40
60
80
100
150
200
300
10
20
40
60
80
100
150 200 300
MCMC steps
Figure 11: Left: Absolute error of EIG computed with the standard reverse anneeled SMC, with downward bias trend elearly visible when levels steps are too low. Right: Error of magnetized evverse-anccale SMC. Values are in comparison to long 250 particle forward SMC estimate, with Monte Carlo standard errors around 0.3.
A.3 Basic reverse-annealed SMC
We also run basic version of our estimator with only MCMC iterations for each temperature and no further adaptivity, shown in Figure 12. As expected, this yields underestimates of the EIG. However, the resulting EIG curve can still be used for BOED as the bias is sufficiently similar across designs. The curve is also still relatively smooth due to the lack of degenerate samples increasing the variance. Even if accurate EIG estimates are required, this could be used to enable coarse first pass to narrow the search space of designs.
15

---

Likelihood evaluations
109
AA SMC 60 MCNC stops
Forward SMC 250 particess
10a
PASMC 10 MCNC staps
14-
13
12
10a
0.2.0.0..6088 1.2 1.4 1.6 1.200 Design (y)
0.20 020.40803310222.1111.8888
Design (y)
Figure 12: Comparison of results using 10 iterations of MCMC per temperature for reverse-annealed SMC on the spring-mass model. The EIGs clearly underestimated but still form usable for BOED.
One caveat is that the reverse-annealed SMC has around a 15% higher Monte Carlo standard error, 0.167 vs. 0.145 for forward SMC. This is not surprising since we are increasing Var(IG(v) 3), and so it is reasonable say that cost of obtaining EIG estimates with the same precision as forward SMC in this case would require about 33% more outer-loop samples than were run in our simulations. However, our main takeaways do change, as an additional 33% cost is small compared reduction in inner-loop likelihood evaluations.
16

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction accurately state that the paper's primary con- tribution is novel EIG estimation method achieving lower computational cost for similar performance across a various BOED problems. These claims are substantiated in the paper's empirical results from several different BOED problems and are supported by thorough methodological discussion.
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The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
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Answer: [Yes]
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: The proofs for the theoretical results are found in Appendix A.I.
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Reviews:
[{"review_id": "oKNyTu9CXz", "reviewer": "Reviewer_LaTk", "summary": "This paper proposes an algorithm for the estimation of the expected information gain (EIG) for Bayesian optimal experimental design via sequential Monte Carlo (SMC). One of the main quantities of interest for Bayesian experimental design algorithms is the marginal likelihood $p(y | d) = \\int p(y|\\theta, d) p(\\theta) d\\theta$ under a given design $d$.  The paper derives a reverse annealing scheme which samples $(y, \\theta_*) \\sim p(y, \\theta)$ from the joint distribution treating the initial $\\theta_* \\sim p(\\theta|y)$ effectively as a posterior sample, and guides the sampling towards the prior by decreasing a temperature parameter, while keeping track of the samples likelihoods. An estimate of the model evidence is them computable via a thermodynamic integral using Simpson's rule. Synthetic experiments evaluate the method against another SMC baseline.", "questions": "* In Definition 4.1, shouldn't the samples be $\\{y_k, \\theta_k)\\}_{k=1}^K$? Currently, $d_k$ is in the sample set.", "limitations": "Limitations are discussed in Sec. 5.4.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n* The paper is mostly well written.\n* The methodology is based on interesting insights which might be useful for the design of other SMC-based EIG estimation algorithms.\n* The algorithm seems to be relatively simple to implement compared to existing alternatives.\n* Theoretical guarantees are provided on the asymptotic unbiasedness of the algorithm.\n\n### Weaknesses\n* The experimental evaluation involves a single SMC baseline, missing comparisons against other EIG estimators (e.g., Foster et al., 2019). Hence, it's hard to assess the significance of the contribution with respect to other methods for BOED.\n* It is not clearly explained how MCMC is applied, even though it's a fundamental part of the algorithm.\n* Eq. 8 might need further detail on its derivation, as it is not immediately clear that the integral on the right-hand side yields the log-marginal likelihood.\n* I missed a reference for background on Simpson's rule.", "was_revised": true, "final_justification": "I've raised my score from 3 to 5 after clarifications and discussions from the rebuttal and other reviews. I'd encourage the authors to include the new experimental results and at least mention them in the main paper, add further details on the derivation of Eq. 8, provide clear references for Simpson's rule, and clarify the use of MCMC within the proposed SMC method, as discussed.", "rebuttal": "We are very grateful to Reviewer LaTK for the detailed feedback and for their positive comments on the paper's writing and novelty.\nWe appreciate the constructive suggestions, which we address below.\n\n1. > The experimental evaluation involves a single SMC baseline, missing comparisons against other EIG estimators ... it’s hard to assess the significance of the contribution with respect to other methods for BOED.\n\nWe agree that positioning RA-SMC against a broader set of estimators would strengthen the paper. \nOur research here is focused on improving the computational efficiency of asymptotically-exact Monte Carlo estimators. \nWithin this class of methods, we found that simpler approaches like Nested Monte Carlo can struggle on problems with complex posteriors. \nTo address the reviewer's point under the time constraints of the rebuttal period and to verify that these problems are indeed challenging for non-tempering methods, we have run a **new experiment during the rebuttal period using the Prior Contrastive Estimation estimator on the spring-mass model** as suggested by Reviewer Z8Ri:\n\n|Design|0.2|0.4|0.6|0.8|1.0|1.2|1.4|1.6|1.8|2.0|\n|------|---|---|---|---|---|---|---|---|---|---|\n|EIG - PCE   |10.205|10.600|10.487|10.469|10.531|10.443|10.343|10.166|10.223|10.121|\n|SE - PCE |0.070 |0.061 |0.063 |0.074 |0.026 |0.044 |0.068 |0.063 |0.048 |0.077|\n|EIG - Forward SMC | 16.065 | 18.455| 20.269| 19.339|19.034|19.648|18.439|18.884|17.349|16.755|\n\nAs shown in the table, the PCE estimates are stable but exhibit a severe downward bias, likely due to the highly informative data in these problems, failing to capture the true EIG or find the optimal design. The failure of both simpler baselines, NMC (as shown in Appendix A.3) and now PCE, provides evidence that these are nontrivial problems that require more complex estimators. \n \nOur contribution is therefore an order-of-magnitude efficiency gain for a powerful tool within this specific class of estimators. With that in mind, comparing RA-SMC to a strong forward SMC is the most logical and direct benchmark for our central claim. However, we agree that other classes of estimators, such as those based on variational inference or neural networks, offer a different set of compelling tradeoffs and a comprehensive comparison across these estimators is an excellent direction for follow-up work. We have **added the new PCE results to the table in Appendix A.3,** and we have **clarified the scope of our contribution in Section 2 (Related Work)** by adding the following paragraph:\n\n\"The methods described above present a diverse set of trade-offs between computational cost, scalability, and asymptotic guarantees. This work specifically focuses on improving the efficiency of tempering-based Monte Carlo estimators, like AIS and SMC, which provide asymptotically-exact estimates crucial for high-fidelity applications, and we show that the RA-SMC estimator makes this powerful but expensive class of methods more practical. To that end, we first review the standard SMC framework, which forms the basis of our proposed approach.\"\n\n2. > It is not clearly explained how MCMC is applied...\n\nWe apologize for the lack of clarity here.\nAt each step in the reverse annealing schedule,\nan MCMC kernel is used to generate a new sample from the current\ntempered distribution using the sample from the previous (higher temperature)\ndistribution as its starting point.\nThis provides the mechanism whereby $E_{\\theta \\sim p_{t_i}}\\log p(y \\mid \\theta)$ is calculated for each $i = 0, \\dots, N$.\n**We will add a short discussion in Section 4.2 to explicitly describe this process.**\n\n3. > Eq. 8 might need further detail on its derivation...\n\nWe agree that a short derivation would improve clarity.\nThe identity follows from the Fundamental Theorem of Calculus, which we include below for convenience.\n\n\\begin{aligned}\n    \\log p(y) \n    &= \\log p(y) - \\log (1)\n    \\\\\\\\\n    &= \\log (z_1(y)) - \\log (z_0(y))\n    \\\\\\\\\n    &= \\int_0^1 \\frac{d}{dt} \\log(z_t(y)) \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\frac{d}{dt} z_t(y) \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\frac{d}{dt} \\int_{\\theta}p(y \\mid \\theta)^t p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\int_{\\theta} \\frac{d}{dt} p(y \\mid \\theta)^t p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\int_{\\theta} \\log p(y \\mid \\theta) p(y \\mid \\theta)^t   p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\int_{\\theta} \\log p(y \\mid \\theta) \\frac{p(y \\mid \\theta)^t   p(\\theta)}{z_t(y)}\\,d\\theta \\,dt.\n\\end{aligned}\n\n**We have added this derivation to Appendix A.1 and have added a reference to \"Simulating Normalizing Constants: From Importance Sampling to Bridge\nSampling to Path Sampling\" by Gelman and Meng (1998) when introducing Eq. 8.**\n\n4. > I missed a reference for background on Simpson's rule.\n\nYes, thank you for catching this omission. **We have added citation for numerical integration methods** on line 158 where we first mention Simpson's rule.\n\n5. > In Definition 4.1, shouldn't the samples be $(y_k, \\theta_k)_{k=1}^K$? Currently, $d_k$ is in the sample set.\n\nYes, this is a typo. Thank you for pointing it out. We have corrected it in the manuscript.\n\n6. > ``acceptance rate using the feedback controller of (Catanach, 2017)'' -> in-text citation should not be within parentheses.\n\nWe have corrected this in the manuscript."}, {"review_id": "z6oqzczUa7", "reviewer": "Reviewer_d56d", "summary": "The manuscript proposes reverse--annealed sequential Monte Carlo for estimating the expected information gain (EIG) that drives Bayesian optimal experimental design.  The standard tempering route—used by sequential Monte Carlo samplers (Chopin 2002; Del Moral et al. 2006) and by many BOED works—starts with the prior and anneals forward toward the posterior with many particles.  Contrary to these forward tempering schemes, RA--SMC begins with an exact posterior draw obtainable in simulation--based design, then anneals backwards towards the prior while accumulating the log--evidence by a thermodynamics inspired integral.  The authors prove the estimator is asymptotically unbiased and demonstrate, on 4--6 dimensional case studies, that RA--SMC can match the accuracy of forward SMC using two orders of magnitude fewer likelihood evaluations.", "questions": "As far as I can understand, RA--SMC relies on the availability of an exact posterior sample, limiting applicability to fully simulator--based design; real--data or sequential--design contexts without simulators are excluded?\n\n Can likelihood--free posterior draws relax the ``exact posterior'' requirement and open RA--SMC to real--data?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n\nThe concept is elegant and practical.  It turns the necessity of prior sampling into an advantage by bootstrapping from a posterior sample naturally available in simulator settings. The theoretical development is solid. Lemma~4.2 establishes consistency under standard ergodicity conditions, and the temperature--schedule corollary offers practical guidance. The empirical study is careful. Each benchmark reports Monte Carlo error bars and raw likelihood budgets, allowing a transparent cost/accuracy comparison. The variance--decomposition analysis strongly justifies the single--particle regime, and the discussion of diagnostic uncertainty is refreshingly candid.\n\nPrior work treats the posterior normalising constant Z(1) as unknown and therefore anneals forward. In BOED the authors observe that a joint prior draw  yields a known posterior density  p(θ∣y), so the reverse normalising constant  Z(0)=1 is known instead. Turning this observation into a practical estimator is, to my knowledge, new.\n\nWeaknesses\n\nAll experiments remain in low--to--moderate dimensions with smooth likelihoods. Scalability to high--dimensional, multimodal or PDE--constrained models is not demonstrated.\n\nBaseline coverage is narrow: annealed importance sampling, bidirectional Monte Carlo bounds, variational MI estimators and neural mutual--information surrogates are omitted despite being cited.\n\nRuntime is measured only in likelihood calls; wall--clock timings and parallel scalability are not evaluated.", "was_revised": true, "final_justification": "Thanks to the authors for patiently addressing questions. I am satisfied with the clarifications provided and maintain my original score: Accept.", "rebuttal": "We sincerely thank the reviewer for their positive assessment and insightful comments, which we address below.\n\n1. >All experiments remain in low–to–moderate dimensions with smooth likelihoods. Scalability to high–dimensional, multimodal or PDE–constrained models is not demonstrated.\n\n\nThe reviewer is correct that we focused on low-to-moderate dimensional problems.\nThis was a deliberate choice to first establish the methodology and test its efficiency on problems known to be challenging due to their multimodal posterior distributions, as demonstrated for the spring-mass system (see Figure 3). \nWe agree that demonstrating scalability to high-dimensional settings is an important step for future research. We are optimistic about our method's potential in this area and have **added the following to the conclusion to highlight high-dimensional scalability as a direction for future work:**\n\n\"Because RA-SMC initializes from a true posterior sample, it may be less susceptible to the particle degeneracy issues that hinder traditional multi-particle SMC methods in high dimensions. Future research could explore ... studying the method's scalability in high dimensions.\"\n\n\n2. >Baseline coverage is narrow.\n\nWe thank the reviewer for this point, which was also raised by others. To strengthen the paper in line with this feedback, we have **run a new experiment during the rebuttal period comparing RA-SMC against Prior Contrastive Estimation on the spring-mass model** as suggested by Reviewer Z8Ri. Our results show that PCE, like standard Nested Monte Carlo, exhibits severe downward bias and is not effective on these challenging problems:\n\n|Design|0.2|0.4|0.6|0.8|1.0|1.2|1.4|1.6|1.8|2.0|\n|------|---|---|---|---|---|---|---|---|---|---|\n|EIG - PCE   |10.205|10.600|10.487|10.469|10.531|10.443|10.343|10.166|10.223|10.121|\n|SE - PCE |0.070 |0.061 |0.063 |0.074 |0.026 |0.044 |0.068 |0.063 |0.048 |0.077|\n|EIG - Forward SMC | 16.065 | 18.455| 20.269| 19.339|19.034|19.648|18.439|18.884|17.349|16.755|\n\nWe chose forward SMC as our primary baseline because it represents the current state-of-the-art for asymptotically-exact EIG estimation in BOED. Our core contribution is showing that RA-SMC achieves comparable accuracy to this gold-standard method while requiring orders of magnitude fewer likelihood evaluations.\n\nA comprehensive comparison across other EIG estimators based on, e.g., variational inference or neural networks represents an excellent direction for future work. We have **added these new results to the appendix.**\n\n\n3. >Runtime is measured only in likelihood calls; wall–clock timings and parallel scalability are not evaluated.\n\nWe agree that parallel scalability is a valid concern. RA-SMC's single-particle nature limits parallelization compared to multi-particle forward SMC, which makes parallel scalability analysis less relevant for this method. However, we agree it would be good to clarify this in the manuscript. We report likelihood evaluations rather than wall-clock time since likelihood evaluation typically dominates runtime by orders of magnitude in simulation-based design. This provides a hardware-agnostic measure that directly translates to computational cost in any simulation-based BOED application, regardless of the specific forward model or computing environment.\n\n4. >Does RA-SMC's reliance on exact posterior samples limit it to simulator-based design, excluding real-data contexts? Can likelihood-free posterior draws relax this requirement?\n\nCurrently the RA-SMC estimator is limited to simulator-based problems.\nThe question of whether likelihood-free methods can relax this requirement is very interesting. In principle, yes. If we can obtain approximate posterior samples, via for example approximate Bayesian computation (ABC), these could potentially serve as starting points for RA-SMC. \n\nWe did some preliminary study on the efficacy of obtaining ABC samples for a different context (extending the method to utilize adaptive tempering).  Our preliminary experiments suggested that the approximation quality\nof the initial posterior samples directly affected RA-SMC's accuracy,\nand the efficiency gains were much less dramatic when accounting for the cost of obtaining the approximate posterior sample. However, our evaluations are still limited, so we are interested in studying this question further and have **added it as a direction for future work**."}, {"review_id": "K2TaAqnDgQ", "reviewer": "Reviewer_2iaK", "summary": "This paper introduces a novel approach called Reverse-Annealed Sequential Monte Carlo (RA-SMC) to estimate the Expected Information Gain (EIG) in Bayesian Optimal Experimental Design (BOED). The method leverages the specific characteristics of BOED to achieve efficient estimation, requiring only a single sample (particle) to provide a good approximation. The paper also provides a comprehensive theoretical foundation for the proposed method and demonstrates its efficacy on various models, showing significant computational savings compared to traditional methods.", "questions": "1. Could you provide a formal proof or further justification for the validity of Formula (8)? This would address any concerns about the theoretical foundation of the method.\n\n2. While the paper demonstrates the sufficiency of using a single particle, could you explore and discuss the potential use of multiple particles (perhaps inspired by particle filters) to further enhance the performance of RA-SMC, especially for complex or multimodal posteriors?\n\n3. Given that \"Sequential Monte Carlo\" is widely used in the context of filtering, would it be possible to reconsider the use of this term for the proposed method? Alternatively, can you provide a clearer distinction between your method and traditional filtering-based SMC techniques?\n\n4. It would strengthen the paper to include a comparison with other modern methods for EIG estimation in BOED. This could help further validate the efficiency and accuracy of RA-SMC.", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. The idea of using reverse annealing for EIG estimation in BOED is highly innovative. The authors effectively exploit the characteristics of BOED, such as known underlying parameters and the goal of estimating EIG, to propose a method that drastically reduces the computational cost.\n\n2. The paper provides a solid theoretical basis for RA-SMC, demonstrating that a single particle can effectively estimate EIG. The proofs and discussions are well-detailed.\n\n3. The method is shown to significantly reduce the number of likelihood evaluations required, offering computational efficiency without sacrificing accuracy. This is demonstrated across several models.\n\n4. The paper is clearly written, with a logical flow from introducing the problem to detailing the RA-SMC approach and providing empirical results. The experimental setup is easy to follow, and the comparison with the forward SMC method is well-executed.\n\nWeaknesses:\n\n1. Eq. (8) is fundamental to the method's theoretical framework, but its validity is not immediately clear, and the authors have not provided a proof. A formal proof or additional justification for this formula would strengthen the theoretical rigor of the paper.\n\n2. While the paper demonstrates that a single particle is sufficient for RA-SMC, there is limited discussion on the potential benefits of using multiple particles. It would be useful to explore whether using multiple particles (similar to particle filter methods) could improve performance, especially in cases with high posterior variance or complex models.\n\n3. The term \"Sequential Monte Carlo (SMC)\" is widely used in various fields, particularly in filtering and dynamic systems. While this term may be appropriate within the context of your work, I would like to point out that this could lead to some confusion for readers coming from other areas. It may be helpful to clearly define the SMC terminology in the context of your work, or alternatively, consider a different name to distinguish your approach from the SMC methods for filtering used in other domains.\n\n4. The paper only compares RA-SMC with forward SMC, but it would be more informative to include comparisons with other state-of-the-art methods for EIG estimation.", "was_revised": true, "final_justification": "Thank authors for their thorough and thoughtful responses. I am satisfied with the clarifications provided and maintain my original score: Strong Accept.", "rebuttal": "We sincerely thank the reviewer for their positive evaluation, high confidence, and insightful feedback. We are encouraged by their assessment and appreciate the constructive suggestions for improvement. We address each point below.\n\n1. > Could you provide a formal proof or further justification for the validity of Formula (8)? \n\nWe agree that a short derivation would improve clarity.\nThe identity follows from the Fundamental Theorem of Calculus, which we include below for convenience.\n\\begin{aligned}\n    \\log p(y) \n    &= \\log p(y) - \\log (1)\n    \\\\\\\\\n    &= \\log (z_1(y)) - \\log (z_0(y))\n    \\\\\\\\\n    &= \\int_0^1 \\frac{d}{dt} \\log(z_t(y)) \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\frac{d}{dt} z_t(y) \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\frac{d}{dt} \\int_{\\theta}p(y \\mid \\theta)^t p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\int_{\\theta} \\frac{d}{dt} p(y \\mid \\theta)^t p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\frac{1}{z_t(y)} \\int_{\\theta} \\log p(y \\mid \\theta) p(y \\mid \\theta)^t   p(\\theta)\\,d\\theta \\,dt\n    \\\\\\\\\n    &= \\int_0^1 \\int_{\\theta} \\log p(y \\mid \\theta) \\frac{p(y \\mid \\theta)^t   p(\\theta)}{z_t(y)}\\,d\\theta \\,dt.\n\\end{aligned}\n\n**We have added this derivation to Appendix A.1 and added a reference to \"Simulating Normalizing Constants: From Importance Sampling to Bridge\nSampling to Path Sampling\" by Gelman and Meng (1998) when introducing Eq. 8.**\n\n2. > Could you explore and discuss the potential use of multiple particles (perhaps inspired by particle filters) to further enhance the performance of RA-SMC?\n\nThis is a great question! We did perform preliminary studies on using multiple particles, primarily as a means to enable adaptive tempering schedules, which require population statistics.\nThe difficulty arises because the initial $\\theta$ sample is generated as a joint sample with $y$, which makes generating additional samples from the same posterior $p(\\theta \\mid y)$ non-trivial. \nWe explored using approximate Bayesian computation (ABC) or MCMC to generate further samples from the same posterior as the initial sample in the path, \nbut our preliminary results suggested that the additional computational cost and complexity of doing this significantly diminished the efficiency gains of the single particle method. \n\nHowever, we agree this is an important point worth discussing. We have **added a note to the conclusion** acknowledging the multi-particle extension as a direction for future work, while noting that the single-particle approach seems to hit a practical sweet spot in terms of accuracy and efficiency for the problems studied. \n\n3. >  Given that \"Sequential Monte Carlo\" is widely used in the context of filtering, would it be possible to reconsider the use of this term for the proposed method? Alternatively, can you provide a clearer distinction between your method and traditional filtering-based SMC techniques?\n\nWe thank the reviewer for raising this point about terminology, as the nomenclature in this area can be ambiguous and we considered this carefully during our work.\nThe core idea of sampling through a sequence of tempered distributions is variously referred to as 'Annealed Importance Sampling' [3], 'Sequential Tempered MCMC' [1], and is related to the 'stepping-stone' algorithm [4]. \n\nWe ultimately chose the 'Sequential Monte Carlo' framework because we follow the convention of [2], which treats tempering as a primary application of the general SMC methodology. We feel this best positions our work within the broader class of modern Monte Carlo samplers.\n\nHowever, we agree that this should be made more explicit. To this end we have **added the following sentence the second paragraph of Section 2:**\n\n\"We note here that in this work, 'sequential' refers to the progression of particles through a sequence of tempered distributions, as distinct from the temporal sequence in state-space and filtering applications.\"\n\n4. > It would strengthen the paper to include a comparison with other modern methods for EIG estimation in BOED.\n\nWe thank the reviewer for this important point. To begin addressing this, we have already run a new experiment during the rebuttal period comparing our method to a Prior Contrastive Estimation (PCE) baseline on the spring-mass model (as suggested by Reviewer Z8Ri), which confirmed the need for robust samplers for these problems. These results as well as a lengthier discussion on this point can be found in the rebuttal to Reviewer LaTK. We agree that a more comprehensive benchmark is a critical direction for future research, and we will highlight this explicitly in the conclusion of the final paper.\n\n\n[1] Catanach, T. A. and Beck, J. L. Bayesian updating and uncertainty quantification using sequential tem-363\npered MCMC with the rank-one modified Metropolis algorithm. arXiv preprint arXiv:1804.08738,364\n2018.\n\n[2] Chopin, N., Papaspiliopoulos, O., et al. An introduction to sequential Monte Carlo, volume 4.366\nSpringer, 2020.367\n\n[3] Neal, R. M. Annealed importance sampling. Statistics and Computing, 11(2):125–139, April423\n2001. ISSN 1573-1375. doi: 10.1023/A:1008923215028\n\n[4] Xie, W., Lewis, P. O., Fan, Y., Kuo, L., and Chen, M.-H. Improving marginal likelihood estimation439\nfor Bayesian phylogenetic model selection. Systematic biology, 60(2):150–160, 2011."}, {"review_id": "LkrGsd88G3", "reviewer": "Reviewer_Z8Ri", "summary": "- EIG estimation is all about computing $E_{y\\sim p(y)}[\\log p(y)]$\n- The variance over $y$ can be as significant as the variance/error coming from the estimate of the partition function\n- A small number of inner particles can be sufficient\n- In fact we can use 1 particle with an improved backwards SMC method, by observing that we get exactly 1 free sample of the posterior since EIG estimation uses data simulated from the current prior\n- the resulting method gives good EIG estimates at a lower likelihood evaluation budget compared to earlier SMC-based approaches", "questions": "- what about EIG gradients w.r.t. $d$? This would be an interesting angle to study as I don't believe it has been attempted using SMC before (though I could be wrong)\n - \"Magnetic reverse-annealed SMC steps\" this sounds like an interesting addition, but could that not be a general part of the algorithm? Were the results for the mass-spring system significantly worse without this amendment?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# Strengths\n - the paper reads very well, it clearly lays out the overall story in a logical manner\n - the idea leads to a more efficient algorithm\n - I believe this approach to be novel\n - theoretical analysis in the form of an asymptotic unbiasedness result is provided\n - clear demonstration in the experiments that this approach requires fewer likelihood evaluations to achieve good quality results compared to some earlier SMC-based algorithms for EIG optimization\n\n# Weaknesses\nThe biggest shortfalls are things that weren't in the paper\n - my question with SMC approaches for BOED is always how they will scale with dimensionality. As far as I can see, scaling with dimensionality of $\\theta$ was not studied here. I think this could be worthy of investigation. In fact, the fact that you are utilizing a single particle initialized from your true theta sample gives me some hope that this might be more scalable than other SMC approaches. I think adding this would have made the paper even better\n - you only compare against other SMC approaches except in Appendix A.3 where you study NMC. I would definitely be interested to see Prior Contrastive Estimation (PCE) https://arxiv.org/pdf/1911.00294 eq (12) on the table in A.3 as well-- it can be a deceptively good baseline for virtually the same cost as NMC. Ideally it would be added to the main plots as well. My motivation to suggest this is that you make an argument that some error in $\\log p(y)$ is fine because of the variance over $y$. In that case, maybe we can start to tolerate these even cruder approximations?", "was_revised": true, "final_justification": "Thank you to the authors for their thorough response to my questions. My score is maintained as \"Accept\"", "rebuttal": "We sincerely thank Reviewer Z8Ri for their thoughtful review and for the insightful questions and suggestions. We were particularly encouraged by the reviewer's optimism about our method's potential.\n\n1. > ...how they will scale with dimensionality...was not studied here.\n\nWe agree that scalability is an important question, and we thank the reviewer for raising this point. Like the reviewer, we **are optimistic about RA-SMC's potential for scalability.** Since we are initializing from a true posterior we are optimistic that we can avoid particle degeneracy issues that hinder traditional multi-particle SMC methods in high dimensions. \n\nOur primary goal in this work was to first establish the core methodology and validate its efficiency on problems known to be challenging due to complex, multimodal posteriors, rather than high dimensionality. A full investigation into dimensionality scaling is an excellent direction for a follow-up study.\n\n2. > ...only compare against other SMC approaches ... I would definitely be interested to see Prior Contrastive Estimation (PCE) on the table in .3 as well...maybe we can start to tolerate these even cruder approximations?\n\nWe thank the reviewer for this suggestion! To address it directly, we **ran a new experiment during the rebuttal period comparing our method to Prior Contrastive Estimation (PCE)** on the spring mass problem. \nOur results show that for these problems, the bias from PCE is too severe to be useful.\n\n|Design|0.2|0.4|0.6|0.8|1.0|1.2|1.4|1.6|1.8|2.0|\n|------|---|---|---|---|---|---|---|---|---|---|\n|EIG - PCE   |10.205|10.600|10.487|10.469|10.531|10.443|10.343|10.166|10.223|10.121|\n|SE - PCE |0.070 |0.061 |0.063 |0.074 |0.026 |0.044 |0.068 |0.063 |0.048 |0.077|\n|EIG - Forward SMC | 16.065 | 18.455| 20.269| 19.339|19.034|19.648|18.439|18.884|17.349|16.755|\n\nAs shown in the table, PCE consistently severely underestimates the EIG and fails to identify the optimal design around $\\gamma = 0.6$. While it performs better than NMC, PCE also demonstrates the need for more powerful estimators.\n\n3. > what about EIG gradients w.r.t. $d$?\n\nWe thank the reviewer for this interesting suggestion. We agree that this would be very interesting to study and to our knowledge it hasn't been studied in the context of SMC samplers before.\n\n\n4. > Magnetic reverse-annealed SMC steps\n\nThe ``magnetic'' steps represent an auxiliary enhancement we explored during development where we nudge the sampler toward the prior mean between temperature levels to potentially aid mixing.\nWhile we wanted to demonstrate another opportunity to take advantage of the unique reverse annealing setting to improve efficiency, we view it as separate from the core algorithm because it's not necessary for good performance (as evidenced by our source-finding experiment where we omitted it entirely) and it complicates the theoretical analysis. The spring-mass results were not significantly worse without the magnetic steps, but we did see a small improvement.\n**We will clarify in the revision that these steps are an optional enhancement rather than a core component of the method.** The fundamental RA-SMC algorithm, starting from an exact posterior sample and tempering backwards using standard MCMC, represents our primary contribution."}]

Author rebuttal:

Meta-review: This paper presents an interesting and novel approach to efficiently estimating EIG (within Bayesian experimental design) using Annealed Sequential Monte Carlo. All reviewers agreed on the significance and novelty of the work. I suggest that the authors incorporate the additional experiments and clarifications raised during the discussions with reviewers in the rebuttal into the final version of the paper.
Decision: Accept (poster)