Reverse Diffusion Sequential Monte Carlo Samplers
Authors: Luhuan Wu
Yi Han
Christian A. Naesseth
John Patrick Cunningham
Venue: NeurIPS 2025 poster ()
Keywords: sampling; sequential monte carlo; diffusion models; probabilistic methods;
Primary area: probabilistic_methods
TL;DR: A training-free diffusion-based sequential monte carlo method for sampling from unnormalized distributions

Abstract: We propose a novel sequential Monte Carlo (SMC) method for sampling from unnormalized target distributions based on a reverse denoising diffusion process. While recent diffusion-based samplers simulate the reverse diffusion using approximate score functions, they can suffer from accumulating errors due to time discretization and imperfect score estimation. In this work, we introduce a principled SMC framework that formalizes diffusion-based samplers as proposals while systematically correcting for their biases. The core idea is to construct informative intermediate target distributions that progressively steer the sampling trajectory toward the final target distribution. 
Although ideal intermediate targets are intractable, we develop \emph{exact approximations} using quantities from the score estimation-based proposal, without requiring additional model training or inference overhead. The resulting sampler, termed \textit{\ourmethodfull}, enables consistent sampling and unbiased estimation of the target's normalization constant under mild conditions. We demonstrate the effectiveness of our method on a range of synthetic targets and real-world Bayesian inference problems.

Reverse Diffusion Sequential Monte Carlo Samplers
Luhuan Wu' Columbia University
Yi Han Columbia University
Christian A. Naesseth University of Amsterdam
John P. Cunningham
Columbia University
Abstract
We propose novel sequential Monte Carlo (SMC) method for sampling from unnormalized target distributions based on a reverse denoising diffusion process. While recent diffusion-based samplers simulate the reverse diffusion using ap- proximate score functions, they can suffer from accumulating errors due to time discretization and imperfect score estimation. In this work, we introduce a princi- pled SMC framework that formalizes diffusion-basee samplers as proposals while systematically correcting for their biases. The core idea construct informative intermediate target distributions that progressively steer the sampling trajectory to- ward the final target distribution. Although ideal intermediate targets are intractable, we develop exact approximations using quantities from the score estimation -based proposal, without requiring additional model training or inference overhead. The resulting sampler, termed Reverse Diffusion Sequential Monte Carlo, enables con- sistent sampling and unbiased estimation of the target's normalization constant under mild conditions. We demonstrate the effectiveness of our method on range synthetic targets and real-world Bayesian inference problems.
Introduction
Sampling from unnormalized target distributions is fundamental problem in many applications, ranging from Bayesian inference [ 12 to simulating molecular systems 3 Classical methods like Markov chain Monte Carlo (MCMC) simulate a Markov chain with the target its stationary distribution, but they can suffer from slow mixing and difficulty traversing between modes for complex distributions. Partiele methods such as importance sampling generate exact samples in large compute limit; yet they struggle with the curse of dimensionality [4]. Alternatively, variational inference (VI) casts inference as an optimization task, though its success depends on the expressiveness of the variational family and the complexity of the optimization landscape.
Recently, diffusion models have emerged as a powerful approach for sampling from complex dis- tributions 6, 77 They define forward noising process that gradually transforms complex target distribution into simple base distribution. A reverse denoising process then reconstructs target samples by simulating the dynamics backward in time, starting from the base distribution and guided by a time-dependent score function. In the generative modeling setting, this score function is approxi- mated by neural network trained on samples from the target distribution. However, in the sampling context, such training data are unavailable and only an unnormalized target density is accessible.
Recent works on diffusion- based samplers explore alternative ways to approximate the score function directly from the target density, enabling sampling without access training data. One line of works,
"Correspondence email: 1w2827/columbia. edu
"Our code is available at /tttt:///gittubb hot.i/bb..ooomuuuuaaa/////SCC
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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known as diffusion Monte Carlo (MC) samplers, estimates the score function using MC methods. Huang et al. [8, Grenioux et al. consider Langevin-style MC algorithms, but they rely on good initialization of the reverse diffusion process to ensure theoretical guarantees. He et al. [10] propose an alternative scheme based on rejection sampling which relaxes prior assumptions and improves sampling efficiency in low-dimensional regimes. These approaches demonstrate both theoretical and
empirical advantages over conventional MCMC methods, particularly for multi-modal distributions.
In contrast to relying on MC estimation during sampling time, a complementary line of research trains a neural network in advance to approximate the score function. To this end, some works propose new score matching objectives [II 12 13, while variational approaches instead optimize divergences between forward and reverse diffusion processes [5 16 [8]
While promising, diffusion- samplers suffer from two sources of bias: discretization error in simulating the reverse diffusion process and approximation error in the estimated or learned score function. An exception is Phillips et al. [1] which mitigates the bias of a trained diffusion- based sampler using Sequential Monte Carlo (SMC), a general inference tool for sequential models 11 20). However, such training-based methods remain computationally complex compared to classical sampling methods and often rely on special neural network preconditioning (2T].
To address these challenges, we develop a new diffusion- based sampler, Reverse Diffusion Sequential Monte Carlo (RDSMC) for sampling from unnormalized target distributions. which is training free and admits theoretical guarantees. Inspired by prior work, we formalize diffusion MC samplers as proposal mechanisms within an SMC framework. At high level, RDSMC generates multiple particles from the reverse diffusion dynamics using MC-based score estimates. To correct the bias in proposals, introduce intermediate target distributions that guide resampling of particles each step, progressively steering them toward the final target distribution of interest.
Crucially, our intermediate targets are efficiently computed using byproducts of MC-based score estimates, incurring no additional cost. Moreover, they form an exact approximation to the ideal intermediate targets that maximize sampling efficiency, defined by the marginal distributions of an extended final target [19 (2). This design helps particles stay closely aligned with the final target throughout the sampling process. In contrast, Phillips et al. [I]] learn neural network-baad -ased surrogates that require additional training and attain the ideal target only at the final step.
inheriting the standard SMC guarantees. In particular, it produces asymptotically exact samples from the target in the limit of many particles, and provides an unbiased estimate of the normalization constant for any fixed size of particles.
RDSMC belongs to a class of nested SMC methods 23 24
Our contributions are summarized follows:
We propose a new SMC algorithm, Reverse Diffusion Sequential Monte Carlo (RDSMC), based on the reverse diffusion process for sampling from unnormalized distributions.
RDSMC is training-free and extends existing diffusion MC samplers to achieve asymptoti- cally exact sampling and provide unbiased estimates of the normalization constant, while incurring almost no computational overhead given the same number of final samples.
Empirically, RDSMC outperforms or matches existing diffusion MC samplers and classical geometric annealing-base based Annealed Importance Sampling (AIS) (25) and SMC samplers (26] on synthetic targets and Bayesian logistic regression benchmarks.
2 Background
Diffusion models. Diffusion models 6 7 evolve a complex target distribution (x) into simple base distribution x1 (z), e.g. Tı (I) (0, 1), via forward stochastic differential equation (SDE)
dr: (t) + g())dB,
0: 0
(1)
where f() and gt) are the drift and diffusion coefficients, and B, the standard Brownian motion. To generate samples from T (2), we simulate a reverse SDE initialized from Φ ~ ग. (x),
dr: [f(t)=; "V 2, log me (=x)] dt g())
t: 0,
(2)
where Va, log (x;) is the score function of the marginal density (x:))) at time induced by the forward process in Eq. ( and B, is the reverse-time Brownian motion.

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Diffusion MC samplers. Building on the diffusion model paradigm, recent works use MC methods to simulate the reverse dynamics in Eq. whose terminal distribution corresponds to the target
distribution सt of interest 8 g 10 While the score function Vx, log Te (are) is generally
intractable, the key idea is to construct MC estimates via the denoising score identity [DSI, 27 28
Vx, log (z)) = /
a(t)zo I1 T x4 xi)dro.
o(t)² -otyyz
(3)
where <300 | 54) x ((ro) (())ar, (1) (
is the denoising posterior, which treats the unnormalized target (xo) as the prior and the forward transition density N(zo ((t) (t) from Eq. (I) as the likelihood. The coefficients (tt) and (t) are determined by the drift and diffusion
terms f(t) and g(t) (see Appendix A.I for details).
Eq. (3) suggests that score estimation can be cast as a posterior inference problem. In practice, one can draw approximate samples from the denoising posterior form an MC estimate of the score, which is then substituted into the reverse dynamics to generate samples from т. Other score identities beyond the DSI can also be leveraged (see Appendix A.2).
Sequential Monte Carlo. SMC 19 20 is particle-based based method for sampling from sequence of distributions defined on variables To.7, terminating at a final target distribution of interest. We consider reverse-time formulation, where SMC evolves a weighted collection of N particles {xt") wi NN from t = T to 0, gradually approximating the final target.
An SMC sampler requires two key design choices [19], sequence of intermediate proposals, (ZT) and (gc ==+1))?-o' and sequence of (unnormalized) intermediate target distributions, ((e())))) such that the final target (0(o)))) recovers the distribution of interest.
SMC initializes N particles of yr (2T) with weights Wiy 1, N. Then for each step ll....., and particle i = I,..., N, it proceeds as follows:
(i)
(5) + "T (2T ) //r((TT) for
1. resample ancestor (i)
r+1 Multinomial (2 (1+41 w ((
(1:N)
2. propagate particle
xt q: (Ire ((())):
3. compute weight w/" (1()(())//// (17 =eff1)]
The final set of weighted particles forms a discrete approximation to the final target Yo. which is asymptotically exact given infinite particles under regularity conditions (20). However, the efficacy of SMC greatly depends on the choice of intermediate targets and proposals the closer these intermediate distributions are to the marginals of the final target, the more effective the SMC sampler.
3 Method
Our goal is sample from a target distribution (xx) = 1 (z). where # (xx) is the unnormalized target and Z = f (z) generally intractable normalization constant. We develop Reverse Diffusion Sequential Monte Carlo (RDSMC), a diffusion- -aaood SMC sampler targeting ((z).
To enable sequential inference with SMC, we define an extended target distribution over discretized trajectory Zo.T = (Io...., .T) of the forward process in Eq.
T
π (ToTT) (FO) II (
where 0 = Ty < <... < Tr = are T 1 discretization times and π (x2 11 1 is the forward transition density from time 7 1 to 7, induced by Eq. (I) (see Appendix A. for analytical expressions). Without loss generality, we assume a uniform discretization step size d = 1/T = т, T. Vt =1.... TT
This construction yields sequence of intermediate targets {n ( ) > whose final marginal π (an) recovers the desired target. These intermediate targets provide a natural setting for SMC, and in fact, optimal intermediate gargets. However, they are intractable to sample from and to evaluate.
3

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To address this challenge, we develop RDSMC leveraging the dual structure of diffusion processes, generating samples in the reverse direction while grounding their targets in the forward direction. In $3.1] we introduce a sequence of proposals based on the reverse diffusion process using MC score estimates. To correct the resulting proposal bias, in $ 33.2 we develop practical intermediate targets that form exact approximations to the optimal targets क (Ta.r)) using byproducts of MC score estimates. Finally in $ 3.3 we present the full RDSMC algorithm and its theoretical guarantees.
Notation, We slightly abuse notation by letting 2; denote both the continuous-timg -immo variable 2t for & E [0, 1J, as used in Eqs. (1) and ( and the discrete-time variable for 1 E (o,..., T), as used throughout this section. We denote St == [((Te) St = g(Te), 0g == a<(T),, 02 == ( for time-dependent diffusion coefficients from Eqs. ( and (3). For generality, define uy as the collection of all auxiliary random variables generated in the MC score estimation at step t.
3.1 Reverse diffusion proposal
We design an extended proposal distribution based on the reverse diffusion process with MC score estimates, jointly modeling diffusion dynamics Iy and auxiliary randomness Ue from score estimation.
MC score estimation. Given a sample Iy at step t. we define a generic MC score estimator s(re, u4) Vx, log (2x) based the DSI in Eq. (3). where u, is the estimation randomness associated with some sampling distribution (uz xz).
As an illustrative example, consider an importance sampling (IS) estimator (Algorithm targeting the posterior (35o (xa)) (xr) logao ofT). We draw M importance samples ui from the
proposal g(u (m) (x)) = (u ((m) ae/ar, oī/aa] for m = -,, ,,, and estimate the score by
M
O4 (m)
log ग (a+)) s(rt, u.)
Σ
Ft
2m w(m))
which corresponds to an MC approximation of the RHS of Eq. (3). The importance weights w (m)) are (m))
(uu
N( a (m)) .o22
w (m)
m m= I, , M.
(6)
(u: xz))
In this case, the auxiliary randomness to the collection of all importance samples de = (u. Jm=1-
In practice, we use more sophisticated AIS scheme to improve the accuracy of score estimates (Algorithm 3). More informative IS proposals or other MC methods such as SMC and rejection sampling may also be used; see Appendix B.I for further details.
Approximate reverse diffusion dynamics. transition kernel of the reverse SDE in Eq. (2 to define a conditional proposal for xt.
With the score estimates in place, we approximate the
( Tt+1: 14+1) == (axx
T++1
[f+11
22 (( M4+1)] 5, 1.
(7)
Extended proposal distributions. The full sampling process starts by drawing .T from a tractable
base distribution q(TT), e.g. ((TT) AO 1). We then iteratively evolve I, for f = T 1,... using the reverse diffusion dynamics q( Zi+1, Mt+1) in Eq. (7) where auxiliary variable 4441
q(u++1 ((+1)) sampled in previous iteration to estimate the score s(Et11, #)-1).
The overall sampling process defines proposal over the extended space of (I,, ue (T
T-1
q(Zo.T ) = g(xx((())))))) ((zT))((u) ZT) II q(x; F+41 (141)g(u, xx).
(8)
While sampling ug at the final step = is not necessary, we retain it for notation simplicity.
This sampling procedure builds on prior work of Huang et al. 8, Grenioux et al. [9] which estimate the score function using MCMC, and by He al. [10] which use rejection sampling. We formalize these ideas by defining an extended proposal distribution that incorporates randomness in MC score estimates. In particular, we adopt an IS or AIS-based MC approach that also provides an unbiased estimate of the normalization constant, property that, in principle, can be achieved by the rejection sampling approach of He al. [0] as well. This property will be leveraged the next part to construct intermediate targets for RDSMC.

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3.2 Intermediate targets
Samples from the reverse diffusion proposal deviate from the desired target и due to two sources of erro: time discretization of the reverse SDE, and bias in score estimation. To correct these errors, we design series of intermediate target distributions. We first characterize the optimal but intractable targets, and then develop practical approximations using byproducts of score estimation. Finally, we extend the intermediate targets to include auxiliary variables. Our construction aligns with the optimal targets intermediate steps and recovers the desired marginal π (2o) at the final step = 0.
Optimal intermediate targets. The optimal intermediate target at step is the marginal distribution of the extended target π (20:T) in Eq.
T
((zrr) =
/ T (Zoer )dzo:- (2;)
II 1=(+1
π < 24-1),
(9)
as it leads to exact samples from xt (Xor)) when combined with locally optimal proposals [9]
However, the marginal <(x)) = (/r))/ (are orto, ofl)d in Eq. (9) involves an intractable
integral, except in the final step = 0, where π (20) known up a normalization constant.
Marginal estimation. Fort > 0, we approximate the marginal (2+) by # uz) using byproducts of MC score estimates, where we recall ū, is the estimation randomness, The key insight that (xx is the normalization constant of the posterior (xo xx)) x (<x)) ag of that is,
/ F(zo)N(xx a,z0, of Z / (xx))N(a agto, o) = ZT(zz))
Hence, we can re-use the posterior inference procedure in MC score estimation to obtain an unbiased estimate of the desired marginal (I,) (up to a factor of 2).
For example, IS or AlS-based score estimation (Algorithms and 3 generates importance weights (w)(m) JM m== targeting T (xro 24) for fixed æy; we then obtain an unbiased marginal estimate
M
a uz)
w (mm), with (x.) [+ ux)l Zx(x)))
(10)
M
At the final step 0, set # (xo, un) = #(xo) (xo) as approximation is needed.
Extended intermediate targets. Finally, we incorporate the auxiliary variables u, by setting their targets to match their sampling distributions and define the extended intermediate target as
T-1
(e (er) = # (Fre u.) ((u xx) II
( ax) g(1441 (441))
(11)
d=t
fort 0,... .T 1, and yT wr) ( ug) q(ur [r).
The structure in Eq. IT] mirrors that of the optimal targets Eq. (9). replacing the intractable
marginal (xx)) with an unbiased estimate A (Iz, u:) (up to factor of Z) and accounting for auxiliary
randomness W.r, Moreover, we make the following two observations.
Observation 1, The final marginal target at = 0 matches the desired T<ro) with no approximations,
(o(2o) f (u(oor, Morrr dzır, uo:r
= - $ fa uo)a(u0 xo) II Txx xz-1) q(u; 12)d2111 Moer
(12)
= 7 (mo) x <(xo),
as A (20, un) #(20) by construction. This implies our final SMC target is correctly specified.
5

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Algorithm 1: Reverse Diffusion Sequential Monte Carlo (RDSMC)
Input: Unnormalized target 7 (xr), number particles N, discretization steps T (with step size 8 = 1/TT,, diffusion schedule (0t, ft, Dt), base distribution ((T)), and additional inputs for score and marginal estimation 1j
(8) N Output: Weighted samples { w Ji=1 and normalization constant estimate for to N do
Sample 277 ~ ((TT);
(t) 400 f Algorithm 3(T, aT, aT, oT, 47 ,7):
Compute score and marginal estimates o7
Compute weight: WT
9(27)
for <-
do
Resample {x/+1,
(1) (6) (1)
s441 "141li= 1 according to weights (wi+1li=1
INN
u
7
f 4- 1 to N do
Sample z!"" g(x:
(8) 2+41 u441 (1) - (x. Eff - [fenafff 1 ó, (,
r then
10
Compute score and marginal estimates 8 #t
Algorithm 3 (a, as 51, ,
else
Compute exact marginal #0 +- #(260)
#f x(a,_,
Compute weight: w!"
13
#441 q(a, 12/ 1 #0:11 (9)
Compute normalization constant estimate 2 ll N Ex w
Observation 2. Marginalizing out uar in (e u:r) recovers the optimal target (xer), Vt,
J TelFer, wer berrddurr / f un) q(ue xx) II ( (x) (us-111111 [
T-1
Eug (ws) (# ue)l
II x ( xx)
(13)
T-1
= Zm(x))
II
ग. (2:+1 wx) x ((Er)),
where the last equality follows from the unbiasedness of marginal estimates, as in Eq. 100 This result shows that intermediate targets "Y (212,, u.r) aligned with the optimal π ((r.)), despite the approximation in ft (22, uz), a property known as the exact approximation [22
While the generic SMC framework remains valid under alternative choices of intermediate targets, provided that Yo (xu) x =<xo). our construction offers a balance between theoretical correctness
and practical efficiency. The marginal estimates #(x,, Uz) in our intermediate targets from Eq. (11) act as !wisting or look-ahead functions 19 Chapter 3] that incorporate future information क (an) into intermediate steps. Together, Observations T [nd ensure that our intermediate targets provide effective guidance throughout the sampling process while recovering the desired target in the end.
3.3 RDSMC: Algorithm and theoretical guarantee
We now derive weighting functions, the final component of RDSMC, which guides the resampling of particles. We then present the complete algorithm and establish its theoretical guarantees.
Weighting functions. Following the standard SMC framework, we define the weighting functions on the extended space (IAT, M.T) for t T -11,- 1, ... ,0 as ratios of intermediate targets and proposals
7 (at: M.:r)
(14)
7+41 (1T41: Me+4 1:7) g( 124 Ue. q(u, xt)

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Substituting the expressions for "/e from Eq. 11 and canceling out common terms yields
# Ue (2(+1122))
Wt 7 (1 We4 (F
(15)
1e+ U241)
At the initial step T, the weight is wT (ær,, ur) (r)((ur] (T)) by construction.
77 (2n.)ur) ((rr)((r(err
#(ayar)) recalling that "r (#r, UT) ((rr)
Notably, although the intermediate targets and proposals involve auxiliary sampling distributions q(2, xx)) for score estimation, the weighting functions wy in Eq. (15) and wy do not depend their evaluation. Hence, generic MC sampler can be used while still retaining computable weights, as long it produces tractable estimates of the score and marginal densities.
The RDSMC algorithm and theoretical guarantees. We summarize the RDSMC algorithm in Algorithm [ It initializes N particles from tractable base distribution. Subsequently, the particles are propagated through a reverse diffusion- based proposal, followed by weighting and resampling using intermediate targets. These steps enabled by running inner-level MC estimation of the score function and the marginal density. In this work, we use AIS (Algorithm 3 while other methods can be incorporated well (see Appendix [.II. The final output of RDSMC is weighted set of samples approximating the target π (xr), along with an estimate of the normalization constant Z.
The inner-level MC estimation involves a sampling procedure (c.g. AIS) targeting the intractable posterior (zo xz). which is then used to assign proper weights 19 Chapter 4.3] proposed samples Fy. Consequently, RDSMC inherits the unbiasedness and asymptotic exactness guarantees of nested SMC 23 19.
RDSMC can be viewed an adaptation of the nested SMC framework 23 24
Theorem (Informal). Under regularity conditions, the RDSMC algorithm provides consistent estimator of the target distribution x (Io) particle size N + co and an unbiased estimator of the normalization constant Z for any N
We provide the formal statement and proof in Appendix C
Theorem T shows that RDSMC effectively wraps diffusion MC-style proposal within an SMC framework, mitigating its bias as the number of particles increases. Moreover, it provides an unbiased estimate of the normalization constant, a property absent in existing diffusion MC methods 9
The bias correction mechanism of RDSMC arises from two key aspects: (1) the final target of RDSMC is explicitly constructed to match the desired target π (Zo)), regardless of the discretization scheme or marginal density approximation (Observation []; and (2) errors in score estimation affect only the proposal steps, which accounted for in the weighting functions. Therefore, the weighting and resampling steps in the outer SMC loop asymptotically correct for proposal bias, ensuring convergence the final target (20) in the limit of many particles.
4 Related works
Diffusion MC samplers. Huang et al. [8] develop the RDMC sampler based on the reverse diffusion process using Langevin-style MCMC estimate the score function. They initialize the sampler at some 7T < via nested Langevin procedure. Grenioux et al. [9] propose a similar SLIPS sampler with iigaa--tonooise-raioo (SNR)-adjusted discretization scheme, and provide further guidance on choosing the initial sampling time Tr under suitable conditions. In contrast, we use AlS-based score estimators to enable marginal estimation and start sampling from the base distribution at 7r 1.
Alternatiell,, He et al. 10 explore rejection sampling for score estimation, improving the sampling efficiency in low-dimensional settings, which may be incorporated into our framework as well.
While these methods provide certain theoretical guarantees, we offer an orthogonal means to improve their accuracy by increasing the number of particles. RDSMC can be viewed as generic SMC wrapper around these diffusion-MC samplers (given suitable score estimators), enabling consistent sampling and unbiased estimation of the normalization constant.
Training- based diffusion samplers In addition to developing MC estimates of the score func- tion, another line of work trains a neural network directly using unnormalized target information.
7

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Variational approaches achieve this goal by minimizing the divergence between forward and reverse processes [4] 17 [6], while Akhound-Sadee -Sadegh et al. [22]. Ou Yang al. [33 exploit various
identities of the score function or related energy function. Similar to their training-free MC-based counterparts, these methods are prone to discretization and score approximation errors.
Closest to our work is Phillips et al. ITI that also use SMC to correct proposal bias. Their method requires training neural networks, while ours is training-free Moreover, their neural network-based intermediate targets incur approximation errors except at the last step, while ours reflect the true target marginals at each SMC iteration. See an empirical comparison in Appendix E.5.3
He et al. [21] show that many training- baaeing-based methods incur heavy computational overhead compared to classical sampling methods and rely special network preconditioning. For this reason, we limit our empirical comparison to training-free samplers. Nonetheless, an interesting direction for future work is to combine our method with training-based approaches to further enhance performance.
SMC for conditional generation from diffusion models. Beyond classical sampling tasks, SMC has been applied to conditional generation for pre-trained diffusion models [29] 30 31 32 While
these works also combine SMC and diffusion dynamics, key distinction is the target distribution: they sample from an existing diffusion model prior tilted with a reward function, whereas our setting involves sampling from any (unnormalized) target distribution. As a result, the way diffusion models are used is different, and we require distinct designs for proposals and intermediate targets in SMC.
Nevertheless, RDSMC uses the estimated marginal densities in Eq. ( as twisting functions to improve SMC sampling efficiency. This strategy is conceptually is similar to that of Wu et al. [30], though in different setting; see Appendix D for further discussion.
Experiments
We evaluate RDSMC on a range of synthetic and real-world target distributions, comparing it to SMC (26), AIS (25). SMS (33, RDMC [8), and SLIPS (9]. AIS and SMC operate over a series of geometric interpolations between the target and a Gaussian proposal with MCMC transitions. SMS samples from the target by sequentially denoising equally noised measurements using Langevin procedure combined with estimated scores. RDMC and SLIPS are described in $
RDSMC uses a crriance-preserving diffusion schedule. To evaluate the effectiveness of our intermedi- ate targets and the importance of resampling, we include two ablations: (i) RDSMC (Proposal), which samples directly from the reverse diffusion proposal in Eq. (8) without any weighting or resampling, and (ii) RDSMC (IS), which applies a final IS correction to samples from RDSMC (Proposal).
Baseline methods follow the implementation of Grenioux et al. [9]. Notably, the information about the target variance (or an estimation) is provided to guide the initialization of the baselines SMC, AIS, SMS and SLIPS. In contrast, our method does not make use of this extra information.
Unless otherwise specified, we use T 100 discretization steps for RDSMC and its variants, and T 1, 024 steps for other methods. We generate N = 4, 096 final samples for all methods, and tune their hyperparameters assuming access either validation dataset
oracle metric.
We provide ablation studies controlling for the discretization steps, the total running time and comparable hyperparameter settings in Appendix [E.5 where we observe RDSMC still remains competitive and, in some cases, superior. All experiment details are included in Appendix B
5.1 Bi-modal gaussian mixtures
We first study bi-modal Gaussian mixtures with an imbalanced weight ratio of wi/(w) + w2) = 0.1 for varying dimensions d. The estimated ratio is obtained by assigning samples to the mode with the highest posterior probability. For each method we select the hyperparameters based on the lowest estimation bias of the weight ratio.
In Figure Ta we compare the marginal histogram of samples from RDSMC, RDSMC (Proposal), and the true target in d = 2. While samples from RDSMC (Proposal) cover both modes, their relative weights are overly balanced. contrast, RDSMC recovers the calibrated mode weights, indicating the importance of the error correction mechanism by SMC.
8

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Wy/(W3 estimation bias
log Z estimation bios
RDSMC RDSMCc/roposal
10*
0.20
RDBBCC
RDBMAI3)
0.15
10
RRMM
5145
(a) d 2: histogram along 1st dimension. While RDSMC's proposal covers the modes, SMC procedure is crucial for producing calibrated weights.
(b) Estimation bias of weight ratio W1/(W1 + W2) and log-normalization constant log Z versus dimension Results averaged over seeds with error showing one standard error. Note that SMS, RDMC,
SLIPS do not provide estimate of log Z. For both metrics, the bias increases with for all methods, while RDSMC outperforms the baselines for most cases.
Figure Bi-modal Gaussian mixture study.
Rings (d = 2)
Funnel (d = 10)
Algorithm
Radius TVD (4)
log Z Bias (4)
Sliced KSD (4)
log Z Bias (4)
AIS
0.10 + 0.00
0.05 + 0.00
0.07 + 0.00
0.28 + 0.01
SMC
0.10 + 0.00
0.05 + 0.00
0.07 0.00
0.28 0.01
SMS
0.24 + 0.01
N/A
0.15 + 0.00
N/A
RDMC
0.37 0.00
N/A
0.13 0.00
N/A
SLIPS
0.19 + 0.00
N/A
0.06 + 0.00
N/A
RDSMC
0.13 = 0.01
0.03 + 0.00
0.11=0000
0.28 E 0.10
RDSMC(IS)
0.15 + 0.01
0.02 + 0.01
0.33 + 0.03
1.61 + 0.14
(DSMO((roposal))
0.09 0.00
N/A
0.32 0.03
N/A
Table 1: Results on Rings and Funnel (mean + standard error over seeds). Bold indicates 95% confidence interval overlap with the best average result. RDSMC has the lowest log Z estimation bias for both targets. On Rings, RDSMC(Proposal) has the lowest radius TVD, followed by AIS, SMC, and RDSMC. On Funnel, SLIPS has the lowest sliced KSD, followed by AIS, SMC, and RDSMC.
Figure [b shows the estimation bias of the weight ratio (left), and that the nogmoaaaaiizttion constant log Z (right). Note that only RDSMC, RDSMC(IS), AIS and SMC provide estimates for log Z. We observe that RDSMC consistently outperforms other methods in both metrics across dimensions. Moreover, RDSMC (Proposal) exhibits consistently high weight ratio estimation bias. While RDSMC (IS) reduces this bias, it still underperforms the full RDSMC procedure, highlighting the effectiveness of intermediate resampling guided by our intermediate target distributions.
In high dimensions, methods exhibit some degree of mode collapse. RDSMC primarily samples from the dominant mode, resulting in average weight ratio estimation biases of 0.09 and 0.08 for 32 and 64, respectively. In contrast, AIS and SMC completely collapse to a single mode, which varies across runs, yielding degenerate estimated ratios of 0 or I and average bias of around 0.4.
5.2 Rings and Funnel distributions
We present results two additional synthetic targets. Rings, introduced by Grenioux et al. 19,, is 2-dimensional distribution constructed via an inverse polar parameterization: the angular component uniformly distributed, while the radius component follows a 4-mode Gaussian mixture. Funnel 34 a 10-dimensional "funnel"-shaped distribution.
For Rings, assess sample quality using total variational distance in the radius component (Radius TVD). For Funnel, use sliced Kolmogorov- Smirroo distance (Sliced KSD) to evaluate sample quality following Grenioux et al. [9]. As both targets admit tractable normalization constants, we also report the estimation bias for methods that compute them. For each method, we select the hyperparameters with lowest Raidus TVD heldout validation set for Rings and lowest Sliced KSD for Funnel.

---

Test LL (T)
Credit (d 25)
Cancer (d
lonosphere (d= 35)
Sonar (d 61)
AIS
/22.73+0..3
-60.45 = 0.31
86.37
-110.11
SMC
-123.17 0.05
-60.28 1: 0.11
-6.37
-110.11
SMS
-527.79 +: 0.85
-215.64 + 0.66
202.56
-275.44
RDMC
-388.24 1.75
-182881
0.43
-108.67 0.09
-128.29 + 0.03
SLIPS
-121.79 + 0.04
56.26 + 0.08
-85.07 0.07
-102.39 + 0.03
RDSMC
-124.00 + 1.96
-62.23
1.96
87.72+1.77
-01152 + 1.84
RDSMC(IS)
-144.38 + 4.63
82.47 + 7.78
2.84
1110.77 + 4.54
RDSMC((roppsal)
-606.03 1.26 244.86
+ 0.97
-92.62 0.04
-134.72 + 0.13
Table 2:
Bayesian logistic regression with test log- lieelikelihodd (mean + standard error) averaged over 5 seeds. Bold indicates 95% confidence interval overlap with that of the best average result. SLIPS achieves the best overall performance, while RDSMC matches or closely approaches it. contrast, RDSMC(Proposal) performs noticeably worse, highlighting the effectiveness of the SMC correction.
As shown in Table 1 RDSMC achieves lower or comparable estimation bias of the log-oormalization constant relative to AIS and SMC on both targets. On the lower-dimensional Rings target, RDSMC (Proposal) has the lowest radius TVD, where AIS, SMC and RDSMC closely match. On the more challenging Funnel target, SLIPS achieves the lowest sliced KSD; while RDSMC performs slightly worse, greatly outperforms its IS and Proposal variants.
5.3 Bayesian logistic regression
Finally, we evaluate inference performance on Bayesian logistic regression models using four datasets. Credit and Cancer (35] involve predicting credit risk and breast cancer recurrence, while Ionosphere [36] and Sonar (37 focus on classifying radar and sonar signals, respectively. The inference is performed on 60% of each dataset, leaving 20% for validation and 20% for testing. Each method is tuned using the validation log- ligelikeloood estimate. We report the test log- bog-liielihood in Table
RDSMC achieves the best or near- best performance across datasets, closely matching SLIPS. which performs the best overall. This comparison suggests that SLIPS's time discretization and initialization strategy may offer complementary benefits despite lacking systematic error correction. RDSMC (IS) underperforms on Credit and Cancer and exhibits higher variance than RDSMC, while RDSMC (Proposal) performs the worst, again highlighting the importance of the SMC mechanism. Compared to baseline methods, RDSMC shows greater variance, likely due to auxiliary randomness its nested structure, suggesting direction for future improvement.
Discussion
This paper presents training-free and diffusion-based SMC sampler, Reverse Diffusion Sequen- tial Monte Carlo (RDSMC), for sampling from unnormalized distributions. By leveraging reverse diffusion dynamics as proposals, we devise informative intermediate targets correct systematic errors within a principled SMC framework. These components rely on Monte Carlo score estimation, without requiring additional training. Our algorithm provides asymptotically exact samples from the target distribution and a Unite-sample unbiased estimate of the normalization constant. Empirical re- sults on both synthetic and Bayesian inference tasks demonstrate competitive or superior performance compared to existing diffusion-baan based and classical geometric annealing- based samplers.
Limitations and future work. As a nested SMC procedure, RDSMC introduces auxiliary variance. Such variance may be mitigated by enhancing autocorrelation within the MC estimation step [c.g. 38). In digeenimmnsional Gaussian mixture experiments, we observe a degree of mode collapse, an issue also seen other samplers. Potential remedies include partial resampling [39], or using more informed proposals. Our method also relies on oracle metrics for hyperparameter tuning, which aligns with existing works; however, developing a more automated strategy remains an important future direction. Finally, to further improve performance, we can incorporate SNR-adapted discretization schemes and informative initializations, as explored by Grenioux et al. [9) as well as combine our approach with maining-based diffusion samplers.
10

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13

---

Test LPPD
Credit (d 25)
Cancer (d = 31)
lonosphere (d 35)
Sonar (d 61)
RDSMC
94.54 + 1.34
-10.59 0.45
-25.97 1 0.73
-18.54 1 0.28
RDSMC
-94.62 0.06
-50.24 0.16
-24.87 + 0.02*
-18.91 = 0.01
RDMC
-138.22 0.35
-78.03 0.12
-44.16 + 0.07
-28.64 0.03
SLIPS
-92.42 + 0.02*
-10.41 0.01
-25.22 + 0.01
-18.40 = 0.01
AIS
-92.11 0.44
-10.13 0.07
-25.09 + 0.02
-18.41 0.01
SMC
-92.55 + 0.05
-10.13 0.01
-25.09 + 0.02
18.41 + 0.01
SMS
-98.00 + 0.27
-20.47 + 0.16
-26.17 = 0.10
223.99 = 0.08
Table 8: Ablation experiments: Bayesian logistic regression with test log pointwise predictive density (Test LPPD, with mean + standard error) averaged over 10 seeds. Bold indicates 95% confidence interval overlap with that of the best average result. RDSMC, AIS, SMC, and SLIPS achieve the best overall performance. RDSMC outperforms RDSMC (Proposal) and RDMC in most cases, highlighting the effectiveness the SMC correction.
where the likelihood p(ys 2ag wr b) is defined in Eq. (43) and the posterior p(w, b (D) is defined in Eq. 44
We use final (weighted) samples from each method to approximate the Test LPPD in Eq. (46).
We report the results in Table 8 We observe that RDSMC, AIS, SMC, and SLIPS achieve the best overall performance. Compared to SLIPS, RDSMC does not require largetvaariacce-asee initialization. Moreover, RDSMC outperforms RDSMC (Proposal) and RDMC in most cases, highlighting the effectiveness of the SMC correction. However, we observe that results for RDSMC exhibit large variance, as the case in the main experiments from Table
The optimal hyperparameters for each method selected based the highest LPPD on heldout validation set, computed analogously to the Test LPPD.
E.5.3 Comparison to Particle Denoising Diffusion Sampler the Funnel target
Finally, we present comparison the Particle Denoising Diffusion Sampler (PDDS) (11, which is also a diffusion- SMC sampler. However, unlike RDSMC, PDDS requires training.
We evaluate PDDS the Funnel target using 10 random seeds. Experiments are conducted on an NVIDIA RTX A6000 GPU, whereas our previous experiments use an NVIDIA A100 GPU. In addition, PDDS is implemented in JAX (following the official implementation at httph::////// https://githuu: thub cont/angusphh1ipp///arrr..o
denoising. wwt.ooiiiinnniissioooooooooooioe hatfffusionssaapleee.m
while RDSMC and the
other basclines implemented in PyTorch.
PDDS requires approximately 82.63 0.43 seconds for training and 15.08 0.073 seconds for final sampling of 4096 particles, resulting in a total runtime of about 97.70 + 0.47 seconds (reported as mean standard error). The runtimes of our method and other training-free baselines are reported in Figure 3 which are all below 25 seconds. However, due to differences in hardware and software frameworks, PDDS's s runtime not directly comparable to those of the previous experiments.
PDDS achieves a log Z bias of 0.26 + 0.04 and a mean Sliced KSD of 0.10 + 0.00. Compared with the results in Table 7 PDDS exhibits slightly lower average log bias than RDSMC, but higher than AIS and SMC. Its Sliced KSD is also higher than that of RDSMC, AIS, and SMC.
34

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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: Our method provides diffusion-based SMC samplers with theoretical guaran- tees. Our claims match our theory results and empirical results.
Guidelines:
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 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.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We discuss limitations of our work in the Experiment section when comparing to other methods, as well in Discussion where we look out for future directions.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sell-speciiiatiinn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
35


Reviews:
[{"review_id": "xxPHUTGeZ0", "reviewer": "Reviewer_oZor", "summary": "This paper combines the sequential Monte Carlo with the reverse diffusion path to provide a novel sampler to draw samples from an irregular target distribution. Different from typical sequential Monte Carlo, this paper introduces auxiliary variables $u_t$ to construct the proposal distribution $q(x_t|x_{t+1},u_{t+1})$ for simulating the transition in the reverse diffusion process and calculating the weights for the resampling. Different from RDMC, this paper will keep a sequence of particles in each update and use them to construct the proposal distribution for resampling. As a result, this paper shows that sequential Monte Carlo will be an unbiased sampler when the number of particles in the sequence approaches infinity and empirically shows the advantages of their methods.", "questions": "Please refer to the weaknesses part.", "limitations": "NA", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strength:**\n\n1. The problem expected to be solved (simulating reverse diffusion path to solve the sampling task) is interesting, and the core technique sequential Monte Carlo helps to approximate the ideal transition kernel $p_{0|t}$ by re-weighting the proposal with $w_i$ is practical since it has bypassed repetitive posterior sampling in previous work, e.g., RDMC.\n2. The core technique, i.e., introducing the auxiliary variables $u_t$ to couple sequential Monte Carlo and RDMC is original and effective, which helps to provide unbiased score estimation in expectation. Although the unbiasedness is conditioned on the infinite particle, it still makes a difference compared with RDMC. \n3. This paper is well written and makes it easy for readers to follow. \n\n**Weaknesses:**\n\n1. The theoretical results provided in this paper are limited. Some important theoretical properties for the proposed algorithm are unknown. For example, can the analysis be extended to the finite particle $N$ case? Will the algorithm converge? What kinds of convergence (high probability or in expectation) will be achieved? What is the size of particles required to be preserved to guarantee the convergence? Since the rejection sampling in high dimensions will bring the dimension curse, while this curse is preserved in the sequential Monte Carlo? \n2. The significance of the advantages of reverse diffusion SMC is limited. In the 2-d case, the improvement of TV distance is limited. Under the 10-d case, neither sliced KSD nor logZ estimation will have advantages. Although the algorithm is elegant and insightful, its effectiveness may be limited.\n3. Authors may add an independent table to clarify the notation, which will improve the readability of this paper. Currently, the notations are usually hidden in different parts of the paper.", "was_revised": true, "final_justification": "In the rebuttal period, the authors have addressed most of my questions on the theory, and therefore, I feel comfortable raising my score.", "rebuttal": "We thank the reviewer for their detailed review, and we are glad that they found the paper well-written and the proposed algorithm elegant and insightful. We find the questions and suggestions helpful and we address them in response below. \n\nNote: _if Latex equations do not render, please consider refreshing the page a couple of times_.\n\n\n> 1. The theoretical results provided in this paper are limited.\n\nOur method can be set up as a standard SMC on an extended space (similar to constructions such as nested SMC (Naesseth et al 2015)). As a result, **theoretical properties of SMC apply directly to our method**: \n\n- **Finite particle guarantee**: for any particle size $N\\geq 1$, any test function $\\phi()$ integrating with respect to the unnormalized target can be estimated unbiasedly (e.g. Naesseth et al, 2019, Theorem 2.3.1). We obtain the unbiased estimate of the normalization constant for $\\phi()=1$. \n\n- **Asymptotic convergence**: as $N\\to \\infty$, under standard regularity assumptions, SMC estimates converge to the true expectation in probability 1 (e.g. Chopin and Papaspiliopoulos, 2020, Proposition 11.4, or Theorem 3 restated in our appendix). Under regularity conditions, the central limit theorem holds for SMC with the same rate ($1/\\sqrt{N}$) as MCMC (Chopin 2004). \n\n- **Curse of dimensionality**: SMC scales well for discretization steps $T$ (usually $N$ needs to scale as $T$ for marginals to be stable, see Proposition 11.13 in Chopin & Papaspiliopoulos (2020) ), but just like other importance sampling methods it requires careful proposal/target choice to scale with dimension $d$ of the latent state. Chatterjee and Diaconis (2018) show that for stability $N$ needs to scale with $\\exp(\\textrm{KL}(p||q))$ where $p$ is the target and $q$ the proposal. In practice, for well-chosen targets and proposals we can still get well-performing algorithms for high-dimensional $x_t$ (e.g. Naesseth et al, 2019, Chapter 3). \n\nWe will include this discussion in the revision to clarify the theoretical properties of our method and their foundations in the classic SMC literature. \n\nAdditional reference:\n_Nicolas Chopin. Central Limit Theorem for Sequential Monte Carlo Methods and Its Application to Bayesian inference. Annals of Stats, 2004._\n\n\n> 2. The significance of the advantages of reverse diffusion SMC is limited.\n\nWe respectfully disagree with the reviewer’s assessment that the effectiveness of RDSMC is limited. In fact, we believe our results demonstrate that RDSMC offers meaningful advantages as it uses fewer discretization steps and does not require access to target information. Moreover, RDSMC extends existing diffusion-based methods by enabling log normalization constant estimates. Below we clarify key points and provide supporting ablation results:\n\n- **RDSMC shows better or competitive performance without access to target variance and with fewer discretization steps:** As described in line 249-255, competing methods including **AIS, SMC and SLIPS have access to the target’s scalar variance and use substantially more discretization steps** (T=1,024); SLIPS also uses an SNR-adapted time discretization schedule and a Langevin-based initialization. In contrast, RDSMC is run with $T=100$ uniformly spaced steps, does not use target variance and starts from a simple Gaussian $N(0,1)$. Despite these differences, RDSMC matches or outperforms these methods in most cases. In particular, RDSMC outperforms SLIPS along other baselines in the GMM weight ratio estimation task across varying dimensions $d \\in [2,64]$  (Figure 2 in appendix).\n\n- As a more neutral comparison, **RDMC is similar to SLIPS but  does not assume target variance, and RDSMC consistently outperforms RDMC across multiple targets**.\n\n- **RDSMC outperforms baselines given target variance**: To further isolate the effect of our core contributions, we include an ablation experiment on rings and funnel targets (see table below), where all methods use the target scalar variance (RDSMC uses it to initialize the proposal for score estimation). In these settings, RDSMC outperforms SMC and SLIPS on most metrics. \n\n\n| Method                            | Rings TVD               | Rings Log Z bias        | Funnel KS               | Funnel Log Z bias        |\n|----------------------------------|--------------------------|--------------------------|--------------------------|---------------------------|\n| RDSMC (T=200, use target var)    | $\\mathbf{0.102 \\pm 0.002}$ | $\\mathbf{0.012 \\pm 0.004}$ | $\\mathbf{0.065 \\pm 0.002}$ | $\\mathbf{0.219 \\pm 0.025}$ |\n| SMC (T=1024, use target var)     | $\\mathbf{0.096 \\pm 0.002}$ | $0.053 \\pm 0.002$         | $0.071 \\pm 0.000$         | $0.283 \\pm 0.005$          |\n| SLIPS (T=1024, use target var)   | $0.189 \\pm 0.003$         | NA                       | $\\mathbf{0.063 \\pm 0.001}$ | NA                        |\n\n\n\n- **Additional capability to estimate normalization constants $Z$**: Unlike SLIPS and RDMC, RDSMC provides $\\log Z$ estimates **without additional compute cost** (given the same number of generated samples, discretization steps and same score estimation strategy). This estimate is useful for downstream tasks including computing free energy in statistical physics, Bayesian model comparison and density estimation. \n\t\nWe will revise the manuscript to clarify these distinctions in settings and include ablation study results.\n\n> 3. Authors may add an independent table to clarify the notation, which will improve the readability of this paper. Currently, the notations are usually hidden in different parts of the paper.\n\nWe thank the reviewer for the suggestion and we will clarify notations in the revised version."}, {"review_id": "VhMdwR0nU2", "reviewer": "Reviewer_5DzY", "summary": "- The paper leverage the diffusion model paradigm to sample from a target distribution, which can be unnormalized.\n- It introduces a training-free Sequential Monte Carlo (SMC) approach where the proposal are derived from a reverse diffusion kernel.\n- The core idea is to augment the state space with an auxiliary variable that accounts for the randomness in the score estimation.", "questions": ".", "limitations": ".", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strength\n\n- The main novelty of the paper is the idea of augmenting the state space with an auxiliary variable to explicitly model the noise in the score estimation process.\n\n## Weaknesses\n\n**Major Theoretical Flaw**\n\nThe central methodology for score estimation appears to be fundamentally flawed:\n\n- The proposed score estimation in Equation (5) is problematic. By examining the weight update in Equation (6), it appears the auxiliary variable $u$ is intended to act as a substitute for $x_0$ within the target density term.\n- However, $u$ is sampled from the proposal distribution $N(x_t / \\alpha_t, \\sigma_t^2 / \\alpha_t^2 I)$, which is a Gaussian with a very large variance, particularly in early steps. This means that samples of $u$ have a low to no chance of falling in regions where the target distribution has high mass.\n- This turns the score calculation into an IS estimate where the proposal has a poor overlap with the target. Consequently, estimating the score in this way is likely as difficult as sampling from the target distribution directly, which defeats the purpose of the proposed method.\n\n**Typos**\n\n- In Line 96: \"initilaizes\" should be corrected to \"initializes\".\n- There is a trailing comma in the denominator of the weight expression in Equation (6) that should be removed.", "was_revised": true, "final_justification": "- The proposed method is novel\n- Following the authors' clarification regarding my concern about the high-variance of the proposal, I have a better view the existing methods\n- I am now updating my rating to reflect this new understanding and the strength of the work", "rebuttal": "We thank the reviewer for their time and effort in evaluating our work, and we appreciate their acknowledgment of the novelty of our work in augmenting the state space to explicitly model the noise in the score estimation process. \n\nHowever, we respectfully disagree with the reviewer's assessment of the score estimation strategy, as it overlooks established prior work and does not fully capture the intended flexibility and central designs of our method. We outline these points below and encourage the reviewer to reconsider the assessment in light of the following clarifications:\n\n- **Eq (5) is built on established methodology:** the IS score estimator (Eq 5) is grounded in a well-established line of works which reverse the forward transition kernel to obtain the proposal $\\mathcal{N}(x_0 \\mid x_t / \\alpha_t, \\sigma_t^2 /\\alpha_t^2)$ to estimate scores (e.g. Huang et al 2023, Akhound-Sadegh et al 2024). While the proposal’s variance is large in earlier steps, it decreases to 0 as $t \\to 0$ and the proposal becomes more informative, making the score estimates more accurate. More concretely, \n\n  - Huang et al 2023 consider a similar importance weighted score estimator (see their Eq 5 in Section 3.3). Their Theorem 1 shows that  sampling from the posterior $p(x_0 \\mid x_t)$ is easier than sampling from the original target $\\pi(x_0)$, as indicated by the improved LSI constant. Empirically, they observed that \"When $t$ is close to 0, we are able to quickly obtain rough score estimates via the importance sampling approach\". \n\n   - Akhound-Sadegh et al 2024 use $\\mathcal{N}(x_0 \\mid x_t, \\sigma_t^2)$ as the proposal distribution to form a self-normalizing estimate (see their Eq 8), based on the Target Score Identity (our Eq 19 in appendix). This proposal is equivalent to ours, as $\\alpha_t$ is set to 1 in their Variance-Exploding diffusion schedule. They also provide bounds on this score estimator. \n\n\n- **Illustrative role of Eq 5 and flexibility of our approach:** As stated in the paper (line 129), the estimator in Eq (5) is meant as an illustrative example and our framework can accommodate different score estimators. We discuss more advanced score estimators in appendix B.1, as suggested in line 133-134 below Eq (5) and (6).  These strategies include annealed importance sampling (Alg. 3 in appendix), more informed IS proposals (appendix B.1.3.) and estimators based on Target Score Identity (Eq 19 in appendix) or Mixed Score Identity (Eq 20 in appendix). \n\n- **Addressing proposal bias is a central design of our work:** The review focuses on the potential bias of a particular score estimator used to construct the proposal, but our method is explicitly designed to account for such bias. A key strength of RDSMC is its ability to incorporate biased proposals within the SMC algorithm and progressively correct for these biases. Theorem 1 provides theoretical guarantees. We also illustrate this point in empirical study. For example, as shown in Figure 1a , using the proposal alone yields biased results, while RDSMC reduces this bias.\n\nWe will clarify these points in the revision to prevent potential misunderstandings and to better highlight the flexibility and theoretical guarantees of our framework. \n\n--- \n\nWe also thank the reviewer for identifying the typos and we will fix them in the revision. \n\nNote: _if Latex equations do not render, please consider refreshing the page a couple of times_."}, {"review_id": "zm0VWbb6jf", "reviewer": "Reviewer_Kk4p", "summary": "The authors developed a training-free diffusion sampler which treated diffusion-based sampler as the simulator within a sequential Monte Carlo framework. The authors proposed to resample the particles in each step to gradually steer the particles to the desired distributions and the issue of discretization bias and score estimation error can be greatly alleviated. The reverse diffusion SMC provides the unbiased estimate of the normalizing constant. Numerous methods show the promising potential of RDSMC compared to other diffusion samplers.", "questions": "1. what is the key step to avoid discretization error/ score error? is it Eq(15)? more clarifications are needed.\n\n2. Is this algo. more like Wu's twisted SMC or vanilla SMC.", "limitations": "Although I appreciate the elegance, I am not convinced if this is a good idea to use SMC to correct the discretization bias and score error. Samplers are well-studied in low-dim distributions, but SMC doesn't scale well unless there is some clever tricks such as Tweedie-like \"look-ahead\" design.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros: \n\n1) using SMC (Line 3 in Alg 2) to estimate score function via denoising score identity seems to be an interesting idea. \n\n2) Eq(15) seems to have some potential alleviate the bias. I am not sure, need more evidence.\n\nCons:\n\n1) I don't fully understand why the SMC setups can correct discretization bias and score estimation errors.\n\n2) I don't understand the motivation to do training-free guidance in sampling setup, do we need a pretrained model in this case? \n\n3) Give more details on the motivation and/or derivation of eq (6) and eq(14).\n\n\n* I didn't carefully check the proof.", "was_revised": true, "final_justification": "look-ahead mechanism seems to be helpful, but I am not fully convinced.", "rebuttal": "We thank the reviewer for their careful and thoughtful evaluation. We are glad that they found our approach elegant and we appreciate their acknowledgement of the score estimation strategy and the potential of the bias correction mechanism. We find the questions and comments noted helpful and we address them below. \n\nNote: _if Latex equations do not render, please consider refreshing the page a couple of times_.\n\n\n> 1. Why can the SMC setups correct discretization bias and score estimation errors?\n\nThe key reasons behind the correctness of RDSMC are: **(1) the final target distribution of RDSMC is explicitly constructed to match the desired target distribution, and (2) the score estimation errors only affect the proposals, which can be asymptotically corrected with SMC**. Our theorem 1 provides a rigorous statement with proof in the appendix. \n\nMore specifically, Eq (13) shows that the final target distribution $\\gamma_0(x_{0:T}, u_{0:T})$ is constructed such that its marginal $\\gamma_0(x_0)$ exactly matches the desired target $\\pi (x_0)$. This property holds regardless of the discretization scheme used, because **the desired target** $\\pi (x_0)$ **is directly included in the final target** $\\gamma_0 (x_{0:T}, u_{0:T})$, and when integrating out all other variables, only $\\pi(x_0)$ remains, ensuring correctness of the marginal. \n\nThe score estimation error also does not affect the correctness of the final target, as **the score estimates are only used in proposals**. As with any SMC method, **the resampling and reweighting steps (Eq 15) asymptotically correct for errors in proposals**, ensuring convergence to the final target in the limit of many particles. \n\nWe will highlight these conceptual motivations and formal guarantees in the revised paper. \n\n> 2. What is the motivation to do training-free guidance in sampling setup, and do we need a pretrained model?\n\n To clarify, **our method does not use training-free guidance, nor does it require a pretrained model**. Our method operates in a fully training-free setting, assuming only access to the unnormalized target density. We will make this point clearer in the revision to avoid confusion. \n\n> 3. Give more details on the motivation and/or derivation of eq (6) and eq(14).\n\n**Eq (6) computes the importance weights required for the score estimation, which reduces to the ratio of unnormalized target density and proposal density.**\n\n- By Tweedie’s formula or the Denoising Score Identity (Eq 3), computing the score is related to sampling from the posterior distribution $p(x_0 \\mid x_t)$. \n\n- Since direct sampling from $p(x_0 \\mid x_t)$ is intractable, we use importance sampling. Eq (6) computes the importance weights targeting $p(x_0 \\mid x_t)$ using proposal $\\mathcal{N}(x_0 \\mid x_t / \\alpha_t, \\sigma_t^2 / \\alpha_t^2)$. \n\n- Note that $p(x_0 \\mid x_t) \\propto \\tilde \\pi (x_0) N(x_t \\mid \\alpha_t x_0, \\sigma_t^2I)$. Eq 6 computes the importance weights as the ratio of the unnormalized target density to the proposal density.\n\n\n**Eq (14) is a direct application of the standard SMC weighting procedure to our specific setting.**\n\n- As described in the background material (line 99), the incremental weight at each step is computed as the ratio of the current intermediate target and the product of the previous intermediate target and the proposal, i.e. $w_t \\gets \\gamma_t(x_{t:T}) / [\\gamma_{t+1}(x_{t+1:T}) q(x_t \\mid x_{t+1})]$. \n\n- By substituting our targets and proposals, we obtain the expression in Eq (14). Note that in our case, the targets and proposals are defined in the extended space which involve $u_{t:T}$. \n\n\nWe will clarify the derivations and motivations of these equations in the revision. \n\n> 4.  I am not convinced if this is a good idea to use SMC to correct the discretization bias and score error. Samplers are well-studied in low-dim distributions, but SMC doesn't scale well unless there is some clever tricks such as Tweedie-like \"look-ahead\" design. \n\nWe appreciate the reviewer’s point about the importance of look-ahead designs in high dimensions. In fact, **our approach does incorporate a look-ahead mechanism through the marginal density** $p(x_t)=\\int p(x_t \\mid x_0) \\pi(x_0) dx_0$​, which integrates the future information $\\pi(x_0)$.\n\nThese marginals serve as the foundation for defining our optimal intermediate target distributions (see Eq. 9), akin to the goals of twisting/ look-ahead adjustments in other SMC literature (e.g. TDS in Wu et al 2023). When the exact marginals are known, one can in principle sample accurately even in high dimension. \n\nIn practice, we estimate the marginal $p(x_t)$ using approximations (Eq. 10) and derive tractable \"twisted\" intermediate targets (Eq 11) and weights (Eq 15). These marginal approximations are increasingly accurate as the sampling trajectory progresses, and provide effective look-ahead guidance to improve sampling efficiency. \n\nIn empirical study, RDSMC consistently outperforms _RDSMC (IS)_ (which does not provide \"look-ahead\" mechanism in intermediate states and only performs IS in the end), and _RDSMC (Proposal)_ (which does not involve any importance weighting correction). **This comparison highlights the effectiveness of the look-ahead mechanism and SMC correction.**\n\nWe will clarify the connection to look-ahead SMC techniques in the revised paper. \n\n\n> 5. What is the key step to avoid discretization error/ score error? is it Eq(15)? more clarifications are needed.\n\nPlease see above response to the first question. \n\n\n> 6. Is this algo. more like Wu's twisted SMC or vanilla SMC.\n\nStructurally, our method shares similarities with Twisted Diffusion Sampler (TDS) (Wu et al 2023), as **both methods use some form of \"look-ahead\"  or \"twisting\" functions to improve the efficiency of SMC.**  These look-ahead functions introduce future information to intermediate states,  promote promising particles through the weighting procedure, and improve the SMC sampling efficiency. \n\nMore concretely, **the incremental weight expression of both TDS and RDSMC takes the same form** (their Eq 11 v.s. Our Eq 15)\n$ \\psi (x_t) \\gamma (x_t, x_{t+1}) / \\psi (x_{t+1}) q(x_t \\mid x_{t+1})$ where $\\psi$ denotes the look-ahead/twisting function, $\\gamma$ is the untwisted target transition and $q$ is the proposal. While the exact definitions of these components are method-specific, this weight structure arises from the expression given optimal proposal and optimal twisting function (Naesseth et al, 2019, Chapter 3). \n\nImportantly,  this weight construction introduces twisting functions in a telescoping manner such that their approximation errors cancel out over the full trajectory. This property ensures that the final target is correct, and **the resulting SMC procedure is asymptotically exact for both TDS and RDSMC**.\n\nHowever, we want to highlight **several important differences between TDS and RDSMC**:\n\n(1) **Problem setting:** The problem we address is fundamentally different from that of TDS. While TDS is designed for sampling from a distribution tilted by a pretrained diffusion model, our goal is to sample from an arbitrary distribution, assuming access to its unnormalized density. \n\n(2) **Design of key components:** Due to the difference in settings, our target distributions, proposal distributions, and the look-ahead functions require significantly different designs. In particular,\n  -  TDS is designed to sample from the conditional $p_\\theta (x_0 \\mid y)\\propto p_\\theta(x_0) p(y\\mid x_0)$ given a pretrained diffusion model $p_\\theta(x_0)$ and a likelihood model $p(y\\mid x_0)$ with observation $y$. **TDS’s twisting function** is $p(y \\mid \\hat x_\\theta (x_t))$ (their Eq 8), **the likelihood evaluated at  the denoising prediction by the pretrained model** $\\hat x_\\theta (\\cdot)$. This twisting function incorporates the observation $y$ into the weighting function (their Eq 11). Consequently, the particles that are better aligned with the observations $y$ are promoted. \n\n  - In our case, RDSMC aims to sample from $\\pi(x_0)$ using a diffusion-based proposal. We use $\\hat p(x_t)$ (Eq 10) as **the twisting function, which approximates the exact marginal** $p(x_t) = \\int p(x_t \\mid x_0) \\pi (x_0) dx_0$. This function integrates out the future information $x_0$, thereby encoding the target information $\\pi(x_0)$ into the intermediate weight. The resulting weight expression (Eq 15) allows us to promote particles that are more consistent with the target $\\pi(x_0)$. \n\n\n(3) **Use of auxiliary variables:** Our method introduces auxiliary randomness through the Monte Carlo estimation of the look-ahead function. Incorporating this randomness within our framework places our approach in the category of nested SMC methods (Naesseth et al 2015). In contrast, TDS does not use auxiliary variables and operates more like a standard SMC algorithm, without requiring nested sampling.\n\nWe will clarify the connections and differences in the revised paper."}, {"review_id": "LbpY67J7rl", "reviewer": "Reviewer_8TQX", "summary": "This paper proposes a novel SMC method to sample from unnormalized target distributions. Utilizing the diffusion framework and building upon diffusion-based SMC methods, this paper designs informative intermediate targets to guide the resampling of the particles at each step, and the new sampler is called RDSMC. Asymptotic consistency to estimate the target distribution is established theoretically as the number of particles diverge. Numerical experiments are performed on synthetic and real-world data.", "questions": "Some questions have been provided in the weakness section. My other questions are as follows.\n\n1. It is claimed that the density $q(u_t|x_t)$ need not be evaluated (line 137). However, that seems to be needed to compute the importance weights (in Alg. 2). Am I missing something?\n\n2. While in a different context, how is the design of the weighting functions similar to TDS?\n\nI would consider raising my score if my questions are properly addressed.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. A new SMC algorithm, RDSMC, for sampling from unnormalized targets\n2. RDSMC enables estimation of the normalization constant\n3. The paper provides a theoretical consistency result for estimating the target and the normalization constant\n\nWeaknesses:\n1. It seems that in most reported experiments (especially with higher dimensional data), SLIPS (sometimes along with other baseline methods) is performing better than the proposed RDSMC.\n2. Also, especially for estimating the normalization constant, there are some other recent diffusion-based samplers that are not included for numerical comparison. For example, while not using information from the unnormalized densities, PDDS also estimates the normalization constant which should be numerically comparable.\n3. Overall, the benefits of using SMC against MCMC (e.g., RDMC) with diffusion-based sampling methods are still not clear to me. I am thinking of benefits other than the final distributional distance (e.g., computation, what is required for estimation, etc.).\n4. There are occasional typos that need to be fixed: \"an exception\" -> \"exceptions\" (line 43); comma -> multiplication (line 99)", "was_revised": true, "final_justification": "The paper is not clear enough and needs calibration. That said, through the discussion with the authors I agree with some merit in the paper's contribution, which I think is worth being published.", "rebuttal": "We thank the reviewer for their thoughtful evaluation and are pleased to see they recognize the our work as a new SMC algorithm that provides unbiased normalization constant along with theoretical consistency guarantees. We also found the questions and suggestions raised helpful. We address these points below. \n\nNote: _if Latex equations do not render, please consider refreshing the page a couple of times_.\n\n\n> 1. It seems that in most reported experiments (especially with higher dimensional data), SLIPS (sometimes along with other baseline methods) is performing better than the proposed RDSMC.\n\nRespectfully, we would like to highlight that RDSMC offers strong performance across a variety of settings. Below we clarify specific points and provide supporting ablation results: \n\n- **RDSMC shows better or competitive performance without access to target variance and with fewer discretization steps:** As mentioned in line 249-255, competing methods including **AIS, SMC and SLIPS have access to the target’s scalar variance and use substantially more discretization steps** (T=1,024); SLIPS also uses an SNR-adapted time discretization schedule and a Langevin-based initialization. In contrast, RDSMC is run with T=100 uniform steps, does not use target variance and starts from a simple Gaussian $N(0,1)$. Despite these differences, RDSMC outperforms or matches these methods in most cases. In particular, RDSMC outperforms SLIPS along other baselines in the GMM weight ratio estimation task across varying dimensions $d \\in [2,64]$ (Figure 2 in appendix).\n\n- As a more neutral comparison, **RDMC is similar to SLIPS but  does not assume target variance, and RDSMC consistently outperforms RDMC across multiple targets.** \n\n- **RDSMC outperforms baselines given target variance**: To further isolate the effect of our core contributions, we include an ablation experiment on rings and funnel targets (see table below), where all methods use the target scalar variance (RDSMC uses it to initialize the proposal for score estimation). In these settings, **RDSMC outperforms SMC and SLIPS on most metrics.**\n\n\n| Method                            | Rings TVD               | Rings Log Z bias        | Funnel KS               | Funnel Log Z bias        |\n|----------------------------------|--------------------------|--------------------------|--------------------------|---------------------------|\n| RDSMC (T=200, use target var)    | $\\mathbf{0.102 \\pm 0.002}$ | $\\mathbf{0.012 \\pm 0.004}$ | $\\mathbf{0.065 \\pm 0.002}$ | $\\mathbf{0.219 \\pm 0.025}$ |\n| SMC (T=1024, use target var)     | $\\mathbf{0.096 \\pm 0.002}$ | $0.053 \\pm 0.002$         | $0.071 \\pm 0.000$         | $0.283 \\pm 0.005$          |\n| SLIPS (T=1024, use target var)   | $0.189 \\pm 0.003$         | NA                       | $\\mathbf{0.063 \\pm 0.001}$ | NA                        |\n\n\nWe will revise the manuscript to highlight these distinctions in settings and include ablation study results.\n\n> 2. There are other recent diffusion-based samplers that are not included for numerical comparison, e.g. PDDS. \n\nWe thank the reviewer for this suggestion. As noted in the Related Works section, **PDDS requires neural network training which introduces significant computation overhead**, while **our method is completely training-free**. For this reason our main comparisons are restricted to other training-free methods. We will include an ablation study comparing to PDDS in the revision. \n\n> 3. The benefits of using SMC against MCMC (e.g. RDMC) with diffusion-based sampling methods are not clear. \n\n- **RDSMC serves as a correction mechanism to RDMC**. When using the same number of samples, discretization steps, and score estimation strategy, **RDSMC runs at the same cost of RDMC but offers asymptotic bias correction** (Theorem 1). In empirical study, we observe that **RDSMC consistently outperforms RDMC across targets**. \n\n- Moreover, **RDSMC provides unbiased estimates of normalization constants while RDMC and typical MCMC methods do not**.  This capability is useful for downstream tasks including computing free energy in statistical physics, Bayesian model comparison and density estimation. \n\nWe will highlight these benefits in the revised paper. \n\n> 4. There are occasional typos that need to be fixed.\n\nThanks for identifying these typos and we will fix them in the revision. \n\n> 5. It is claimed that the density $q(u_t \\mid x_t)$ need not be evaluated (line 137). However, that seems to be needed to compute the importance weights (in Alg. 2). \n\nWe thank the reviewer for pointing this confusion out. You are absolutely correct that the sampling distribution $q(u_t \\mid x_t)$ is evaluated as part of the importance sampling-based score estimation procedure (Alg. 2). **What we intended to emphasize is that this distribution does not need to be evaluated in the outer SMC loop.** Specifically, $q(u_t \\mid x_t)$ does not appear explicitly in the weight computation in Eq (15), as it cancels out between the proposal and target. We will revise the manuscript to clarify our intention.\n\n >6. While in a different context, how is the design of the weighting functions similar to TDS?\n\n\nWe thank the reviewer for pointing out this insightful connection between RDSMC and TDS (Wu et al 2023). While the goals and problem settings differ, **both methods incorporate “look-ahead” or \"twisting\" functions into the weighting functions.** These twisting functions introduce future information to intermediate states,  promote promising particles through the weighting procedure, and improve the SMC sampling efficiency. \n\n**In both TDS and RDSMC, the incremental weight expression takes the same form** (their Eq 11 v.s. Our Eq 15)\n$ \\psi (x_t) \\gamma (x_t, x_{t+1}) / \\psi (x_{t+1}) q(x_t \\mid x_{t+1})$ where $\\psi$ denotes the twisting function, $\\gamma$ is the untwisted target transition and $q$ is the proposal. While the exact definitions of these components are method-specific, this weight structure arises from the expression given optimal proposal and optimal twisting function (Naesseth et al, 2019, Chapter 3). \n\nImportantly,  this weight construction introduces twisting functions in a telescoping manner such that their approximation errors cancel out over the full trajectory. This property ensures that the final target is correct, and **the resulting SMC procedure is asymptotically exact for both TDS and RDSMC**.\n\nTo further compare TDS and RDSMC, we clarify their exact settings:  \n- TDS is designed to sample from the conditional $p_\\theta (x_0 \\mid y)\\propto p_\\theta(x_0) p(y\\mid x_0)$ given a pretrained diffusion model $p_\\theta(x_0)$ and a likelihood model $p(y\\mid x_0)$ with observation $y$. **TDS’s twisting function** is $p(y \\mid \\hat x_\\theta (x_t))$ (their Eq 8), **the likelihood evaluated at  the denoising prediction by the pretrained model** $\\hat x_\\theta (\\cdot)$. This twisting function incorporates the observation $y$ into the weighting function (their Eq 11). Consequently, the particles that are better aligned with the observations $y$ are promoted. \n\n- In our case, RDSMC aims to sample from $\\pi(x_0)$ using a diffusion-based proposal. We use $\\hat p(x_t)$ (Eq 10) as **the twisting function, which approximates the exact marginal** $p(x_t) = \\int p(x_t \\mid x_0) \\pi (x_0) dx_0$. This function integrates out the future information $x_0$, thereby encoding the target information $\\pi(x_0)$ into the intermediate weight. The resulting weight expression (Eq 15) allows us to promote particles that are more consistent with the target $\\pi(x_0)$. \n\nFinally, while both methods use twisting functions, **RDSMC introduces auxiliary randomness $u_t$ for estimating $\\hat p(x_t)$, whereas TDS relies on a pretrained model**. This auxiliary randomness requires careful weight construction in RDSMC such that its influence cancels out properly (Eq 14 -> Eq 15). As a result, our method falls into the class of nested SMC methods, where TDS operates more in line with standard SMC. \n\nWe will incorporate this discussion into the revised manuscript to clarify the similarities and distinctions between TDS and RDSMC."}]

Author rebuttal:

Meta-review: The motivation of the work is the weakness of current diffusion-based samplers for unnormalized targets. Reverse diffusion methods rely on approximate scores and discretization, which introduce bias, and existing Monte Carlo corrections remain inefficient. The paper proposes Sequential Monte Carlo (SMC) as a natural framework to correct these issues, by reweighting and resampling biased diffusion-based proposals.

The main contribution is the design of a reverse diffusion sequential Monte Carlo algorithm. The method augments the state space with auxiliary variables that capture randomness in score estimation and incorporates them into the proposal and weights. This allows diffusion-based proposals to be used while their bias is corrected by SMC. The algorithm inherits standard and well-studied SMC guarantees and enables unbiased estimation of normalization constants, something not offered by competing samplers. Empirically, the method is competitive with or stronger than baselines, especially when fewer steps or no target variance are assumed.

The discussion clarified open questions. Theoretical guarantees were confirmed to follow from standard SMC theory on an extended space. Empirical concerns were addressed by pointing out that competing methods often use more computation or extra information, while ablations showed competitive or superior performance under matched conditions. The relation to twisted SMC was explained, with similarities in weight structure but differences in setting, design, and use of auxiliary randomness. The authors also committed to improve clarity and notation. Most reviewers raised their scores after these clarifications.

In conclusion, the paper provides a principled and original integration of diffusion-based proposals with sequential Monte Carlo, supported by theory and empirical results. Despite some clarity issues, the contribution is sound and of value to the ML / Monte-Carlo community.
Decision: Accept (poster)