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Elements of Sequential Monte Carlo

Christian A. Naesseth, Fredrik Lindsten, Thomas B. Schön

arXiv:1903.04797v3stat.MLcs.LGstat.CO

TL;DR

SMC addresses challenges in approximate inference by recursively approximating posterior distributions with weighted samples. The tutorial reviews SMC methods and design choices, including a result that importance-weighted independent SMC samplers give consistent expectation estimates as the number of samplers grows.

  • Problem

    As the number of iterations increases, importance sampling can suffer weight degeneracy, leaving the target distribution effectively approximated by a single sample.

    Source: §2.1.2 · p. 23

  • Method

    SMC recursively approximates posterior distributions with weighted samples, using proposals informed by the previous target approximation and resampling particles.

    Source: Introduction · p. 5, §2.1.2 · p. 23, §2.2 · p. 25

  • Results

    Importance-weighted independent SMC samplers give consistent estimates of expectations as the number of samplers M grows, regardless of the particle count N.

    Source: §4.4.1 · p. 83

  • Takeaways & Limitations

    SMC has been applied to inference in complex models across areas including probabilistic programming, variational inference, graphical models, and Bayesian nonparametrics.

    Source: Introduction · p. 5

    High-dimensional targets remain difficult for SMC, where the difficulty manifests as weight degeneracy and path degeneracy.

    Source: §2.2 · p. 28

Abstract

from arXiv · show

A core problem in statistics and probabilistic machine learning is to compute probability distributions and expectations. This is the fundamental problem of Bayesian statistics and machine learning, which frames all inference as expectations with respect to the posterior distribution. The key challenge is to approximate these intractable expectations. In this tutorial, we review sequential Monte Carlo (SMC), a random-sampling-based class of methods for approximate inference. First, we explain the basics of SMC, discuss practical issues, and review theoretical results. We then examine two of the main user design choices: the proposal distributions and the so called intermediate target distributions. We review recent results on how variational inference and amortization can be used to learn efficient proposals and target distributions. Next, we discuss the SMC estimate of the normalizing constant, how this can be used for pseudo-marginal inference and inference evaluation. Throughout the tutorial we illustrate the use of SMC on various models commonly used in machine learning, such as stochastic recurrent neural networks, probabilistic graphical models, and probabilistic programs.

Introduction

This tutorial presents sequential Monte Carlo as a method for approximating Bayesian inference with weighted samples, unifying its methods and illustrating applications in broader offline problems and machine learning. It covers probabilistic models and target distributions, SMC design choices, pseudo-marginal and properly weighted methods, conditional SMC and its applications, and concludes with the tutorial’s structure and code examples.

1.1 Historical Background

  • SMC began as particle filters for online inference in nonlinear state space models [Gordon et al., 1993, Stewart and McCarty, 1992, Kitagawa, 1993] and now applies to broader models.Sequential importance sampling and sampling/importance resampling laid foundations for SMC.Source: §1.1 · p. 6

1.2 Probabilistic Models and Target Distributions

  • SMC can address offline inference beyond traditional filtering when a sequence of targets ends at the distribution of interest.The final target may equal the posterior or contain it as a marginal; intermediate targets serve to help approximate it, and introducing data earlier can often improve inference accuracy.Source: §1.2 · pp. 6, 7, 9, 11

1.3 Applications

  • SMC has been successfully applied to probabilistic graphical models, where exact inference is often intractable and approximation is challenging.The graph structure encodes conditional independencies and lets users express prior information.Source: §1.3 · pp. 12–13
  • SMC-based approaches can serve as inference back ends for probabilistic programs, addressing the challenge of general inference across models expressed in a programming language [Wood et al., 2014].Probabilistic programs combine random draws and observations to define joint probabilistic models.Source: §1.3 · p. 13
  • SMC can sample latent stochastic processes for Bayesian nonparametric inference, including Dirichlet process mixture clustering [Fearnhead, 2004].The mixture model can handle an unknown, conceptually infinite number of components.Source: §1.3 · p. 14
  • SMC and importance sampling can support inference evaluation by bounding or estimating symmetric KL divergence between approximate and true or gold-standard posteriors [Grosse et al., 2015, 2016; Cusumano-Towner and Mansinghka, 2017].These results concern simulated data for the bounds and a gold-standard algorithm for the estimate.Source: §1.3 · pp. 14–15

1.4 Example Code

  • The tutorial uses inline Python code snippets to illustrate its algorithms and methods.It lists the modules needed to run the examples.Source: §1.4 · p. 15

1.5 Outline

  • The tutorial’s next chapter introduces importance sampling as a foundation for SMC, then covers its limitations, SMC’s remedies, practical issues, and theoretical results.Source: §1.5 · p. 15
  • Chapter 3 examines proposal adaptation and learning, then intermediate target distributions that help approximate the posterior.These are presented as the two key SMC design choices.Source: §1.5 · pp. 15–16
  • Chapter 4 proves unbiasedness of the SMC normalization-constant estimate and covers pseudo-marginal methods, properly weighted SMC, and distributed or parallel approaches.It also describes combining MCMC and SMC through pseudo-marginal algorithms; properly weighted SMC unites and extends random weights and nested SMC.Source: §1.5 · p. 16
  • Chapter 5 introduces conditional SMC, proves unbiasedness of the inverse normalization-constant estimate, and discusses IPMCMC and applications to evaluating approximate inference.IPMCMC combines SMC and CSMC to use multi-core and parallel computing architectures.Source: §1.5 · p. 16
  • Chapter 6 concludes the tutorial with a discussion and outlook.Source: §1.5 · p. 16

Importance Sampling to Sequential Monte Carlo

This section traces the progression from importance sampling and sequential importance sampling to sequential Monte Carlo, describing how resampling and propagation address weight imbalance while noting remaining challenges and the role of resampling choices and model and proposal conditions. It also covers properties of SMC approximations, including their convergence in distribution to the target as particle counts increase.

2.1 Importance Sampling

  • IS estimates target expectations by correcting proposal samples with importance weights, yielding a strongly consistent empirical approximation and an unbiased estimate of the normalization constant.The proposal is user chosen and assumed simple to sample from and evaluate.Source: §2.1 · pp. 17–19

2.1.1 Sequential Importance Sampling

  • SIS uses an autoregressive proposal and recursively updates weights, decomposing proposal design into T conditionals; its normalization-constant estimate is unbiased and consistent, but the ratio estimate is only consistent.The ratio estimate is strongly consistent but is not generally unbiased.Source: §2.1.1 · pp. 20–21

2.1.2 Shortcomings of Importance Sampling

  • IS and SIS variance generally increases exponentially with T, while normalized weights can collapse onto one particle; proposals that are not sufficiently heavy-tailed can also yield infinite weight variance.A proposal should cover the target’s bulk probability mass; designing one for high-dimensional models can be challenging.Source: §2.1 · p. 19, §2.1.2 · p. 23

2.2 Sequential Monte Carlo

  • SMC resamples from the previous empirical target approximation and propagates particles with conditional proposals, mitigating degeneracy and, for some sequence models, preventing final-marginal estimates from deteriorating as T grows.For other sequence models, SMC still tends to provide more accurate estimates in practice than IS.Source: §2.2 · pp. 24–25
  • SMC weights tend to be more evenly distributed than SIS weights because resampling focuses computation on promising particles, while low-weight particles are less likely to persist.The sample-diversity illustration shows this resampling behavior.Source: §2.2 · p. 27, Figure 2.2 · p. 28
  • Resampling alleviates weight degeneracy over time, but high-dimensional latent variables can still cause weight degeneracy, and resampling can collapse early trajectories to a single ancestor.Avoiding weight degeneracy can require particle counts that scale exponentially with log-weight variance; the tutorial focuses on proposals and targets that can ameliorate, but not solve, the problem.Source: §2.2 · pp. 28–29
  • With N = 20 and T = 100, increasing β to represent stronger long-range dependence reduces SMC log-normalizer estimation efficiency, increasing Monte Carlo variance and negative bias.The example varies β across 0.001, 0.1, 0.5, 0.7, and 0.99.Source: §2.2 · p. 29, Figure 2.4 · p. 31

2.2.1 Low-variance Resampling

  • Stratified and systematic resampling reduce resampling variance by correlating draws; systematic resampling uses one shared uniform and is more computationally efficient, but may fail to converge depending on sample ordering.Stratified resampling draws one uniform within each of N strata.Source: §2.2.1 · pp. 30–32

2.2.2 Effective Sample Size and Adaptive Resampling

  • Adaptive resampling uses effective sample size to trigger resampling only when weight variability crosses a threshold, with a corrected weight update when resampling is omitted.A common threshold is ESS below N/2, and adaptive resampling is often combined with low-variance resampling.Source: §2.2.2 · p. 33

2.3 Analysis and Convergence

  • SMC estimates integrals of functions under the unnormalized target and the normalizing constant unbiasedly, but the log-normalizer estimate is negatively biased; normalized-target expectations are not unbiased, though SMC is strongly consistent.The central limit theorem characterizes the limiting distribution and asymptotic variance, informing the precision of estimates.Source: §2.3 · pp. 34–35
  • Asymptotic variance can grow exponentially with latent dimension, while filtering variance can remain stable over time under forgetting conditions; normalizing-constant variance grows linearly with time, requiring N to scale at least linearly in t.The stable-in-time result applies under assumptions on the model and proposal, generally for test functions of the most recent latent variable.Source: §2.3 · pp. 36–37
  • Theorem 2.3.4 restates a generic sample bound on KL divergence from the expected SMC approximation to the target distribution.Source: §2.3 · p. 37
  • The expected SMC approximation may be continuous even though each constructed approximation is a point mass.Source: §2.3 · p. 37
  • As the number of particles increases, the SMC approximation tends in distribution to the true target.Source: §2.3 · p. 38

Learning Proposals and Twisting Targets

This section reviews how SMC proposals and intermediate or twisting targets can be designed or learned, including optimal and approximate methods that use future information. It also covers iterated auxiliary SMC for learning twisting potentials and the assumptions and possible limitations of these approaches.

3.1 Designing the Proposal Distribution

  • The proposal is a central SMC design choice: prior proposals can yield poor approximations with few particles, especially in high-dimensional latent spaces.Using the model prior as proposal is known as the bootstrap particle filter or bootstrap SMC.Source: §3.1 · p. 39

3.1.1 Locally Optimal Proposal Distribution

  • The locally optimal proposal minimizes the KL divergence between successive targets, equivalently minimizing incremental-weight variance, but is often computationally infeasible.Its importance weights may be intractable, or sampling from it may be too costly.Source: §3.1.1 · pp. 40–41

3.1.2 Approximations to the Optimal Proposal Distribution

  • In a bimodal-posterior example, Laplace, EKF, and UKF Gaussian approximations all miss bimodality, while EKF and UKF proposals are likely to outperform Laplace.Nested Monte Carlo approximations can approach the locally optimal proposal’s asymptotic variance as sample count increases, but increase complexity by a factor of M; they can offer lower memory use, parallelization, and sometimes better accuracy at equal compute.Source: §3.1.2 · pp. 43, 46

3.1.3 Learning a Proposal Distribution

  • Adaptive proposal learning uses the inclusive KL divergence to encourage coverage of target high-probability regions, while amortized inference learns a data-to-proposal mapping from simulated data.SMC-based stochastic gradients are biased and do not guarantee convergence to a local minimum, though they have proved useful in practice; amortized gradients need not use SMC.Source: §3.1.3 · pp. 48–50
  • Amortized proposal learning uses an unbiased gradient estimate that supports convergence to a local minimum, but requires a proposal effective across possible datasets and an accurate model for the observed data.The convergence claim follows standard stochastic approximation results.Source: §3.1.3 · p. 50
  • VSMC [Naesseth et al., 2018, Le et al., 2018, Maddison et al., 2017] fits the complete SMC approximation to the target, including resampling, but its objective and gradient are intractable.A practical gradient approximation ignores the resampling-gradient contribution and has been shown to work well in practice [Naesseth et al., 2018, Le et al., 2018, Maddison et al., 2017].Source: §3.1.3 · pp. 51–52

3.2 Adapting the Target Distribution

  • Because only the final target matters, intermediate targets can use future observations to improve its SMC approximation; these auxiliary or twisted methods address finite-particle path impoverishment from reweighting and resampling [Guarniero et al., 2017, Heng et al., 2017].The usual latent-variable targets omit future observations at each iteration, limiting how earlier particle states can represent later information.Source: §3.2 · p. 53, §3.2.1 · p. 53

3.2.1 Twisting the Target

  • Using each iteration’s marginal of the final target with the locally optimal proposal yields perfect samples from the final target and makes that proposal globally optimal.The optimal twisting potentials are typically unavailable analytically because they require difficult integration.Source: §3.2.1 · pp. 53–54

3.2.2 Designing the Twisting Potentials

  • Fully adapted SMC [Pitt and Shephard, 1999, Johansen and Doucet, 2008] uses one-step-ahead twisting; when optimal potentials are unavailable, parameterized positive potentials can be learned from their recursive condition.The learning construction defines potentials as expectations over the next transition and assumes this expectation can be computed analytically.Source: §3.2.2 · pp. 57–59
  • Iterated auxiliary SMC alternates running SMC with parameter updates based on the current approximation, repeating until a stopping criterion is met.The proposed stopping criterion uses the normalizing-constant estimate; parameter updates proceed recursively from t = T−1 to 1.Source: §3.2.2 · pp. 59–60
  • For a stochastic RNN, Gaussian-like twisting potentials satisfy the criteria for applying Algorithm 8 to this non-Markovian latent-variable model.Its transition and observation model parameters are defined by RNNs.Source: §3.2.2 · p. 60
  • The method makes strong assumptions on the model, and end-to-end optimization of independent twisting potentials and proposals might produce more accurate algorithms.Twisting target distributions remains an active research area [Guarniero et al., 2017, Heng et al., 2017, Lawson et al., 2018, Lindsten et al., 2018].Source: §3.2.2 · p. 61

Taylor and Unscented Transforms

Taylor and Unscented transforms approximate the conditional distribution of (x_t, y_t) given earlier latent states, using linearization or sigma points, respectively.

Taylor Transform

  • Taylor linearizes the transition and observation functions around zero-mean noise values to form a Gaussian approximation of (x_t, y_t) given x_1:t−1.The approximation identifies its mean and covariance blocks using v_t ∼ N(0,Q) and e_t ∼ N(0,R).Source: Taylor and Unscented Transforms · p. 62

Unscented Transform

  • The Unscented Transform jointly represents the noises as z_t=(v_t,e_t) and uses sigma points to approximate the conditional moments of (x_t,y_t).It then approximates x_t given x_1:t−1 and y_t as Gaussian; the sigma points are based on the first two moments of z_t.Source: Taylor and Unscented Transforms · p. 63
  • The Unscented Transform uses 2n_x+2n_y+1 sigma points, with weights determined using λ and the dimensions n_x and n_y.The design parameter α, β, and κ include λ = α²(n_x+n_y+κ)−n_x−n_y.Source: Taylor and Unscented Transforms · pp. 63–64
  • Like the EKF-based approximation, the UKF-based approximation still needs access to densities corresponding to Eq. (3.3), but unlike EKF it does not need derivative access and requires choosing α, β, and κ.α and κ control sigma-point spread, while β is related to the distribution of z_t; typical values are (α, κ, β) = (10^-3, 0, 2).Source: Taylor and Unscented Transforms · p. 64

Nested Monte Carlo: Algorithms and Applications

This section presents nested Monte Carlo methods for approximating idealized algorithms that are intractable because of unevaluable likelihoods or weights, or unsampleable proposals, while retaining favorable properties of the methods they emulate. It covers SMC normalization-estimate unbiasedness and its use in pseudo-marginal and proper-weight methods, nested and distributed SMC, and random-weights SMC.

4.1 The Unbiasedness of the Normalization Constant Estimate

  • The SMC normalization-constant estimate is non-negative and unbiased, a property that underpins several nested Monte Carlo methods.The chapter formalizes this result in Proposition 4.1.1 and provides a self-contained proof.Source: §4.1 · p. 66
  • The normalization estimate can be viewed as a function of the SMC algorithm’s random seed, a representation used in PMH and nested SMC.The seed comprises base auxiliary random variables from which particles and ancestor variables are generated.Source: §4.1 · p. 66

4.2 Particle Metropolis-Hastings

  • Particle-based methods combine importance sampling or SMC with Metropolis-Hastings to support approximate Bayesian inference when likelihoods are intractable.The chapter presents particle MCMC methods developed mainly in Andrieu et al. [2010].Source: §4.2 · p. 68

4.2.1 Particle Marginal Metropolis-Hastings

  • PMMH is Metropolis-Hastings on an extended space with the random seed, and its parameter samples have the posterior as their marginal distribution.It replaces the exact likelihood with a non-negative unbiased SMC estimate; the extended-space target has the required posterior marginal.Source: §4.2.1 · pp. 69, 71

4.2.2 Sampling the Latent Variables

  • With T=20 and 10,000 samples, PIMH estimates using N=5 and 10 yield distinctly non-Gaussian marginals, while N=20 gives more satisfactory finite-run results.PIMH uses an SMC approximation to propose latent-variable trajectories, accepting according to the ratio of normalization-constant estimates.Source: §4.2.2 · pp. 72–73, Figure 4.2 · p. 74

4.2.3 Correlated Pseudo-Marginal Methods

  • Correlating successive random seeds can reduce SMC computation by enabling fewer particles, addressing the high particle counts needed for accurate normalization estimates.The seed proposal must scale favorably with its dimension, which typically grows linearly with particle count; examples include preconditioned Crank-Nicolson, elliptical slice sampling, and Hamiltonian Monte Carlo.Source: §4.2.3 · p. 75

4.3 Proper Weights and Sequential Monte Carlo

  • Proper weighting permits particles and weights to be jointly random and supports algorithms including random-weights particle filters, SMC2, and NSMC methods [Naesseth et al., 2015, 2019].Under proper weighting, the numerator converges to ZC Eγ[h(x)] and the denominator to ZC, so their ratio converges to Eγ[h(x)].Source: §4.3 · pp. 75–76

4.3.1 Random Weights Sequential Monte Carlo

  • Random-weights SMC replaces intractable weights, such as exponentiated integrals in SDE models, with non-negative unbiased estimates.The method extends to SMC when unbiased weight estimates are conditional on previous samples.Source: §4.3.1 · pp. 77–78

4.3.2 Nested Sequential Monte Carlo

  • NSMC uses an internal SMC sampler over latent-state components to approximate the locally optimal proposal, enabling more effective proposals for high-dimensional states.For many models, this costs O(NM) operations and can be more accurate than standard SMC with NM particles using the prior proposal.Source: §4.3.2 · pp. 78–79

4.4 Distributed Sequential Monte Carlo

  • Distributed SMC uses nested Monte Carlo to parallelize computation, while particle-level parallelization must address resampling’s need for interaction between particles.Approaches discussed include MH or rejection sampling for resampling, or modifying resampling to limit interaction.Source: §4.4 · p. 82

4.4.1 Importance Weighted Independent SMC Samplers

  • Importance-weighting independent SMC samplers yields a consistent estimate as the number of nodes M grows, regardless of the particle count N per sampler.Weighting by each sampler’s normalizing-constant estimate avoids the finite-N bias of simply averaging sampler estimates; the aggregate is not expected to outperform one sampler with NM particles.Source: §4.4.1 · p. 83

4.4.2 The Island Particle Filter

  • The island particle filter [Vergé et al., 2015] runs an outer SMC sampler over M nodes, each with an inner SMC sampler of N particles, and recommends adaptive outer resampling.Resampling when the outer effective sample size falls below a threshold can reduce communication costs; increasing N can make outer resampling rare.Source: §4.4.2 · pp. 83–84

Proof of Unbiasedness

The proof establishes unbiasedness of the normalizing-constant estimate for Algorithm 3 with multinomial resampling and notes that the property also holds with low-variance and adaptive resampling.

Proof of Unbiasedness

  • The unbiasedness property applies beyond multinomial resampling, including low-variance and adaptive resampling schemes.The appendix proves the multinomial-resampling case and says the proof extends straightforwardly to those other cases.Source: Proof of Unbiasedness · p. 87
  • Nonnegative particle weights imply a nonnegative normalizing-constant estimate, bZ_T.The proof introduces ancestor-index variables when handling particle paths.Source: Proof of Unbiasedness · p. 87
  • The proof reduces an average of N^(T−1) values, each equal to 1, to establish the result.The stated result holds for any T.Source: Proof of Unbiasedness · pp. 87, 89

Conditional SMC: Algorithms and Applications

This section describes conditional SMC as a reference-particle method combining SMC and MCMC to produce target-stationary samplers for further inference. It also covers AIDE’s use of SMC or CSMC meta-inference to estimate PDFs or reciprocals, distinguishing approximate from gold-standard SMC and comparing its importance-weighted inference-evaluation bound with related work.

5.1 Conditional Sequential Monte Carlo

  • At N=5, iterated CSMC approximates γT(x1) using effectively a single sample, while ancestor sampling is already accurate; its extra step may cost proportional to T² versus standard CSMC’s often O(T).Ancestor sampling does not avoid path degeneracy, and whether its benefit outweighs its cost is case-dependent.Source: §5.1 · p. 93

5.2 The Unbiasedness of the Inverse Normalization Constant Estimate

  • CSMC's product-of-average-weights normalizer estimate is biased but consistent, while its inverse is non-negative and unbiased when the retained trajectory is target-distributed.The consistency claim holds as particle count tends to infinity; the inverse-estimate unbiasedness assumption requires a target-distributed retained trajectory, such as after the Markov chain reaches stationarity.Source: §5.2 · p. 95

5.3 The Expected SMC Empirical Distribution

  • CSMC provides an alternate representation of the expected SMC empirical distribution, useful in applications including variational SMC and inference evaluation.The representation expresses the expected SMC empirical distribution through a CSMC run with the sample trajectory retained.Source: §5.3 · pp. 95–96

5.4 Parallelization

  • IPMCMC [Rainforth et al., 2016] runs interacting SMC and CSMC nodes in parallel, generating several approximate target samples and supporting inference for complex probabilistic programs.Nodes update which ones run CSMC, trading off SMC exploration and CSMC exploitation; computation is parallel, and nodes communicate only a normalization-constant estimate each iteration.Source: §5.4 · pp. 96–98

5.5 Evaluating Inference

  • Inference evaluation targets symmetrized KL divergence (Jeffreys divergence), with BREAD [Grosse et al., 2016] bounding it for AIS on simulated data and AIDE [Cusumano-Towner and Mansinghka, 2017] covering general approximations.BREAD's stochastic upper bound uses forward and reverse AIS; AIDE compares an approximation with a gold-standard inference approximation rather than requiring simulated data.Source: §5.5 · pp. 98–100
  • SMC- or CSMC-based meta-inference estimates a distribution’s PDF or reciprocal without bias, up to a proportionality constant that cancels in Jeffrey’s divergence.AIDE uses these estimates as approximations of the PDFs and their reciprocals.Source: §5.5 · p. 101
  • The approach of Cusumano-Towner and Mansinghka [2017] is essentially equivalent to the bound in Eqs. (5.14)–(5.16), but uses an SMC approximation as the gold standard instead of the exact target.Superscripts q and p distinguish the approximation from the gold-standard algorithm; the divergence uses their CSMC distributions and normalizing-constant estimates, with q(x) and p(x|y) denoting the expected SMC empirical distributions.Source: §5.5 · p. 101
  • AIDE’s procedure uses an importance-weighted version of Eq. (5.18), equivalent to the bound based on one meta-inference run for each algorithm (Mt = 1 and Mg = 1).The bound differs slightly from that derived by Cusumano-Towner and Mansinghka [2017].Source: §5.5 · p. 101

5.A Proof of Unbiasedness

The proof fixes a reference trajectory as one particle and focuses on multinomial resampling at each iteration.

5.A Proof of Unbiasedness

N T QCSMC

The proof identifies terms in the expression with the SMC distribution and normalization-constant estimator, then simplifies the result using QSMC.

N T QCSMC

  • Multiplying and dividing by q1 identifies the SMC distribution in the integrand.Source: N T QCSMC · pp. 102–103
  • Rewriting normalized weights in the denominator together with N^-T identifies the normalization-constant estimator.Source: N T QCSMC · p. 103
  • The normalization-constant estimators cancel, simplifying the remaining denominator.Source: N T QCSMC · p. 103
  • The proof marginalizes bT and uses QSMC as a distribution over particles to complete the derivation.Source: N T QCSMC · p. 103

Proof of SMC Distribution Representation

The proof establishes an alternate representation of the expected SMC empirical distribution by matching expectations of bounded continuous functions.

Proof of SMC Distribution Representation

Jeffreys Divergence Bound

The bound is derived under multinomial resampling, using conditional Jensen inequalities. The argument extends to a Jeffreys divergence bound between generic expected SMC empirical distributions.

Jeffreys Divergence Bound

Discussion

The discussion presents SMC as a weighted-sample approach to approximate inference and summarizes extensions spanning theory, learned proposals, and parallel algorithms. It highlights open challenges in jointly learning proposals and twisting potentials for complex models.

Discussion

  • SMC approximates latent-variable distributions by sampling from a simpler proposal and correcting proposal–posterior discrepancy with weights.The discussion characterizes SMC as an alternative or complement to MCMC.Source: Discussion · p. 106
  • The tutorial summarizes SMC’s roots, general algorithm, practical and theoretical results, proposal and target learning, and applications including PMH, nested SMC, and distributed computing.It reports consistency and a central limit theorem for the SMC estimate under appropriate assumptions, and provides a proof of unbiasedness for the normalization-constant estimate.Source: Discussion · p. 106
  • CSMC provides unbiased estimates of the inverse normalization constant; IPMCMC combines SMC and CSMC to use parallel computing and evaluate approximate inference.The tutorial also discusses recent CSMC applications for evaluating approximate inference methods.Source: Discussion · p. 107
  • Efficient proposal learning remains an open research area, and jointly learning twisting potentials and proposals for complex models is challenging and largely unsolved.The discussion says twisting potentials can drastically improve accuracy.Source: Discussion · p. 107
  • Parallel and distributed SMC remains an active research direction, with possible opportunities in new algorithm designs and probabilistic programming.The discussion cites Lee and Whiteley [2016], Lee et al. [2010], and divide-and-conquer SMC.Source: Discussion · p. 107
  • SMC is used within larger iterative algorithms, while unbiased likelihood estimates have been used to construct tighter ELBO lower bounds for deep architectures with stochastic latent models.Examples include particle methods within MCMC, iterated filtering, expectation-maximization, SMC2, and nested SMC.Source: Discussion · pp. 107–108
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