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Inference Networks for Sequential Monte Carlo in Graphical Models
Brooks Paige, Frank Wood
TL;DR
Designing effective SMC proposals is difficult, and prior-based proposals can be inefficient for informative observations. The paper constructs and learns a structured inverse model to provide proposals offline; these learned proposals can be reused for inference on new observations, provided the fixed generative model plausibly describes the data.
Problem
SMC requires proposal distributions that can be difficult to design, and prior proposals may be arbitrarily inefficient with tightly peaked likelihoods.
Source: §1 · p. 1Method
The method constructs an inverse factorization of a known generative model and learns a conditional density estimator for latent variables given observations, entirely offline.
Source: §3 · p. 3, §3.1 · p. 4, §3.2 · p. 5Results
The learned proposals can be reused to perform SMC inference for new settings of the observed variables.
Source: §1 · p. 1Takeaways & Limitations
The approach can automate proposal design and amortize SMC inference costs, while also serving as a structured neural-network training method when direct sampling is satisfactory.
Source: §5 · p. 8The method assumes a known fixed generative model that could plausibly generate the data later observed.
Source: §5 · p. 9
Abstract
from arXiv · showhide
We introduce a new approach for amortizing inference in directed graphical models by learning heuristic approximations to stochastic inverses, designed specifically for use as proposal distributions in sequential Monte Carlo methods. We describe a procedure for constructing and learning a structured neural network which represents an inverse factorization of the graphical model, resulting in a conditional density estimator that takes as input particular values of the observed random variables, and returns an approximation to the distribution of the latent variables. This recognition model can be learned offline, independent from any particular dataset, prior to performing inference. The output of these networks can be used as automatically-learned high-quality proposal distributions to accelerate sequential Monte Carlo across a diverse range of problem settings.
1. Introduction
Sequential Monte Carlo depends on effective proposal distributions, which are difficult to design and can be inefficient when based on the prior. The paper learns model-specific proposals offline so they can be reused for inference on new observations.
1. Introduction
- The method builds on SMC, which has been applied beyond state-space filtering to factor graphs, hierarchical Bayesian models, procedural graphics, and probabilistic programs.The paper situates its approach amid complementary memory-efficient and highly parallel SMC implementations.Source: §1 · p. 1
- Poorly chosen proposals can make SMC inefficient, while designing suitable proposals is a practical impediment to applying it to new models.Prior proposals may be arbitrarily inefficient with tightly peaked likelihoods, and some settings lack a natural or available prior.Source: §1 · p. 1
- The approach learns model-specific proposal distributions offline, before inference on a particular dataset, for reuse with new observations.The paper frames this as amortized inference: an artifact learned from the model can accelerate later inference tasks.Source: §1 · pp. 1–2
2. Preliminaries
The preliminaries define Bayesian-network inference as estimating latent variables given observations, then describe SMC’s sequential particle procedure and the proposal choices it requires. They also introduce autoregressive neural density estimation as a flexible model for those proposals.
2. Preliminaries
- A Bayesian network specifies a joint distribution and conditional-independence structure; inference estimates the latent-variable posterior given observed values, generally using approximations when exact inference is intractable.The unnormalized target is the joint density of latent and observed variables, with an unknown normalizing constant.Source: §2 · p. 2
2.1. Sequential Monte Carlo
- SMC incrementally samples and weights particles across a sequence of targets on increasing subsets of latent variables, with resampling favoring higher-weight particles.Each step’s proposal approximates a lower-dimensional transition to the next latent variable.Source: §2.1 · p. 2
2.2. Target densities and proposal kernels
- The ideal proposal follows the incremental change between target densities, but this conditional is usually intractable and must be approximated.Choosing intermediate target densities is application-specific, and selecting them does not remove the need to supply a proposal.Source: §2.2 · pp. 2–3
2.3. Neural autoregressive distribution estimation
- Autoregressive density estimators factor a high-dimensional distribution into one-dimensional conditionals; MADE’s masks ensure each output depends only on preceding inputs.With a sigmoid output, MADE represents p(x_n|x_1, . . . , x_{n−1}) for binary data.Source: §2.3 · p. 3
3. Approach
The approach reverses a graphical model’s factorization to make observed values inputs to an approximation of the latent posterior, then learns neural conditional densities offline from synthetic samples. The resulting proposal can preserve local dependencies, model jointly dependent latent variables, and be reused across datasets or repeated time steps.
3.1. Defining the inverse model
- The inverse model reverses the generative ordering so observed values become inputs, while its factorization preserves local conditional dependencies from the original graph.Its parent sets are formed from a variable’s Markov blanket restricted to later latent variables and observations; the resulting inverse graph need not be fully connected.Source: §3.1 · p. 4
3.2. Learning a family of approximating densities
- Offline training uses synthetic samples from the tractable joint model and requires gradients only for the recognition model, not the generative model.The learned mapping from observations to proposal parameters is intended to work across datasets; training can use stochastic gradient descent.Source: §3.2 · p. 5
- The inverse model’s conditionals are approximated with a product of smaller neural conditional densities, reducing the joint optimization to expectations requiring samples only of each variable and its inverse-model parents.The inverse graph is designed not to introduce conditional independencies absent from the original model; Proposition 1 states that certain local conditional independencies in the inverse graph also hold in the original.Source: §3.1 · p. 4, §3.2 · p. 5
3.3. Joint conditional neural density estimation
- For head-to-head structures, fully dependent latent variables sharing parents are modeled as a joint conditional density using a conditional extension of MADE.The model extends MADE by passing shared parents through its hidden layers; for real-valued data, its outputs parameterize a mixture of Gaussians for each dimension.Source: §3.3 · pp. 5–6
3.4. Training the neural network
- A hybrid procedure trains on synthetic data and refreshes training and validation sets when validation error rises or a maximum number of steps is reached.The authors report this appeared more efficient than drawing a new synthetic dataset for every gradient update.Source: §3.4 · p. 6
4. Examples
The examples apply learned inverse-factor proposals to polynomial regression, a hierarchical pump-failure model, and a factorial HMM. They show posterior approximations, marginal-likelihood estimation, and reduced particle degeneracy across these settings.
4.1. Inverting a single factor
- In polynomial regression, the pretrained neural-network proposal produces curves close to the MCMC posterior, though slightly more diffuse.The plotted proposals use identical learned network weights; the MCMC reference follows 10,000 burn-in samples and 10,000 thinned samples.Source: Figure 4 · p. 7
- The polynomial-regression example uses a joint proposal for all three latent weights because explaining away makes them all depend on the observations.The network has two hidden layers of 300 units and a three-Gaussian mixture output.Source: §4.1 · p. 7
4.2. A hierarchical Bayesian model
- The hierarchical model’s repeated inverse-factor structure lets one learned factor represent θ_n across replicates, simplifying learning relative to fitting the joint posterior.The divide-and-conquer SMC scheme samples replicated structures in parallel with independent particle sets, then samples α and β jointly to evaluate the final target terms.Source: §4.2 · pp. 7–8
- With as few as 5 samples, divide-and-conquer SMC with neural-network proposals achieves reasonable estimates of the normalizing constant.Figure 5 compares marginal-likelihood convergence against likelihood-weighted and neural-network importance sampling; it reports means over 10 runs with error bars of two standard deviations.Source: Figure 5 · p. 8, §4.2 · p. 8
4.3. Factorial hidden Markov model
- Across 30 time steps using 100 particles, transition-dynamics proposals nearly immediately degenerate to a single possible solution, while learned proposals increase effective sample size at each stage and reduce resampling.Different device combinations can yield identical total energy usage, making active-device combinations impossible to disambiguate from a single observation. Figure 6 reports means and standard deviations over 10 runs.Source: Figure 6 · p. 8, §4.3 · p. 8
5. Discussion
The discussion frames the approach as shifting SMC proposal design and inference costs out of application time. It relies on a trusted, fixed generative model and uses an objective chosen for SMC refinement.
5. Discussion
- The approach aims to compile away application-time SMC inference costs and automate manual proposal-density design.When direct model sampling is satisfactory, it can also be viewed as a graphical-model-regularized way to train interpretable neural networks, with an emulator chosen to approximate the generative model.Source: §5 · pp. 8–9
- Unlike variational autoencoders, the method assumes a known generative model with fixed parameters and structured latent variables, avoiding reliance on representative training data.This requires trusting that the generative model could plausibly generate the data later observed and conditioned on.Source: §5 · p. 9
- The authors choose D_KL(π||q), arguing it yields approximations more appropriate for SMC refinement than a variational Bayes objective.The passage points to prior discussions of zero-forcing behavior and pathological learned importance-sampling distributions.Source: §5 · p. 9