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Particle Learning and Smoothing

Carlos M. Carvalho, Michael S. Johannes, Hedibert F. Lopes, Nicholas G. Polson

arXiv:1011.1098v1stat.ME

TL;DR

State-space models require sequential inference for states and static parameters, along with smoothing under parameter uncertainty. The paper proposes particle learning, a fully adapted particle method using conditional sufficient statistics, and reports superior performance to standard particle-filtering alternatives and competitiveness with MCMC. Its approach relies on conditional sufficient-statistic structure and faces additional limitations for nonlinear state evolution and Monte Carlo accuracy.

  • Problem

    State-space inference must estimate states and fixed parameters sequentially while also smoothing states over all data, but fixed-parameter learning with particles is notoriously difficult.

  • Method

    Particle learning samples jointly represented states and conditional sufficient statistics in a fully adapted resample–propagate filter for filtering, parameter learning, and smoothing.

  • Results

    PL dominates standard particle-filtering alternatives in estimation accuracy and is reported as a competitor to MCMC strategies.

  • Takeaways & Limitations

    PL provides an alternative to FFBS/MCMC for conditional DLMs and to MCMC for nonlinear non-Gaussian models, while generalizing the mixture Kalman filter to parameter learning and smoothing.

  • Takeaways & Limitations

    The method requires conditional sufficient-statistic structure, and nonlinear state evolution prevents use of state sufficient statistics.

Abstract

from arXiv · show

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.

1. INTRODUCTION

The paper introduces particle learning (PL) to jointly address sequential state filtering, parameter learning, and smoothing under parameter uncertainty. It extends particle methods through sufficient statistics and reports strong performance against particle-filtering alternatives and MCMC.

  • PL approximates filtering and smoothing distributions while accounting for uncertainty in static parameters across a broad class of state space models.
  • Its central algorithm samples from particles representing the joint posterior of states and conditional sufficient statistics in a fully adapted resample–propagate framework.
  • For Gaussian and conditionally Gaussian dynamic linear models, PL is defined over both state and parameter sufficient statistics.
  • PL generalizes the mixture Kalman filter by adding parameter learning and extends its applicability to nonlinear state evolutions and sequential smoothing.
  • Simulation studies report that PL dominates standard particle-filtering alternatives in estimation accuracy and competes with MCMC strategies.

2. PARTICLE FILTERING IN STATE SPACE MODELS

State-space inference targets the joint posterior of states and static parameters as observations arrive. Particle methods approximate this distribution sequentially, but state–parameter codependence complicates joint sampling.

  • Sequential filtering and parameter learning are represented by the joint posterior p(x_t,θ|y_t), where y_t contains observations through time t.
  • Particle methods use a discrete particle representation of the joint filtering distribution and update it as new data arrive.
  • Sequential sampling is difficult because state propagation depends on parameters while parameter inference depends on states.
  • The traditional pure-filtering setting assumes known parameters and includes the Kalman filter and original particle-filter applications.

The bootstrap filter

The bootstrap filter propagates state particles using the evolution model and weights them by the new observation likelihood. Its resampling step is used to limit particle degradation, producing a propagate–resample ordering.

  • The state predictive distribution is the basis for the bootstrap filter’s forward update.
  • Starting from particles approximating p(x_t|y_t), the filter propagates through p(x_{t+1}|x_t) and assigns weights proportional to p(y_{t+1}|x_{t+1}).
  • Resampling is optional because weighted particles can estimate quantities directly, but it is used to avoid decay in the particle approximation.
  • The bootstrap filter is called a propagate–resample filter because propagation precedes resampling.

Auxiliary particle filter (APF).

The auxiliary particle filter uses the next observation to select promising particles before propagation. This fully adapted resample–propagate design avoids importance sampling when predictive and propagation densities are tractable.

  • The APF represents the joint filtering distribution of consecutive states and performs resampling before propagation.
  • Its first resampling step uses weights proportional to the predictive p(y_{t+1}|x_t), selecting particles informed by the current observation.
  • When predictive and propagation densities can be evaluated and sampled, the APF obtains exact samples from p^N(x_{t+1}|y_{t+1}) without an importance-sampling step.
  • When exact adaptation is unavailable, the APF uses a guessed future state g(x_t) to define the first-stage importance function and then propagates via p(x_{t+1}|x_t).
  • Pre-selection ensures that only “good” particles are propagated, a feature identified as important for the proposed method’s success.
  • Sequential estimation of fixed parameters remains difficult because static parameters lack state evolution, causing particle atoms for p(θ|y_t) to decrease over time.

Storvik’s filter.

Storvik’s filter recursively updates low-dimensional sufficient statistics while using a propagate–resample strategy for sequential parameter learning.

  • Storvik’s filter: Storvik’s filter assumes p(θ|x_t,y_t) depends on recursively updated low-dimensional sufficient statistics.The statistics follow s_t+1 = S(s_t,x_t+1,y_t+1).
  • Storvik’s filter: Its proposal conditions on y_t+1 but still uses a propagate–resample structure.
  • Liu and West’s filter: Liu and West’s alternative approximates p(θ|y_t) with a mixture of multivariate normals.
  • Liu and West’s filter: The mixture uses shrinkage and overdispersion constants to generate fresh parameter samples and reduce particle decay.

3. PARTICLE LEARNING AND SMOOTHING

Particle learning combines conditional sufficient statistics with fully adapted resample–propagate updates to perform sequential filtering, parameter learning, and smoothing under parameter uncertainty.

  • Particle learning: PL represents parameter uncertainty through conditional sufficient statistics and optionally represents latent states with their sufficient statistics.The state representation supports Rao–Blackwellization when available.
  • Particle learning: The algorithm first resamples particles using predictive observation probabilities, then propagates states informed by the new observation.This ordering is intended to reduce compounding approximation errors.
  • Particle learning: Fully adapted PL can produce exact draws from the particle approximation when p(y_t+1|z_t) and p(x_t+1|z_t,y_t+1) are analytically tractable.
  • State sufficient statistics: Rao–Blackwellization produces flatter first-stage weights and increases algorithmic efficiency.
  • Smoothing: The same filtering pass retains particle approximations for all earlier times, enabling a direct backward sampling pass for smoothing.Backward smoothing samples filtered states with weights proportional to p(x_t+1|x_t,θ,y_t).
  • Model monitoring: PL directly provides filtered joint posteriors and predictive distributions, whereas MCMC would require repetition at each time point.

4. CONDITIONAL DYNAMIC LINEAR MODELS

For conditional dynamic linear models, PL filters joint sufficient statistics for states and parameters, extending mixture Kalman filtering to parameter learning and smoothing.

  • Conditional dynamic linear models: PL is more efficient for conditional dynamic linear models because it is a perfectly adapted filter.It also extends MKF with fixed-parameter learning and state smoothing.
  • Model class: The CDLM formulation accommodates normal, scale-mixture, and discrete-mixture observation and state-shock distributions.Hidden Markov auxiliary-state specifications are also discussed.
  • Conditional dynamic linear models: In CDLMs, joint state filtering and parameter learning reduce to filtering the joint distribution of their respective sufficient statistics.
  • PL algorithm: For discrete auxiliary states, PL resamples using predictive probabilities, propagates states, and updates state and parameter sufficient statistics.The required conditional densities are available for evaluation and sampling in the considered models.
  • Smoothing: Backward propagation combines with both PL algorithms to draw the full marginal smoothing distribution for all states.
  • Dynamic factor example: In the dynamic factor example, PL tracks time-varying loadings through a discrete Markov switching state and learns its transition probabilities.

5. NONLINEAR FILTERING AND LEARNING

PL extends to nonlinear conditional Gaussian state-space models by retaining predictive evaluation and parameter sufficient-statistic learning when state sufficient statistics are unavailable.

  • Nonlinear filtering: PL extends to conditional Gaussian dynamic models with nonlinear evolution equations.
  • Nonlinear filtering: Nonlinearity prevents use of state sufficient statistics, but PL can still evaluate the predictive distribution.
  • Smoothing: The backward smoothing strategy extends to obtain samples from p(x_T|y_T) in the nonlinear setting.
  • Nonlinear example: The heavy-tailed nonlinear example uses a t-Student observation distribution with ν degrees of freedom.
  • PL algorithm: The nonlinear algorithm resamples particles by predictive likelihood, propagates states and parameters, and retains parameter sufficient-statistic updates.
  • Simulation evidence: A simulated example reports that sequential parameter learning efficiently covers the true parameters and compares PL with MCMC and a benchmark particle filter.

6. COMPARING PARTICLE LEARNING TO EXISTING METHODS

The simulations compare PL with state-sufficient-statistics variants, standard particle filters, and MCMC, assessing filtering, parameter learning, smoothing, accuracy, and computational behavior. Across these experiments, PL is competitive or superior, while PLsuff can reduce Monte Carlo error in smaller particle sets.

  • PL versus state sufficient statistics: PLsuff has smaller Monte Carlo error and less bias than PL for smaller particle sets, particularly when estimating σ2 and τ 2.With one long set, differences dissipate; comparisons use one N = 100,000 set versus 20 N = 5000 sets.
  • PL versus particle filters: PL is uniformly better than BF, FABF, and APF in logarithm relative mean square error for pure filtering.The comparison uses MSE relative to BF, and PL differs algorithmically from FABF by reversing propagate–resample steps.
  • PL versus particle filters: Across quantiles and error-variance combinations, PL is at least as good as FABF and often significantly better than BF when learning τ 2.The evaluation reports mean absolute errors for sequential distributions of x_t and τ 2.
  • PL versus Liu–West: PL favors Liu–West filtering uniformly across scenarios, with larger discrepancies when the signal-to-noise ratio τ/σ is small.The comparison estimates β and x_t with known σ2 and τ 2 across τ/σ values 0.1, 0.5, and 1.0.
  • PL versus FFBS: For smoothing with known parameters, PL and FFBS have costs that increase linearly with T, while PL outperforms FFBS in filtering times.For fixed T, FFBS cost increases linearly with N, whereas PL cost increases exponentially with N.
  • PL versus MCMC: PL combined with backward smoothing provides an alternative to MCMC, with significant improvements in computational time and Monte Carlo variability for parameter estimation.The paper contrasts PL’s one-pass particle computation with MCMC procedures involving high-dimensional Markov-chain convergence.

7. FINAL REMARKS

The paper presents particle learning (PL) tools that jointly support sequential parameter learning, state filtering, and smoothing across a broad class of state space models. Extensive simulations find that PL compares favorably with existing methods in accuracy and computational performance.

  • PL incorporates sequential parameter learning, state filtering, and smoothing for a large class of state space models.It extends mixture Kalman filtering to include parameter learning and supports nonlinear state evolutions.
  • PL provides an alternative to FFBS/MCMC methods for conditional dynamic linear models and nonlinear non-Gaussian models.The paper also positions PL as a generalization of the mixture Kalman filter approach.
  • Extensive simulation evidence evaluates PL against standard methods using computational time and accuracy.The comparisons include particle-filtering alternatives and MCMC-based strategies.
  • PL compares favorably with existing strategies and remains robust to particle degeneracies as sample size grows.The authors additionally describe the scheme as intuitive and easy to implement.
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