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The iterated auxiliary particle filter

Pieralberto Guarniero, Adam M. Johansen, Anthony Lee

arXiv:1511.06286v2stat.CO

TL;DR

Particle-filter likelihood estimates can vary substantially across models and observation sequences, limiting their practical robustness. The paper iteratively approximates an ideal zero-variance filter through twisted models, and reports improved performance over the bootstrap filter, including in parameter-estimation applications.

  • Problem

    Particle-filter estimates of marginal likelihood can vary considerably with the model or observations, although robust likelihood estimates are important for parameter inference using particle Markov chain Monte Carlo.

    Source: §1 · p. 3, §5 · p. 11

  • Method

    The iterated auxiliary particle filter repeatedly approximates the optimal positive-function sequence ψ* using particle-filter output, then uses the resulting filter to estimate the marginal likelihood.

    Source: §1 · p. 3, §3.4 · p. 8

  • Results

    The resulting estimates significantly outperform the bootstrap particle filter in the reported applications and show robustness to changes in latent-state dimension and model parameters.

    Source: §1 · p. 3, §6 · p. 21

  • Takeaways & Limitations

    The iAPF can be used as an ingredient in PMMH for Bayesian parameter estimation in general state-space HMMs.

    Source: §5 · p. 11, §6 · p. 21

    Practical use of ψ* is generally unavailable because computing the required functions and normalizers pointwise is not possible explicitly in general models.

    Source: §2 · p. 5

Abstract

from arXiv · show

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions psi and has an associated psi-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence psi* that is optimal in the sense that the psi*-auxiliary particle filter's estimate of L has zero variance. In practical applications, psi* is unknown so the psi*-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate psi*, and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm.

1 Introduction

The paper targets accurate marginal-likelihood estimation in hidden Markov models, where particle-filter estimates can be sensitive to the model and observations. It introduces an iterated auxiliary particle filter that approximates an ideal zero-variance filter and reports improved performance over the bootstrap filter.

1 Introduction

  • The marginal likelihood L associated with observations y1:T is the main statistical quantity of interest and serves as a likelihood function for unknown model parameters.The model is specified through an initial density, transition density, and observation density.Source: §1 · pp. 2–3
  • Particle-filter likelihood estimates are unbiased and converge almost surely, but their practical accuracy can vary with the model or observation sequence.That variability is undesirable for particle Markov chain Monte Carlo, where robust accuracy to small model changes is useful.Source: §1 · p. 3
  • The iterated auxiliary particle filter approximates an ideal sequence ψ* whose particle-filter estimate of L has zero variance, and significantly outperforms the bootstrap filter in the reported applications.The paper also reports robustness to increases in latent-state dimension and changes in model parameters.Source: §1 · p. 3
  • The paper places its method among twisted particle filters, auxiliary particle filters, block sampling, and look-ahead schemes, and restricts its exposition to time-homogeneous HMMs.The authors note that generalizing to time-inhomogeneous HMMs is fairly straightforward.Source: §1 · p. 3

2 Twisted models and the ψ-auxiliary particle filter

The section constructs alternative twisted models indexed by positive functions and uses their associated particle filters to estimate the original marginal likelihood. It identifies an optimal sequence yielding an exact estimate, then explains why that sequence is generally impractical and motivates approximation.

2 Twisted models and the ψ-auxiliary particle filter

  • Twisted models are formed from bounded, continuous, positive functions ψ, while an invariance result preserves the quantity of interest across the resulting model family.The associated BPF can also be viewed as an auxiliary particle filter, termed the ψ-APF.Source: §2 · pp. 3–4
  • For the sequence ψ* specified by the proposition, the ψ*-auxiliary particle filter estimate ZNψ* is exactly equal to L, with probability 1.Source: §2 · p. 5
  • Implementing the ψ-APF requires pointwise computation of ψ_t and its normalizing function, which generally prevents direct use of ψ*.The section therefore frames filter design as a sequence of function-approximation problems.Source: §2 · p. 5
  • Constant ψ recovers the bootstrap filter, while ψ_t(x_t)=g(x_t,y_t) gives the fully adapted auxiliary particle filter.Look-ahead choices also connect the family to delayed-sample and block-sampling methods, though exact strategies may be impractical.Source: §2 · p. 6

3 Function approximations and the iterated APF

The section motivates approximating the zero-variance choice by showing how the ψ-APF’s asymptotic variance behaves as ψ approaches ψ*. It then develops implementable function classes, a recursive approximation, and an iterative filter that updates ψ* estimates from particle-filter output.

3.1 Asymptotic variance of the ψ-APF

  • The ψ-APF asymptotic variance tends to zero as ψ approaches ψ* in an appropriate sense.This supports designing filters by approximating ψ*, although the optimal sequence is generally unavailable in practice.Source: §3.1 · p. 7

3.2 Classes of f and ψ

  • Practical implementation is restricted to Gaussian-mixture initial distributions and transition and ψ function classes that allow the required quantities to be computed explicitly.For the specified classes, ψt, f(x, ψt), and the resulting mixture transition components can be computed.Source: §3.2 · p. 7

3.3 Recursive approximation of ψ*

  • Algorithm 3 approximates ψ* recursively using a backward sweep that mirrors forward filtering backward smoothing.The exact backward sweep would yield ψ*, while the proposed procedure uses particle-filter output to construct approximations.Source: §3.3 · p. 8

3.4 The iterated auxiliary particle filter

  • The iAPF repeatedly runs a ψ-APF, re-estimates ψ* from its particles, and stops when the specified stopping rule is met.Particle numbers can be increased when estimates oscillate and have high relative variability; a monotonically increasing sequence is treated as evidence that approximations may be improving.Source: §3.4 · pp. 8–9

4 Approximations of smoothing expectations

The section extends particle-filter estimates to smoothing expectations, defined as conditional expectations of functions of the latent trajectory. It describes a consistent ratio estimate and clarifies that the likelihood-optimal ψ* need not optimize general smoothing expectations.

4 Approximations of smoothing expectations

  • A smoothing expectation π(ϕ) is the conditional expectation of a bounded, continuous function ϕ of the latent trajectory given the observations.It is written as γ(ϕ)/γ(1).Source: §4 · p. 9
  • The particle-filter estimate γN(ϕ) is unbiased and strongly consistent, and γN(ϕ)/γN(1) gives a strongly consistent approximation of π(ϕ).The construction uses an estimate of γ(ϕ) obtained from the particle filter.Source: §4 · pp. 9–10
  • The ψ*-APF is optimal for approximating γ(1) ≡ Z, but not for π(ϕ) in general.Source: §4 · p. 10

5 Applications and examples

The applications test the iAPF’s likelihood estimates and parameter-inference performance across linear Gaussian and stochastic-volatility models. Across these settings, the iAPF often improves on the BPF, including lower likelihood-estimate variability and better PMMH chain performance, with results depending on the model and setting.

5.1 Implementation details

  • The implementation used adaptive resampling with κ = 0.5 and a mixture transition proposal to guard against large or unbounded asymptotic-variance terms.The stopping rule used k = 5 for the linear Gaussian application and k = 3 for the stochastic-volatility applications; τ was 0.5 except for the multivariate model, where it was 1.Source: §5.1 · pp. 11–12

5.2 Linear Gaussian model

  • For d > 10, the BPF could not provide reasonable estimates in feasible time, and the FA-APF had the same difficulty for d > 20.The BPF and FA-APF took slightly longer on average than the iAPF; the iAPF’s final iteration used more than its 1,000 starting particles only at d = 40 and d = 80.Source: Relative variance of approximations of… · p. 13
  • Across α values from 0.3 to 0.5, iAPF likelihood-estimate variability was more consistent than the BPF’s, with the BPF taking at least five times longer.The comparison fixed d = 10 and used 1,000 replications; separate unreported simulations found the FA-APF slightly worse than the iAPF at double the computational time.Source: Relative variance of approximations of… · p. 14, Figure 2 · p. 14
  • iAPF-PMMH and Kalman-MH chains were practically indistinguishable in integrated autocorrelation time, while BPF-PMMH performed 3–4 times worse, depending on the parameter.The iAPF’s advantage over the BPF was less pronounced in experiments with larger δ than with δ = 0.25.Source: Particle marginal Metropolis–Hastings · p. 15, Figure 4 · p. 16

5.3 Univariate stochastic volatility model

  • In univariate stochastic volatility, PMMH chains showed an improvement with the iAPF, although the BPF-PMMH appeared fairly robust to likelihood-estimate variability in this application.Near the approximate MLE, iAPF likelihood estimates were significantly less variable than BPF estimates and may be more suitable for simulated maximum likelihood approximations.Source: §5.3 · pp. 17–18, Figure 6 · p. 18

5.4 Multivariate stochastic volatility model

  • In the 20-currency multivariate model, most Pound Sterling parameter posterior densities differed from their priors, and the pre- and post-crisis regimes showed qualitative differences.Differences in posterior expectations of m were positive for close to all currencies, suggesting generally higher post-crisis volatility; the analysis used a 79-dimensional parameter space.Source: §5.4 · pp. 19–20, Figure 7 · p. 20, Figure 8 · p. 21

6 Discussion

The iAPF iteratively approximates an idealized zero-variance particle filter, with empirical performance depending on the approximation approach and application setting. The discussion identifies possible extensions and computational trade-offs.

6 Discussion

  • The iAPF approximates an idealized particle filter with zero-variance marginal-likelihood estimates, and empirical studies used parametric optimization for models with Gaussian transitions.It performed reasonably in some regimes where the BPF could not provide adequate approximations and was applied to Bayesian parameter estimation using PMMH.Source: §6 · p. 21
  • For similar marginal-likelihood accuracy, the iAPF used orders of magnitude fewer particles than the BPF, potentially benefiting models with space-complexity constraints.The authors also suggest possible use in simulated maximum-likelihood procedures.Source: §6 · p. 21
  • Viewing particle-filter design as function approximation may improve performance, and the authors identify more sophisticated methods for estimating ψ* as a direction for future work.They note alternatives to the parametric-optimization approach described in Section 5.1.Source: §6 · p. 21
  • Nonparametric ψ* estimates succeeded but cost much more than parametric optimization; diagonal Σ also performed fairly well when ψ* could not be perfectly represented by the chosen function class.The iAPF stopping rule could use variance estimators instead of multiple independent likelihood estimates, and resampling need not be multinomial.Source: §6 · p. 22

A Expression for the asymptotic variance in the CLT

The appendix proof of the asymptotic-variance expression defines a sequence of densities and uses an identity from Doucet and Johansen with elementary manipulations.

A Expression for the asymptotic variance in the CLT

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