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Bayesian Optimization of Noisy Log-Likelihoods Evaluated by Particle Filters -- One Parameter Case --

Genshiro Kitagawa

TL;DR

The paper tackles maximizing a noisy log-likelihood estimated by a particle filter, where Monte Carlo variability and non-differentiability hinder conventional optimization. It adopts Bayesian optimization with a Gaussian process surrogate and an iteration-dependent Upper Confidence Bound to locate the maximizer from a limited number of evaluations, incorporating an explicit observation-noise model. Through a one-parameter state-space example and extensive Monte Carlo experiments, it demonstrates robust estimation of both the maximizer and the maximum log-likelihood, aided by a normalization step that stabilizes the surrogate. The findings support using Bayesian optimization for likelihood maximization in settings where gradient-based or exhaustive search methods are impractical, and they establish a practical evaluation framework based on mean-squared error and convergence criteria.

Abstract

Likelihood functions evaluated using particle filters are typically noisy, computationally expensive, and non-differentiable due to Monte Carlo variability. These characteristics make conventional optimization methods difficult to apply directly or potentially unreliable. This paper investigates the use of Bayesian optimization for maximizing log-likelihood functions estimated by particle filters. By modeling the noisy log-likelihood surface with a Gaussian process surrogate and employing an acquisition function that balances exploration and exploitation, the proposed approach identifies the maximizer using a limited number of likelihood evaluations. Through numerical experiments, we demonstrate that Bayesian optimization provides robust and stable estimation in the presence of observation noise. The results suggest that Bayesian optimization is a promising alternative for likelihood maximization problems where exhaustive search or gradient-based methods are impractical. The estimation accuracy is quantitatively assessed using mean squared error metrics by comparison with the exact maximum likelihood solution obtained via the Kalman filter.

Bayesian Optimization of Noisy Log-Likelihoods Evaluated by Particle Filters -- One Parameter Case --

TL;DR

The paper tackles maximizing a noisy log-likelihood estimated by a particle filter, where Monte Carlo variability and non-differentiability hinder conventional optimization. It adopts Bayesian optimization with a Gaussian process surrogate and an iteration-dependent Upper Confidence Bound to locate the maximizer from a limited number of evaluations, incorporating an explicit observation-noise model. Through a one-parameter state-space example and extensive Monte Carlo experiments, it demonstrates robust estimation of both the maximizer and the maximum log-likelihood, aided by a normalization step that stabilizes the surrogate. The findings support using Bayesian optimization for likelihood maximization in settings where gradient-based or exhaustive search methods are impractical, and they establish a practical evaluation framework based on mean-squared error and convergence criteria.

Abstract

Likelihood functions evaluated using particle filters are typically noisy, computationally expensive, and non-differentiable due to Monte Carlo variability. These characteristics make conventional optimization methods difficult to apply directly or potentially unreliable. This paper investigates the use of Bayesian optimization for maximizing log-likelihood functions estimated by particle filters. By modeling the noisy log-likelihood surface with a Gaussian process surrogate and employing an acquisition function that balances exploration and exploitation, the proposed approach identifies the maximizer using a limited number of likelihood evaluations. Through numerical experiments, we demonstrate that Bayesian optimization provides robust and stable estimation in the presence of observation noise. The results suggest that Bayesian optimization is a promising alternative for likelihood maximization problems where exhaustive search or gradient-based methods are impractical. The estimation accuracy is quantitatively assessed using mean squared error metrics by comparison with the exact maximum likelihood solution obtained via the Kalman filter.
Paper Structure (21 sections, 18 equations, 4 figures, 2 tables)

This paper contains 21 sections, 18 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. State-Space Model and State Estimation
  4. Log-likelihood Evaluation
  5. Properties of Log-Likelihood Obtaine by Particle Filter
  6. Normalization of Log-Likelihood
  7. An illustrative Example
  8. Bayesian Optimization with Noisy Likelihoods
  9. Posterior of Gaussian Process
  10. Acqusition Function
  11. Iteration-Dependent Exploration Parameter $\kappa_t$
  12. Convergence Criterion
  13. Summary of Bayesian Optimization Procedure
  14. Numerical Experiments
  15. Experimental Setup
  16. ...and 6 more sections

Figures (4)

  • Figure 1: Upper panels: Log-likelihood obtained by particle filters with the praticle number. $m$=1,000, 10,000 and 100,000, 0.005$\leq \tau^2 \leq$ 0.025. Red curve indicates the true log-likelihood function obtained by the Kalman filter. Lower panels: standardized log-likelihoods.
  • Figure 2: Squared error of Bayesian optimization of log-likelihood computed by particle filter obtained by 100 repetition of optimization. Upper panels: $\log_{10}$MSE($x_i$) versus iter, lower panels: $\log_{10}$MSE($\ell(x_i)$) versus iter.
  • Figure 3: Posterior distribution of Bayesian process. iter=1, 3, 5, 10, 30 and 100. $\sigma_f=1$, $\sigma_n$=0.3, $\ell$=0.1, 0.2 and 0.3.
  • Figure 4: Convergence of $|x_{t+1}-x_t|$ and $|\ell(x_{t+1})-\ell(x_t)|$, Bayesian process parameters are $\sigma_f$=1, $\sigma_n$=0.3, $\ell$=0.2 and the number of particle $m$ is 100,000.