Monte Carlo State-Space Likelihoods by Weighted Posterior Kernel Density Estimation

Abstract

Maximum likelihood estimation and likelihood ratio tests for nonlinear, non-Gaussian state-space models require numerical integration for likelihood calculations. Several methods, including Monte Carlo (MC) expectation maximization, MC likelihood ratios, direct MC integration, and particle filter likelihoods, are inefficient for the motivating problem of stage-structured population dynamics models in experimental settings. An MC kernel likelihood (MCKL) method is presented that estimates classical likelihoods up to a constant by weighted kernel density estimates of Bayesian posteriors. MCKL is derived by using Bayesian posteriors as importance sampling densities for unnormalized kernel smoothing integrals. MC error and mode bias due to kernel smoothing are discussed and two methods for reducing mode bias are proposed: “zooming in” on the maximum likelihood parameters using a focused prior based on an initial estimate and using a posterior cumulant-based approximation of mode bias. A simulated example shows that MCKL can be much more efficient than previous approaches for the population dynamics problem. The zooming-in and cumulant-based corrections are illustrated with a multivariate variance estimation problem for which accurate results are obtained even in 20 parameter dimensions.