A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models

Abstract

This paper addresses the problem of Monte Carlo approximation of posterior probability distributions. In particular, we have considered a recently proposed technique known as population Monte Carlo (PMC), which is based on an iterative importance sampling approach. An important drawback of this methodology is the degeneracy of the importance weights when the dimension of either the observations or the variables of interest is high. To alleviate this difficulty, we propose a novel method that performs a nonlinear transformation on the importance weights. This operation reduces the weight variation, hence it avoids their degeneracy and increases the efficiency of the importance sampling scheme, specially when drawing from a proposal functions which are poorly adapted to the true posterior.For the sake of illustration, we have applied the proposed algorithm to the estimation of the parameters of a Gaussian mixture model. This is a very simple problem that enables us to clearly show and discuss the main features of the proposed technique. As a practical application, we have also considered the popular (and challenging) problem of estimating the rate parameters of stochastic kinetic models (SKM). SKMs are highly multivariate systems that model molecular interactions in biological and chemical problems. We introduce a particularization of the proposed algorithm to SKMs and present numerical results.

Particle filtering with transformed weights

“…In that case classical particle filters become numerically unstable and fail to converge. This is a typical phenomenon in highdimensional models [6] but also, contrary to intuition, in systems where the observations present a high signal-to-noise ratio and hence the posterior probability mass is confined within a very small region of the space of the system variables [7].…”

“…The nonlinear IS (NIS) method of [7] entails the application of a nonlinear transformation to the ratio in Eq. (4), in such a way that the resulting transformed IWs (TIWs) have a smaller range of variation (and hence a smaller empirical variance) than the original IWs.…”

“…(4), in such a way that the resulting transformed IWs (TIWs) have a smaller range of variation (and hence a smaller empirical variance) than the original IWs. While various possibilities exist, we restrict our attention here to the clipping transformation of [7]. Specifically, let i 1 , ..., i N be a permutation of the indices {1, ..., N } such thatŵ …”

“…We propose to tackle the problem of weight degeneracy using a recently introduced variation of the IS technique called nonlinear importance sampling (NIS) [7]. In classical IS schemes, standard importance weights (IWs) are computed proportional to the ratio of the target probability density function (pdf) and the proposal density used to generate samples, and then they are normalized (so they add up to 1).…”

“…In a NIS scheme, transformed importance weights (TIWs) are additionally computed by applying a nonlinearity to the standard IWs before their normalization. The nonlinearity is chosen to reduce the variation of the IWs and hence mitigate their degeneracy, at the expense of introducing a certain distortion in the approximation of the target probability distribution 1 The benefit of performing NIS has been advocated for the population Monte Carlo class of methods in [7]. In this paper, we investigate the application of this technique to PFs.…”