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.
where N is the number of Monte Carlo samples in the parameter space and N × M is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC 2 algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.
In recent years, particle filtering has become a powerful tool for tracking signals and time-varying parameters of random dynamic systems. These methods require a mathematical representation of the dynamics of the system evolution, together with assumptions of probabilistic models. In this paper, we present a new class of particle filtering methods that do not assume explicit mathematical forms of the probability distributions of the noise in the system. As a consequence, the proposed techniques are simpler, more robust, and more flexible than standard particle filters. Apart from the theoretical development of specific methods in the new class, we provide computer simulation results that demonstrate the performance of the algorithms in the problem of autonomous positioning of a vehicle in a -dimensional space.
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