CALCULATING PROBABILITY DENSITIES WITH HOMOTOPY, AND APPLICATIONS TO PARTICLE FILTERS

Abstract

We explore a homotopy sampling procedure and its generalization, loosely based on importance sampling, known as annealed importance sampling. The procedure makes use of a known probability distribution to find, via homotopy, the unknown normalization of a target distribution, as well as samples of the target distribution. In the context of stationary distributions that are associated with physical systems the method is an alternative way to estimate an unknown microcanonical ensemble. We make the connection between the homotopy and a dynamics problem explicit. Further, we propose a reformulation of the method that leads to a rejection sampling alternative. We derive the error incurred in computing the target distribution normalization, when sample inversion is not possible. The error in the procedure depends on the errors incurred in sample averaging and the number of stages used in the computational implementation of the process. However, we show that it is possible to exchange the number of homotopy stages and the total number of samples needed at each stage in order to enhance the computational efficiency of the implemented algorithm. Estimates of the error as a function of stages and sample averages are derived. These could guide computational efficiency decisions on how the calculation would be mapped to a given computer architecture. Consideration is given to how the procedure can be adapted to Bayesian estimation problems, both stationary and non-stationary. The connection between homotopy sampling and thermodynamic integration is made. Emphasis is placed on the non-stationary problems, and in particular, on a sequential estimation technique know