An Interacting Particle Method for Approximate Bayes Computations
Abstract
Approximate Bayes Computations (ABC) are used for parameter inference when the likelihood function is expensive to evaluate but relatively cheap to sample from. In ABC, a population of particles in the product space of outputs and parameters is propagated in such a way that its output marginal approaches a delta function at the measured output and its parameter marginal approaches the posterior distribution. Inspired by simulated annealing, we present a new class of particle algorithms for ABC, based on a sequence of Metropolis kernels, associated with a decreasing sequence of tolerances w.r.t. the measured output. Unlike other algorithms, our class of algorithms is not based on importance sampling. Hence, it does not suffer from a loss of effective sample size due to re-sampling. We prove convergence under a condition on the speed at which the tolerance is decreased. Furthermore, we present a scheme that adapts the tolerance according to the mean and the standard deviation of the distance of the particles from the measured output, and the jump distribution in parameter space according to the covariance of the population. These adaptations can be interpreted as mean-field interactions between the particles. Thus, the statistical independence of the particles is preserved, in the limit of infinite sample size. The performance of this new class of algorithms is investigated with a toy example, for which we have an analytical solution.Keywords
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