Convergence properties of incremental Bayesian evolutionary algorithms with single Markov chains

Bayesian evolutionary algorithms (BEAs) are a probabilistic model of evolutionary computation for learning and optimization. Starting from a population of individuals drawn from a prior distribution, a Bayesian evolutionary algorithm iteratively generates a new population by estimating the posterior fitness distribution of parent individuals and then sampling from the distribution offspring individuals by variation and selection operators. Due to the non-homogeneity of their Markov chains, the convergence properties of the full BEAs are difficult to analyze. However, recent developments in Markov chain analysis for dynamic Monte Carlo methods provide a useful tool for studying asymptotic behaviors of adaptive Markov chain Monte Carlo methods including evolutionary algorithms. We apply these results to Investigate the convergence properties of Bayesian evolutionary algorithms with incremental data growth. We study the case of BEAs that generate single chains or have populations of size one. It is shown that under regularity conditions the incremental BEA asymptotically converges to a maximum a posteriori (MAP) estimate which is concentrated around the maximum likelihood estimate. This result relies on the observation that increasing the number of data items has an equivalent effect of reducing the temperature in simulated annealing.