Optimal search efficiency of Barker's algorithm with an exponential fitness function

Markov Chain Monte Carlo (MCMC) methods may be employed to search for a probability distribution over a bounded space of function arguments to estimate which argument(s) optimize(s) an objective function. This search-based optimization requires sampling the suitability, or fitness, of arguments in the search space. When the objective function or the fitness of arguments vary with time, significant exploration of the search space is required. Search efficiency then becomes a more relevant measure of the usefulness of an MCMC method than traditional measures such as convergence speed to the stationary distribution and asymptotic variance of stationary distribution estimates. Search efficiency refers to how quickly prior information about the search space is traded-off for search effort savings. Optimal search efficiency occurs when the entropy of the probability distribution over the space during search is maximized. Whereas the Metropolis case of the Hastings MCMC algorithm with fixed candidate generation is optimal with respect to asymptotic variance of stationary distribution estimates, this paper proves that Barker's case is optimal with respect to search efficiency if the fitness of the arguments in the search space is characterized by an exponential function. The latter instance of optimality is beneficial for time-varying optimization that is also model-independent.