Adaptive Monte Carlo on multivariate binary sampling spaces

A Monte Carlo algorithm is said to be adaptive if it can adjust automatically its current proposal distribution, using past simulations. The choice of the parametric family that defines the set of proposal distributions is critical for a good performance. We treat the problem of constructing such parametric families for adaptive sampling on multivariate binary spaces. A practical motivation for this problem is variable selection in a linear regression context, where we need to either find the best model, with respect to some criterion, or to sample from a Bayesian posterior distribution on the model space. In terms of adaptive algorithms, we focus on the Cross-Entropy (CE) method for optimisation, and the Sequential Monte Carlo (SMC) methods for sampling. Raw versions of both SMC and CE algorithms are easily implemented using binary vectors with independent components. However, for high-dimensional model choice problems, these straightforward proposals do not yields satisfactory results. The key to advanced adaptive algorithms are binary parametric families which take at least the linear dependencies between components into account. We review suitable multivariate binary models and make them work in the context of SMC and CE. Extensive computational studies on real life data with a hundred covariates seem to prove the necessity of more advanced binary families, to make adaptive Monte Carlo procedures efficient. Besides, our numerical results encourage the use of SMC and CE methods as alternatives to techniques based on Markov chain exploration.