Adaptive Monte Carlo on Multivariate Binary Sampling Spaces

A Monte Carlo algorithm is said to be adaptive if it can adjust automaticallyits current proposal distribution, using past simulations. The choice of the para-metric family that defines the set of proposal distributions is critical for a goodperformance. We treat the problem of constructing such parametric families foradaptive sampling on multivariate binary spaces.A practical motivation for this problem is variable selection in a linear regres-sion context, where we need to either find the best model, with respect to somecriterion, 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 foroptimisation, and the Sequential Monte Carlo (SMC) methods for sampling.Raw versions of both SMC and CE algorithms are easily implemented using bi-nary vectors with independent components. However, for high-dimensional modelchoice problems, these straightforward proposals do not yields satisfactory re-sults. The key to advanced adaptive algorithms are binary parametric familieswhich take at least the linear dependencies between components into account.We review suitable multivariate binary models and make them work in thecontext of SMC and CE. Extensive computational studies on real life data with ahundred covariates seem to prove the necessity of more advanced binary families,to make adaptive Monte Carlo procedures efficient. Besides, our numerical resultsencourage the use of SMC and CE methods as alternatives to techniques basedon Markov chain exploration.