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Individual adaptation: an adaptive MCMC scheme for variable selection problems

The increasing size of data sets has lead to variable selection in regression becoming increasingly important. Bayesian approaches are attractive since they allow uncertainty about the choice of variables to be formally included in the analysis. The application of fully Bayesian variable selection methods to large data sets is computationally challenging. We describe an adaptive Markov chain Monte Carlo approach called Individual Adaptation which adjusts a general proposal to the data. We show that the algorithm is ergodic and discuss its use within parallel tempering and sequential Monte Carlo approaches. We illustrate the use of the method on two data sets including a gene expression analysis with 22 577 variables.