The famous probabilist and statistician Persi Diaconis wrote an article not too long ago about the "Markov chain Monte Carlo (MCMC) Revolution." The paper describes how we are able to solve a diverse set of problems with MCMC. The first example he gives is a text decryption problem solved with a simple Metropolis Hastings sampler.
I was always stumped by those cryptograms in the newspaper and thought it would be pretty cool if I could crack them with statistics.

Finding the best subset of variables for a regression is a very common task in statistics and machine learning. There are statistical methods based on asymptotic normal theory that can help you decide whether to add or remove a variable at a time. The problem with this is that it is a greedy approach and you can easily get stuck in a local mode. Another approach is to look at all possible subsets of the variables and see which one maximizes an objective function (accuracy on a test set, for example).

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