Here is the next problem set, due on Thursday, October 11.

Today we spent some time discussing Markov chains and stationary distributions. It is our goal to simulate a sample by constructing our Markov chain so that we approach the stationary distribution (hopefully rapidly).

We then looked at a simple Metropolis-Hastings MCMC sampler for generating a sample from a standard normal distribution. The code is in the charts. We saw how varying the proposal distribution’s width can affect the samples: Too narrow, and although the acceptance rated is high, it takes a long time for the sampler to get anywhere because successive samples are highly correlated; too broad and the acceptance rate goes way down to the point where the sampler hardly moves at all. Something in the middle works best. We also saw that if we start very far away from the peak of the distribution, it takes a while for the program, which is “hill-climbing”, to get to where there is a significant amount of probability. This is why people frequently discard the first part of the Markov chain as “burn-in”

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