The proposal distribution will yield a random walk

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Unformatted text preview: en with hierarchical models, even with conjugate priors we are unable to express the posterior analytically. The reason Bayesian statistics is so widely used is because of the development of computational methods for simulating draws from the posterior distribution. Even though we are unable to express the posterior analytically, with a large enough sample of simulated draws we can compute statistics of the posterior with arbitrary precision. This approach is called Monte Carlo simulation. We will describe the two most commonly used Monte Carlo methods, which both fall under the umbrella of Markov Chain Monte Carlo (MCMC) methods: the MetropolisHastings algorithm, and Gibbs’ sampling. 6.1 Markov chains The results from MCMC depend on some key results from the theory of Markov chains. We will not do a thorough review of Markov Chains and will rather present the necessary results as fact. A continuous state Markov chain is a sequence θ0 , θ1 , . . . with θt ∈ Jd that satisfies the Marko...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

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