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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.
- Spring '12