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lec25 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Acceptance-Rejection Method (AR) 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 29, 2007 Lecture 25 MCMC: Metropolis Hastings Algorithm A good reference is Chib and Greenberg ( The American Statistician 1995). Recall that the key object in Bayesian econometrics is the posterior distribution: f ( Y T | θ ) p ( θ ) p ( θ |Y T ) = R f ˜ ( Y T | θ ) dθ ˜ It is often difficult to compute this distribution. In particular, the integral in the denominator is difficult. So far, we have gotten around this by using conjugate priors – classes of distributions for which we know the form of the posterior. Generally, it’s easy to compute the numerator, f ( Y T | θ ) p ( θ ), but it is hard to compute ˜ ˜ the normalizing constant, the integral in the denominator, f ( Y T | θ ) dθ . One approach is to try to compute this integral in some clever way. Another, more common approac R h is Markov Chain Monte-Carlo (MCMC). The goal here is to generate a random sample θ 1 , .., θ N from p ( θ |Y T ). We can then use moments from this sample to approximate moments of the posterior distribution. For example, 1 E ( θ |Y T ) ≈ θ N X n There are a number of methods for generating random samples from an arbitrary distribution. Acceptance-Rejection Method (AR) The goal is to simulate ξ ∼ π ( x ). We can calculate for each the value of a function, f ( x ), such that ) ( x ) = f ( x π h k . The constant k is unknown. We have some candidate pdf ( x ) that we can simulate draws from, and there is a known constant c such that f ( x ) ≤ ch ( x ) We simulate draws from π ( x ) as follows: 1. Draw z ∼ h ( x ), u ∼ U [0 , 1] ) 2. If ≤ f ( z u z ch , ξ ( z then = . Otherwise repeat (1)....
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lec25 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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