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Unformatted text preview: CSE 6740 Lecture 24 How Do I Evaluate HighDimensional Integrals? (Sampling) Alexander Gray [email protected] Georgia Institute of Technology CSE 6740 Lecture 24 – p. 1/ ? ? Today 1. Integration and Sampling 2. Monte Carlo Variance Reduction 3. Markov Chain Monte Carlo CSE 6740 Lecture 24 – p. 2/ ? ? Integration and Sampling Why integration, and why sampling. CSE 6740 Lecture 24 – p. 3/ ? ? Integration Suppose we want to find I = integraldisplay b ( x ) dx. (1) If x is lowdimensional, we can use standard quadrature techniques. However, quadrature techniques effectively grid up the space, so that their cost is exponential in the dimensionality D of x . CSE 6740 Lecture 24 – p. 4/ ? ? Integration Now suppose we have the form b ( x ) = a ( x ) f ( x ) , (2) where f is a probability density function. We get this form whenever we want to compute the expected value of a function a ( x ) , where x ∼ f : I = E ( a ) = integraldisplay a ( x ) f ( x ) dx. (3) CSE 6740 Lecture 24 – p. 5/ ? ? Integration and Sampling The law of large numbers ensures that the sample mean over iid samples from f converges to the integral: hatwide I = 1 S S summationdisplay s a ( x s ) → E ( a ) (4) as S → ∞ . hatwide I is an unbiased estimator of I . This is called Monte Carlo integration . CSE 6740 Lecture 24 – p. 6/ ? ? Integration and Sampling Its error is effectively its variance, which is 1 S integraldisplay ( a ( x ) E ( a )) 2 dx = σ 2 a /S. (5) An estimate of this is hatwide σ 2 = 1 S 1 S summationdisplay s parenleftBig a ( x s ) hatwide I parenrightBig 2 . (6) CSE 6740 Lecture 24 – p. 7/ ? ? Integration and Sampling Expectations are ubiquitous in statistics but this idea happens to be critical for making Bayesian statistics practicable. Recall that for a dataset { x } ≡ { x 1 , . . . , x N } , the likelihood is f ( { x } θ ) = f ( x 1 , . . . , x N  θ ) = N productdisplay i =1 f ( x i  θ ) = L ( θ ) , (7) and the posterior is f ( θ { x } ) = f ( { x } θ ) f ( θ ) integraltext f ( { x } θ ) f ( θ ) dθ = L ( θ ) f ( θ ) c ∝ L ( θ ) f ( θ ) (8) where c = integraltext f ( { x } θ ) f ( θ ) dθ . CSE 6740 Lecture 24 – p. 8/ ? ? Integration and Sampling Bayesians want to compute the posterior mean θ = E ( θ ) = integraldisplay θf ( θ { x } ) dθ. (9) Note that this has the form we specified, where the integrand θ ∼ f ( θ { x } ) and a ( θ ) = θ . So if we can draw samples θ 1 , . . . , θ S from the posterior f ( θ { x } ) , 1 S S summationdisplay s θ s → E ( θ ) (10) as S → ∞ . CSE 6740 Lecture 24 – p. 9/ ? ? Integration and Sampling Bayesians also want to compute the 1 α posterior interval ( a, b ) such that integraltext a −∞ f ( θ { x } ) dθ = integraltext ∞ b f ( θ { x } ) dθ = α/ 2 and P ( θ ∈ ( a, b ) { x } ) = integraldisplay b a f ( θ { x } ) dθ = 1 α. (11) This can also be done by drawing samples θ s from the posterior f ( θ { x } ) . We can approximate the posterior....
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 Fall '08
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 Markov chain, Markov chain Monte Carlo

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