Lecture 21 Notes

2 04 06 08 1 posterior after 4 h 7 t 5 4 3 2 1 0 0 02

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Unformatted text preview: 1 p(1 − p) pxi (1 − p)1−xi Z i P 1 1+Pi xi 1+ i (1−xi ) = p (1 − p) Z ￿ ￿ = Beta(2 + i xi , 2 + i (1 − xi )) Prior for p 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Posterior after 4 H, 7 T 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Posterior after 10 H, 19 T 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Predictive distribution Posterior is nice, but doesn’t tell us directly what we need to know We care more about P(xN+1 | x1, …, xN) By law of total probability, conditional independence: P (xN +1 | D) = = ￿ ￿ P (xN +1 , θ | D)dθ P (xN +1 | θ)P (θ | D)dθ Coin flip example After 10 H, 19 T: p ~ Beta(12, 21) E(xN+1 | p) = p E(xN+1 | θ) = E(p | θ) = a/(a+b) = 12/33 So, predict 36.4% chance of H on next flip Approximate Bayes Approximate Bayes Coin flip example was easy In general, computing posterior (or predictive distribution) may be hard Solution: use the approximate integration techniques we’ve studied! Bayes as numerical integration Parameters θ, data D P(θ | D) = P(D | θ) P(θ) / P(D) Usually, P(θ) is simple; so is Z P(D | θ) So, P(θ | D) ! P(D | θ) P(θ) ‣ similarly for conditional model: if X ⊥ θ, ‣ P(θ | X, Y) ! P(Y | θ, X) P(θ) Perfect for MH P(I....
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This note was uploaded on 01/24/2014 for the course CS 15-780 taught by Professor Bryant during the Fall '09 term at Carnegie Mellon.

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