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Lecture 21 Notes - 15-780 Grad AI Lecture 21 Bayesian...

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15-780: Grad AI Lecture 21: Bayesian learning, MDPs Geoff Gordon (this lecture) Tuomas Sandholm TAs Erik Zawadzki, Abe Othman
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Admin Reminder: project milestone reports due today Reminder: HW5 out
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Review: numerical integration Parallel importance sampling allows ZR(x) instead of R(x) biased, but asymptotically unbiased Sequential sampling (for chains, trees) Parallel IS + resampling for sequential problems = particle filter
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Review: MCMC Metropolis-Hastings: randomized search procedure for high R(x) Leads to stationary distribution = R(x) Repeatedly tweak current x to get x’ If R(x’) ! R(x), move to x’ If R(x’) << R(x), stay at x randomize in between Requires good one-step proposal Q(x’ | x) to get acceptable acceptance rate and mixing rate
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Review: Gibbs Special case of MH for X divided into blocks Proposal Q: pick a block i uniformly (or round robin, or any other fair schedule) sample X B(i) ~ P( X B(i) | X ¬B(i) ) Acceptance rate = 100%
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Review: Learning P(M | X ) = P( X | M) P(M) / P( X ) P(M | X , Y ) = P( Y | X , M) P( X | M) / P( Y | M) Example: framlings Version space algorithm: when prior is uniform and likelihood is 0 or 1
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Bayesian Learning
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Recall iris example H = factor graphs of given structure Need to specify entries of ϕ s ϕ 0 ϕ 0 ϕ 0 ϕ 0 ϕ 4 ϕ 3 ϕ 2 ϕ 1 Φ 1 params X 1 X 2 X 3 X 4
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Factors lo m hi set. vers. vir. p i q i 1–p i –q i r i s i 1–r i –s i u i v i 1–u i –v i setosa p versicolor q virginica 1–p–q ϕ 0 ϕ 1 ϕ 4
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Continuous factors lo m hi set. vers. vir. p 1 q 1 1–p 1 –q 1 r 1 s 1 1–r 1 –s 1 u 1 v 1 1–u 1 –v 1 ϕ 1 Discretized petal length Continuous petal length Φ 1 ( , s ) = exp( ( s ) 2 / 2 σ 2 ) parameters set , vers , vir ; constant σ 2
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Simpler example H T p 1–p Coin toss
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Parametric model class H is a parametric model class: each H in H corresponds to a vector of parameters θ = (p) or θ = (p, q, p 1 , q 1 , r 1 , s 1 , …) H θ : X ~ P( X | θ ) (or, Y ~ P(Y | X , θ )) Contrast to discrete H , as in version space Could also have mixed H : discrete choice among parametric (sub)classes
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Continuous prior E.g., for coin toss, p ~ Beta(a, b): Specifying, e.g., a = 2, b = 2: P ( p | a, b ) = 1 B ( a, b ) p a 1 (1 p ) b 1 P ( p ) = 6 p (1 p )
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Prior for p 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5
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Coin toss, cont’d Joint dist’n of parameter p and data x i : P ( p, x ) = P ( p ) i P ( x i | p ) = 6 p (1 p ) i p x i (1 p ) 1 x i
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Coin flip posterior P ( p | x ) = P ( p ) i P ( x i | p ) /P ( x ) = 1 Z p (1
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