Lecture 21 Notes

# Pi qi 1piqi vers ri si 1risi vir ui vi 1uivi

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Unformatted text preview: actors ϕ0 ϕ1–ϕ4 setosa p versicolor q virginica 1–p–q lo m hi set. pi qi 1–pi–qi vers. ri si 1–ri–si vir. ui vi 1–ui–vi Continuous factors ϕ1 lo set. m p1 q1 vers. r1 s1 vir. u1 v1 hi Φ1 (￿, s) = 1–p1–q1 2 2 exp(−(￿ − ￿s ) /2σ ) 1–r1–s1 parameters ￿ , ￿ , ￿ ; 1–u1–v1 Discretized petal length constant σ 2 set vers vir Continuous petal length Simpler example H p T 1–p Coin toss 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, p1, q1, r1, s1, …) Hθ: 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 Continuous prior E.g., for coin toss, p ~ Beta(a, b): 1 P (p | a, b) = pa−1 (1 − p)b−1 B (a, b) Specifying, e.g., a = 2, b = 2: P (p) = 6p(1 − p) Prior for p 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Coin toss, cont’d Joint dist’n of parameter p and data xi: P (p, x) = P (p) ￿ i = P (xi | p) 6p(1 − p) ￿ i p (1 − p) xi 1−xi Coin ﬂip posterior P (p | x) = P (p) = ￿ i P (xi | p)/P (x) ￿...
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