COMS 4771 Spring 2015 Expectation - Maximization
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چکیده
Example 1 (Mixture of K Poisson distributions). The sample spaces are X = Z+ := {0, 1, 2, . . . } = and Y = [K] := {1, 2, . . . ,K}. The parameter space is Θ = ∆K−1 × R++, where ∆K−1 := {π = (π1, π2, . . . , πK) ∈ R+ : ∑K j=1 πj = 1}, R+ := {t ∈ R : t ≥ 0}, and R++ := {t ∈ R : t > 0}. Each distribution Pθ in P = {Pθ : θ ∈ Θ} is as follows. If (X,Y ) ∼ Pθ for θ = (π, λ1, λ2, . . . , λK), then Y ∼ Categorical(π), X |Y = j ∼ Pois(λj), j = 1, 2, . . . ,K. In other words, Prθ(Y = j) = πj , and Prθ(X = k |Y = j) = λj e−λj/k!. The joint probability of (x, y) ∈ X × Y under Pθ is
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