Introduzione alla Statistica bayesiana - Politecnico di Milano

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Previsione di nuove osservazioni Observe n tosses of a coin: (y 1 , . . . , y n ) (y i = 1 if the coin was H at the i-th toss) What is the probability that the next toss will be H? Bayesian prediction: P(Y n+1 = 1|Y 1 = y 1 , . . . , Y n = y n ) Posterior predictive distribution: L(Y new |Y = y) A. Guglielmi Bayesian Statistics 6
Prediction What is the probability that the next toss will be H? Likelihood: θ ∑ y i (1 − θ) n−∑ y i ; prior π(θ); posterior π(θ|y) ∫ P(Y n+1 = 1|Y 1 = y 1 , . . . , Y n = y n ) = ∫ = (0,1) (0,1) θπ(θ|y)dθ = E(θ|y) = α + ∑ y i α + β + n Ex: n = 10, ∑ y i = 3, α = β = 1: P(Y n+1 = 1|θ)π(θ|y)dθ P(Y 11 = 1|Y 1 = y 1 , . . . , Y n = y n ) = E(θ|y) = 4 12 = 1 3 ≠ 1 2 = E(θ) A. Guglielmi Bayesian Statistics 7
Page 1 and 2: . . Introduzione alla Statistica ba
Page 3 and 4: Bayes’ Theorem Once we obtain y,
Page 5: Stima dei parametri Viene fatta sin
Page 9 and 10: Bayesian Hierarchical Models Multil
Page 11 and 12: Bayesian Gaussian hierarchical mode

Introduzione alla Statistica bayesiana - Politecnico di Milano

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