bayes - Bayesian Estimation Bayes theorem If A1 A2 An is a...

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Bayesian Estimation Bayes’ theorem If A 1 ,A 2 , ··· n is a partition of the sample space, and B is any set, then for each i , P ( A i | B )= P ( B | A i ) P ( A i ) n j =1 P ( B | A j ) P ( A j ) = P ( B | A i ) P ( A i ) P ( B ) . f Y | X ( y | x f X,Y ( x, y ) f X ( x ) = f X,Y ( x, y ) R -∞ f X,Y ( x, y ) dy = f X | Y ( x | y ) f Y ( y ) R -∞ f X | Y ( x | y ) f Y ( y ) dy 1 θ is a random variable! X f ( x | θ ) Ω Θ h ( θ )(p r io r) X :ar v x : observed value Θ: arv θ : observed value Prior Data Posterior 2 X 1 , ,X n is a random sample from f ( x | θ ). Prior distribution of Θ: h ( θ ) Joint conditional pdf of X 1 , n given Θ= θ is: f ( x 1 | θ ) f ( x 2 | θ ) f ( x n | θ ) Joint pdf of X 1 , n and Θ is: g ( x 1 , ,x n f ( x 1 | θ ) f ( x 2 | θ ) f ( x n | θ ) h ( θ ) If Θ is a continous random variable, the joint marginal pdf of X 1 , n is: g 1 ( x 1 , n Z -∞ g ( x 1 , n ) 3 The conditional pdf of Θ, given X 1 = x 1 , n = x n is: (posterior) k ( θ | x 1 , n g ( x 1 , n ) g 1 ( x 1 , n ) = f ( x 1 | θ ) f ( x 2 | θ ) f ( x n | θ ) h ( θ ) g 1 ( x 1 , n ) 4

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Example : X 1 , ··· ,X n Bernoulli( θ ), 0 <θ< 1 f ( x 1 , ,x n | θ )= θ x i (1 - θ ) n - x i ∈{ 0 , 1 } Consider h ( θ )=6 θ (1 - θ ) , 0 1 Prior distribution theta prior 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 g 1 ( x 1 , n 6( n ¯ x + 1)!( n - n ¯ x + 1)! ( n + 3)! k ( θ | x 1 , n Γ( n +4) θ n ¯ x +1 (1 - θ ) n - n ¯ x +1 Γ( n ¯ x + 2)Γ( n - n ¯ x +2) This is Beta( n ¯ x +2 ,n - n ¯ x + 2).
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue.

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bayes - Bayesian Estimation Bayes theorem If A1 A2 An is a...

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