NotesNov29 - Example A sample of Bernoulli random variable...

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Example A sample of Bernoulli random vari- able with a uniform prior on the probability for success θ . The posterior distribution of θ is Beta ( m +1 , n - m + 1). The two Bayes estimators of θ : The mean of the posterior distribution: ˆ θ 1 = n i =1 X i + 1 n + 2 The maximizer of the posterior density: ˆ θ 2 = n i =1 X i n . Continuous observations Suppose that X 1 , X 2 , . . . , X n are iid continuous random variables with a pdf f X ( x ; θ ).
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Suppose that the unknown parameter θ can take values in a continuous set. We view θ as a value of a continuous ran- dom variable, Θ , with a prior density p .
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The posterior density of Θ is f ( θ | x 1 , . . . , x n ) = p ( θ ) Q n i =1 f X ( x i ; θ ) f X 1 ,...,X n ( x 1 , . . . , x n ) . The denominator f X 1 ,...,X n of the observations can be computed using the law of total prob- ability: f X 1 ,...,X n ( x 1 , . . . , x n ) = Z -∞ n Y i =1 f X ( x i ; θ ) p ( θ ) dθ.
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Example : Suppose that X 1 , X 2 , . . . , X n are iid normal random variables with an unknown mean θ and a known variance σ 2 0 .
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