Lecture10.pdf - Stat210A Theoretical Statistics Lecture Date Sept 30 2014 Bayesian Inference Lecturer Tamara Broderick 1 Scribe Ross Boczar Being

# Lecture10.pdf - Stat210A Theoretical Statistics Lecture...

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Stat210A: Theoretical Statistics Lecture Date: Sept. 30, 2014 Bayesian Inference Lecturer: Tamara Broderick Scribe: Ross Boczar 1 Being Bayesian Bayes’ Theorem (for densities): X, Θ are random variables p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x ) Joint p ( x, θ ) Marginal p ( θ ) Conditionals p ( x | θ ) , p ( θ | x ) Want to use these to do inference One school of thought: statistics ⇐⇒ “inverse probability” Probabilistic model/simulation: fixed θ random X Statistical inference: observe x knowledge of Θ Bayesian thinking: Represent knowledge of Θ with a distribution: p ( θ ) (prior); p ( θ | x ) (posterior) Example Θ: true proportion of U.S. residents that watch Game of Thrones X : # who watch out of n residents Bayesian model: p ( x | θ ) = Bin( x | n, θ ) = n x θ x (1 - θ ) n - x p ( θ ) = Beta( θ | a, b ) = Γ( a + b ) Γ( a )Γ( b ) θ a - 1 (1 - θ ) b - 1 1

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2 Bayesian Inference For the prior distribution, a, b are known as hyperparameters . Note that when a = b = 1, we have a uniform distribution. In general, a, b changing can represent us having more information about θ . Some useful identities: E [Θ] = a a + b Var(Θ) =
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