lec3 - C260A Lecture 3 A Recipe for Inference Christopher...

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C260A Lecture 3: A Recipe for Inference Christopher Lee October 1, 2009
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What’s the probability the sun will rise tomorrow? Pierre-Simon Laplace worked out a clever solution to this problem. .. 1
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The Binomial Likelihood Two outcomes: “success” vs. “failure” p ( s | θ , n ) = ± n s θ s ( 1 - θ ) n - s for s successes in n trials; p ( success ) = θ ; ( n s ) = n ! s ! ( n - s ) ! 2
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Pr ( θ | n = N ) = Pr ( n = N | θ ) Pr ( θ ) R 1 0 Pr ( n = N | θ ) Pr ( θ ) d θ = θ N R 1 0 θ N d θ = ( N + 1 ) θ N E ( θ | n = N ) = Z 1 0 θ Pr ( θ | n = N ) d θ = Z 1 0 θ ( N + 1 ) θ N d θ = N + 1 N + 2 3
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Pseudocount Principle Intuitively, we’d estimate θ = s s + f , where s = n , f = 0 Our Bayesian result is equivalent to s = n + 1 and f = 1 As if we added one “pseudocount” to each category. 4
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Estimating Disease Carrier Risk Women have two X chromosomes ( χχ ); men have one X and one Y ( χ Y ) Say a disease is caused by a mutation on the X chromosome ( χ d ), but only if you have no normal X chromosome(s). Estimate risk that a woman is a disease carrier ( χ d χ ) if she has a son with no disease symptoms ( S - ). 5
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A Recipe for Inference What is observable? What is hidden? What is the likelihood model that connects them? What is the prior? Sum over all possible models! 6
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What makes a “powerful” observation? p ( H | O ) = p ( O | H ) p ( H ) h p ( O | h ) p ( h ) 7
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highly unlikely under a very likely model. p
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This note was uploaded on 04/12/2010 for the course CHEM CHEM 260A taught by Professor Chrislee during the Spring '10 term at UCLA.

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lec3 - C260A Lecture 3 A Recipe for Inference Christopher...

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