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lec2 - C260A Lecture 2 Intro to Inference Christopher Lee...

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C260A Lecture 2: Intro to Inference Christopher Lee September 29, 2009
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What is the fundamental difference between math and science ? 1
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A Diagnostic Test (T) for a Disease (D) T - T + total D + 1 9 10 D - 960 30 990 total 961 39 1000 p ( T + | D + ) = 9 / 10 ; p ( T - | D - ) = 960 / 990 2
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Why We Need Conditional Probability For any probability claim, we need to know its condi- tions! p ( A | B ) totally different meaning than p ( B | A ) ! Real life has a crucial asymmetry: we go from observ- able variables to hidden variables. Mixing these up is deadly! 3
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How can we define observable vs. hidden ? 4
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An observable is a variable whose value has zero uncertainty. 5
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An observable is a measurement whose operational definition yields a unique value with zero uncertainty (by definition). Everything else is hidden (by definition)! 6
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For each of the following items, state one observable and one hidden parameter associated with it: a six-sided die; a photon; your friend; a completed paper ballot; an email you received; a cryptographic signature. Explain precisely how each “observed parameter” fulfills the operational definition. 7
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Conditional Probability as Set Theory The basic set operations are intersection, union and negation . 8
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p ( C ) | S C | | S | p ( A B ) | S A B | | S | p ( A B ) = | S A B | | S A | | S A | | S | p ( B | A ) p ( A ) 9
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Bayes Law p ( H | O ) p ( O ) p ( O | H ) p ( H ) p ( H | O ) = p ( O | H ) p ( H ) p ( O ) posterior = likelihood × prior average likelihood 10
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Claim: This Equation Captures the Scientific Method p ( H | O ) = p ( O | H ) p ( H ) p ( O ) 11
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Hypothesis Testing p ( H | O ) = p ( O | H ) p ( H ) p ( O ) High likelihood high posterior 12
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Occam’s Razor p ( H | O ) = p ( O | H ) p ( H ) p ( O ) Given two models with equal likelihood, the one with a higher prior will have a higher posterior. What can we say in general about priors of complex hypotheses vs. simple hypotheses?
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