Probability

# Probability - Probability Overview Definitions Jargon Blood...

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Probability: Overview, Definitions, Jargon; Blood Feuds BioE131/231

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Probability vs Statistics Statistics is a hard sell (on a good day) Most palatable approach that I know: Concentrate on modeling rather than tests “Bayesian” vs “Frequentist” schools Emphasize connection to information theory Physical limits on information storage & transmission Where probability meets signals & systems engineering
Bayesians and Frequentists Believe it or not, statisticians fight Frequentists (old school) Emphasis on tests : t, χ 2 , ANOVA… Write down competing hypotheses, but only analyze null hypothesis (!) Report “significance” (actually improbability) Bayesians (new school) Emphasis on modeling Build a model for all competing hypotheses Use probabilities to represent levels of belief

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A word on notation
Distributions & densities

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Discrete vs continuous Binomial Gaussian
Cumulative distributions Density function: Cumulative distribution function:

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More definitions
Similarly for probability density functions: etc. (replace sums by integrals) Normalization

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Independence
“I.I.D.”

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“Uniform”
Let’s get Bayesian

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Example: Fall ’05 admissions Accept ed (A=1) Rejecte d (A=0) Total California resident (C=1) 8,493 22,206 30,699 California nonresiden t (C=0) 1,162 5,098 6,260 Total 9,655 27,304 36,959 P(A=1, C=1) P(A=1 | C=1) P(A=1) P(C=1) P(C=1 | A=1)
Example: Fall ’05 admissions Accept ed (A=1) Rejecte d (A=0) Total California resident (C=1) 0.23 0.60 0.83 California nonresiden t (C=0) 0.03 0.14 0.17 Total 0.26 0.74 1.00 = 0.23 / 0.26 = 0.23 / 0.83 P(A=1, C=1) P(A=1 | C=1) P(A=1) P(C=1) P(C=1 | A=1)

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Bayesian inference Probabilities & frequencies are essentially alternative ways of looking at the same thing However... frequencies are sometimes more intuitive We will return to more examples of Bayes’ Theorem and inference
Experimental error

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Experimental error (cont.)
Approximate errors

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Shannon information Can also be interpreted as number of bits that an “ideal” data compression algorithm needs to encode message ‘x’
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