Probability

Probability - Probability Overview Definitions Jargon Blood...

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Unformatted text preview: Probability: Overview, Definitions, Jargon; Blood Feuds BioE131/231 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 A word on notation Distributions & densities Discrete vs continuous Binomial Gaussian Cumulative distributions • Density function: • Cumulative distribution function: More definitions Similarly for probability density functions: etc. (replace sums by integrals) Normalization Independence “I.I.D.” “Uniform” Let’s get Bayesian 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) 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 Experimental error (cont.)(cont....
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This note was uploaded on 12/03/2011 for the course BIO 118 taught by Professor Staff during the Fall '08 term at Rutgers.

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Probability - Probability Overview Definitions Jargon Blood...

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