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Introduction to Probability Theory—Bayesian Inference Illustrated by Pass-Fail Trials
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Wolfram Technology Conference 2011
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Champaign, Illinois, USA
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This presentation illustrates Bayes' rule as the tool for inductive reasoning using the context of single parameter binary trials like: a head/tail coin toss, a pass/fail regulatory inspection, or a guilty/not guilty jury decision. The binomial distribution quantifies the probability of m passes in n trials, given p equals the passing probability in each trial. Bayes' rule infers the probability distribution function (PDF) for p from the data ( m passes in n trials) and easily generates stopping criteria by signaling during sequential trials when desired precision for the inferred value of p is achieved. When precision after n trials is insufficient, the Bayesian may perform k additional trials, while the frequentist must start over with k + n additional trials for a total of k + 2n trials. This is because the Bayesian prior uses available data while inferring the implications of new data. As used here, a likelihood function (LKF) differs from its underlying PDF by a factor, kk : LKF = kk*PDF and PDF = LKF/kk. Bayes' rule, in terms of likelihood functions, says the posterior LKF is the product of a prior LKF and a data LKF.
LKFpost[ p | m, n, c ] = LKFprior[ p ]*LKFdata[ m, n | c ].
The Bayesian credible interval (the range of p given m passes in n trials) is contrasted with the frequentist confidence interval (the range of p in a stated proportion of repeated experiments, each with n trials). The maximum likelihood estimate for p is shown to be a special case of the more general Bayesian inference for p.
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http://www.wolfram.com/events/technology-conference-2011
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| IntroductionToProbabilityTheory.cdf (278.9 KB) - CDF Document | | IntroductionToProbabilityTheory.nb (278.2 KB) - Mathematica Notebook |
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