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goodtalk - Introduction Goods 1967 paper Example Good...

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Introduction Good’s 1967 paper Example Good smoothing Good Smoothing Jim Albert Bowling Green State University December 8, 2009
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Introduction Good’s 1967 paper Example Good smoothing Outline Introduction Good’s 1967 paper Example Good smoothing
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Introduction Good’s 1967 paper Example Good smoothing Introduction General problem in categorical data analysis is how to handle small counts. Wald confidence interval for a proportion ˆ p - 1 . 96 r ˆ p (1 - ˆ p ) n , ˆ p + 1 . 96 r ˆ p (1 - ˆ p ) n ! does not work well for small n . P ( interval covers p ) is not uniformly 0.95.
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Introduction Good’s 1967 paper Example Good smoothing Ad-hoc solution Add small counts to data, and apply frequentist methods to the adjusted data. John Tukey suggested “starting” counts by 1/6. Agresti and Coull suggest adding “2 successes and 2 failures” to data, and then apply Wald interval estimate. In contingency tables with zero counts, common to add 1/2 to each cell.
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Introduction Good’s 1967 paper Example Good smoothing Why not Bayes? Adding imaginary counts corresponds to prior information. Leads to a Bayesian analysis. I. J. Good was one of the first to discuss the choice of imaginary counts in smoothing categorical data. Famous 1967 paper by Good “A Bayesian significance test for multinomial distributions” discusses his general approach.
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Introduction Good’s 1967 paper Example Good smoothing Testing problem Observe y = ( y 1 , ..., y t ) from multinomial distribution with sample size n and probabilities p = ( p 1 , ..., p t ). Test hypothesis H : p 1 = ... = p t = 1 t Usual test procedure is Pearson’s statistic: X 2 = t X j =1 ( y j - n t ) 2 n t which is asymptotically χ 2 ( t - 1).
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Introduction Good’s 1967 paper Example Good smoothing Motivation for Bayes Accuracy of chi-square approximation for small counts is questionable. Desirable to develop an “exact” Bayesian test free from asymptotic theory. Use procedure with confidence for all t and n .
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Introduction Good’s 1967 paper Example Good smoothing Bayes factor Ratio of marginal densities under the hypotheses H and A (not H ).
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