86,502 research outputs found

    Fundamental Flaws in Feller's Classical Derivation of Benford's Law

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    Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable XX to be approximately Benford distributed, that is, for log⁥10X\log_{10}X to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of "spread" or measure of dispersionComment: 7 page

    Bayesian Posteriors Without Bayes' Theorem

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    The classical Bayesian posterior arises naturally as the unique solution of several different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. For example, the classical Bayesian posterior is the unique posterior that minimizes the loss of Shannon information in combining the prior and the likelihood distributions. These results, direct corollaries of recent results about conflations of probability distributions, reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.Comment: 6 pages, no figure

    Unauthorized Immigrants in California: Estimates for Counties

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    Estimates changes in unauthorized immigrant populations between 2001 and 2008 by county and zip code, including percentage of total population. Discusses the challenges of obtaining accurate counts and implications for policy
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