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Beta distribution
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=====Mean and sample size===== The beta distribution may also be reparameterized in terms of its mean ''μ'' {{nowrap|1=(0 < ''μ'' < 1)}} and the sum of the two shape parameters {{nowrap|1= ''ν'' = ''α'' + ''β'' > 0}}(<ref name=Kruschke2011>{{cite book|last=Kruschke|first=John K.|author-link=John K. Kruschke|title=Doing Bayesian data analysis: A tutorial with R and BUGS|year=2011|publisher=Academic Press / Elsevier|page=83|isbn=978-0123814852}}</ref> p. 83). Denoting by αPosterior and βPosterior the shape parameters of the posterior beta distribution resulting from applying Bayes' theorem to a binomial likelihood function and a prior probability, the interpretation of the addition of both shape parameters to be sample size = ''ν'' = ''α''·Posterior + ''β''·Posterior is only correct for the Haldane prior probability Beta(0,0). Specifically, for the Bayes (uniform) prior Beta(1,1) the correct interpretation would be sample size = ''α''·Posterior + ''β'' Posterior − 2, or ''ν'' = (sample size) + 2. For sample size much larger than 2, the difference between these two priors becomes negligible. (See section [[#Bayesian inference|Bayesian inference]] for further details.) ''ν'' = ''α'' + ''β'' is referred to as the "sample size" of a beta distribution, but one should remember that it is, strictly speaking, the "sample size" of a binomial likelihood function only when using a Haldane Beta(0,0) prior in Bayes' theorem. This parametrization may be useful in Bayesian parameter estimation. For example, one may administer a test to a number of individuals. If it is assumed that each person's score (0 ≤ ''θ'' ≤ 1) is drawn from a population-level beta distribution, then an important statistic is the mean of this population-level distribution. The mean and sample size parameters are related to the shape parameters ''α'' and ''β'' via<ref name=Kruschke2011/> : ''α'' = ''μν'', ''β'' = (1 − ''μ'')''ν'' Under this [[Statistical parameter|parametrization]], one may place an [[uninformative prior]] probability over the mean, and a vague prior probability (such as an [[exponential distribution|exponential]] or [[gamma distribution]]) over the positive reals for the sample size, if they are independent, and prior data and/or beliefs justify it.
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