Beta Distribution - Alternative Parametrizations - Two Parameters - Mean and Sample Size

Mean and Sample Size

The beta distribution may also be reparameterized in terms of its mean μ (0 < μ < 1) and the addition of both shape parameters ν = α + β (ν > 0)( 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 = αPosterior + βPosterior 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= αPosterior + βPosterior - 2, or ν=(sample size)+2. Of course, for sample size much larger than 2, the difference between these two priors becomes negligible. (See section titled "Bayesian inference" for further details.) In the rest of this article ν = α + β will be referred to as "sample size", 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

Under this parametrization, one may place an uninformative prior probability over the mean, and a vague prior probability (such as an exponential or gamma distribution) over the positive reals for the sample size, if they are independent, and prior data and/or beliefs justify it.