Betaverval

Betaverval is a term used in statistics to denote an interval estimate for a probability parameter that is derived from the Beta distribution, typically as the posterior distribution for a binomial probability in Bayesian inference. In practice, a betaverval represents a credible interval for the probability of success, formed from the Beta posterior rather than a frequentist confidence interval. The precise interpretation follows Bayesian principles: the interval is a region of parameter values that have a specified posterior probability mass given the observed data and prior information.

Calculation generally follows Bayesian updating. Starting with a prior Beta(α0, β0) and observing x successes in n trials, the posterior is Beta(α0+x, β0+n−x). The 95% betaverval is obtained by taking the 2.5th and 97.5th percentiles of this posterior distribution. In equal-tailed form, the endpoints are the inverse Beta cumulative distribution values at 0.025 and 0.975, with α' = α0+x and β' = β0+n−x. Other definitions, such as the highest posterior density interval, yield endpoints that may differ from the equal-tailed version.

Notes and context clarify that betaverval is not universally standardized; some sources simply call this a Beta-derived credible interval or a posterior interval. It is widely used in Bayesian data analysis tools and is closely related to concepts such as the Beta distribution, credible intervals, and Bayesian intervals for binomial proportions. As a concrete example, with a flat prior α0=β0=1 and x=7, n=12, the posterior is Beta(8,6) and the 95% betaverval is the interval between the 2.5th and 97.5th percentiles of Beta(8,6).