15.5 Beta

Beta distribution The continuous Beta distribution is the posterior probability distribution for the parameter $\theta$ , where one has observed $h$ successes in a given $N$ events, and each event is assumed to have a $\theta$ probability of success.

$\begin{eqnarray*} P(\theta|h,N) = (N+1)\cdot \binom{N}{h} \theta^{h} (1-\theta)^{N-h} \end{eqnarray*}$

Although it may look like a binomial, the Beta distribution is used to find the best estimate for the parameter $\theta$ where the number of successes and events, $h$ and $N$ are given.

The posterior density for θ given 3 heads and 9 tails overlaid with dashed vertical quartile lines: 1% at 0.07, 5% at 0.11, 25% at 0.20, 50% at 0.28, 75% at 0.36, 95% at 0.49 and 99% at 0.59. — **Figure C.5.** Posterior probability distribution for the $\theta$ values of the bent coin - the probability that the coin will land heads. The distribution is shown for data 3 heads and 9 tails. The various quartiles are shown in the plot.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.