Moment matching for a beta prior ($\alpha =m\kappa ,\beta =(1-m)\kappa$)

Moment matching for a beta prior turns a desired mean and variance for an unknown probability into the parameters of a beta distribution prior. Choose $0<m<1$ andSolving the beta distribution mean and variance identities givesThe strict upper bound on the variance makes $\kappa >0$, so both parameters are positive. Substitution gives mean $\mathbb{E}[P]=m$ and variance $\mathrm{Var}(P)=v$ exactly. A smaller specified variance at fixed mean produces a more concentrated prior distribution.

For example, a desired prior mean $m=0.3$ and variance $v=0.01$ give $\kappa =0.21/0.01-1=20$, hence the beta distribution prior $\mathrm{Beta}(6,14)$. With the same mean and variance $v=0.035$, the calculation instead gives $\kappa =5$ and the broader beta distribution prior $\mathrm{Beta}(1.5,3.5)$. Both appear in the concentration diagram above.

The excluded boundary cases have clear meanings. Zero variance would require $P=m$ with certainty, not a beta distribution with finite positive parameters. At the other boundary, any random variable $P\in [0,1]$ satisfies $P^2\le P$, hence variance at most $m(1-m)$; equality requires $P$ to take only the endpoint values zero and one. This is a Bernoulli distribution, not a beta distribution. Thus $v\ge m(1-m)$ cannot be matched by positive finite parameters, and $m=0$ or $m=1$ is also excluded.

After observing $n=h+t>0$ Bernoulli trials, the Beta-binomial conjugacy makes the posterior mean a weighted average of the prior mean $m$ and observed binomial proportion $h/n$:This formula gives $\kappa$ a precise interpretation as the prior's weight relative to the sample size in this posterior mean. It does not require fictitious observed counts: the beta distribution parameters can be any positive real numbers. For the eight-head, two-tail example above, $\kappa =4$ and $m=1/2$, so the posterior mean is $(4\cdot 1/2+10\cdot 0.8)/14=5/7$.

// Target: statistical-inference.bigb