Demo: Prior Design and Predictive Checks
Before seeing data, a beta prior implies a beta-binomial distribution for future counts. Use this page as a prior calibration check before committing to an analysis.
Mathematical setup
Let $\Theta\sim\mathrm{Beta}(\alpha,\beta)$ and, conditional on $\Theta$, let $Y\mid\Theta\sim\mathrm{Binomial}(m,\Theta)$ for $m$ future trials. Integrating over the prior gives the beta-binomial prior predictive distribution:
\[\Pr(Y=k)= \binom{m}{k} \frac{B(k+\alpha,m-k+\beta)}{B(\alpha,\beta)}, \qquad k=0,\ldots,m.\]Its mean is $m\alpha/(\alpha+\beta)$. At a fixed prior mean, increasing $\alpha+\beta$ makes the prior for $\Theta$ more concentrated, which usually makes the predictive count distribution less overdispersed.
What to try
- Hold the prior mean near 0.2 and compare $\mathrm{Beta}(2,8)$ with a stronger prior such as $\mathrm{Beta}(8,32)$.
- Increase $m$. The predictive distribution spreads over more possible counts, but its center stays tied to the prior mean.
- Check whether extreme counts, such as no successes, look plausible before data are collected.
Use this as a quick calibration check: do the exact beta-binomial future-count probabilities match what would be plausible before data are collected?