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:

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?

Try it in Python

This cell computes the exact beta-binomial prior predictive distribution for future success counts.

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

alpha = 2.0
beta = 8.0
m = 20

k = np.arange(m + 1)
pmf = stats.betabinom.pmf(k, m, alpha, beta)
prior_mean = alpha / (alpha + beta)
predictive_mean = m * prior_mean

plt.bar(k, pmf)
plt.axvline(predictive_mean, color="black", linestyle="--", label="predictive mean")
plt.xlabel("future successes")
plt.ylabel("prior predictive probability")
plt.legend()
plt.show()

print(f"prior mean for theta = {prior_mean:.3f}")
print(f"predictive mean count = {predictive_mean:.2f}")
print(f"P(Y = 0) = {pmf[0]:.3f}")
print(f"P(Y >= 8) = {stats.betabinom.sf(7, m, alpha, beta):.3f}")

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