Demo: Bayesian Estimation
For Bernoulli data with a beta prior, posterior updating is parameter updating. The widget shows how prior strength and observed counts combine.
Mathematical setup
Let $X_i\mid\theta\sim\mathrm{Bernoulli}(\theta)$ and let $\theta\sim\mathrm{Beta}(\alpha,\beta)$. If $s=\sum_i x_i$ successes are observed in $n$ trials, then conjugacy gives
\[\Theta\mid x\sim\mathrm{Beta}(\alpha+s,\beta+n-s).\]The posterior mean and, when the posterior mode is interior, the MAP estimate are
\[E[\Theta\mid x]=\frac{\alpha+s}{\alpha+\beta+n}, \qquad \hat\theta_{\mathrm{MAP}}=\frac{\alpha+s-1}{\alpha+\beta+n-2}.\]The posterior mean behaves like a weighted average of the prior mean $\alpha/(\alpha+\beta)$ and the sample proportion $s/n$.
What to try
- Keep $s/n$ fixed and increase $\alpha+\beta$. The posterior moves less because the prior has more effective strength.
- Compare a symmetric prior with a skeptical prior such as $\alpha=2,\beta=8$. The same data can lead to different posterior compromise.
- Try boundary data, such as $s=0$ or $s=n$. The page reports boundary MAP behavior rather than using the interior formula blindly.
When $s>n$, the script caps the effective number of successes at $n$ so the posterior remains meaningful. When the beta posterior is not maximized in the interior, the MAP statistic is reported as a boundary value rather than by the interior formula.