A conjugate prior is a prior distribution that, when combined with a given likelihood, produces a posterior in the same family. This enables closed-form Bayesian updating.
Beta-Binomial: The Beta(α,β) prior is conjugate to the Binomial likelihood.
Gamma-Poisson: Gamma(α,β) prior conjugate to Poisson(λ).
As more data arrives, the posterior narrows and shifts toward the true parameter value. The prior's influence shrinks — it contributes "pseudo-counts" α and β that become negligible for large n.
The prior (blue), likelihood (orange), and posterior (green) are shown. Click observations to update!