Conjugate priors are a powerful concept in Bayesian statistics. A prior is called conjugate to a likelihood if, after observing data and applying Bayes' theorem, the resulting posterior distribution is in the same family as the prior. This mathematical relationship makes it much easier to update beliefs about parameters as new data arrives, since both the prior and posterior share the same functional form—only the parameters change.

Consider the Beta-Bernoulli model. Suppose you want to estimate the probability of success $p$ in a series of independent Bernoulli trials (think of repeated coin flips). If you use a Beta distribution as your prior for $p$, and your likelihood comes from observing the outcomes of those coin flips (modeled with a Bernoulli distribution), the posterior for $p$ will also be a Beta distribution. This is because the Beta prior and the Bernoulli likelihood are conjugate pairs.

Mathematically, if your prior is $p \sim \operatorname{Beta}(α, β)$ and you observe $p \mid \text{data} \sim \operatorname{Beta}(\alpha + k,\; \beta + n - k)$ trials with $k$ successes, the posterior is:

Another classic example is the Normal-Normal model. If your data are assumed to be drawn from a Normal distribution with unknown mean $μ$ (and known variance), and you use a Normal prior for $μ$, then the posterior for $μ$ after observing the data is also Normal. This conjugacy allows you to update your beliefs about the mean efficiently as you collect more measurements.

Conjugate priors are not just a mathematical curiosity — they allow for closed-form, analytical solutions to posterior inference, avoiding the need for complex numerical methods in many practical problems.