As shown in the figure above, as the number of observations grows, the Bayesian coverage intervals (BCIs) become narrower and narrower around $0.4$.

However, if you take a closer look, you will find that the centers of the BCIs are not exactly $0.4$, due to the persistent influence of the prior distribution and the randomness of the simulation path.

## Role of a Conjugate Prior

We have made assumptions that link functional forms of our likelihood function and our prior in a way that has eased our calculations considerably.

In particular, our assumptions that the likelihood function is **binomial** and that the prior distribution is a **beta distribution** have the consequence that the posterior distribution implied by Bayes' Law is also a **beta distribution**.

So posterior and prior are both beta distributions, albeit ones with different parameters.

When a likelihood function and prior fit together like hand and glove in this way, we can say that the prior and posterior are **conjugate distributions**.

In this situation, we also sometimes say that we have **conjugate prior** for the likelihood function $\textrm{Prob}(X | \theta)$.

Typically, the functional form of the likelihood function determines the functional form of a **conjugate prior**.

A natural question to ask is why should a person's personal prior about a parameter $\theta$ be restricted to be described by a conjugate prior?

Why not some other functional form that more sincerely describes the person's beliefs.

To be argumentative, one could ask, why should the form of the likelihood function have *anything* to say about my

personal beliefs about $\theta$?

A dignified response to that question is, well, it shouldn't, but if you want to compute a posterior easily you'll just be happier if your prior is conjugate to your likelihood.

Otherwise, your posterior won't have a convenient analytical form and you'll be in the situation of wanting to

apply the Markov chain Monte Carlo techniques deployed in {doc}`this quantecon lecture <bayes_nonconj>`.

We also apply these powerful methods to approximating Bayesian posteriors for non-conjugate priors in

{doc}`this quantecon lecture <ar1_bayes>` and {doc}`this quantecon lecture <ar1_turningpts>`