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This illustrates how the prior, likelihood, and posterior behave for inference for a normal mean (\(\mu\)) from normal-distributed data, with a conjugate prior on \(\mu\).

Specifically the prior on \(\mu\) is N(\(\mu_0\), \(\tau_0^2\)) [dotted line] and the data is sampled from a normal distribution N(\(\mu\), \(\sigma^2\)), which gives the likelihood [black line]. Note that the likelihood is scaled so it fits nicely on the graph (remember, likelihoods only matter up to a constant, so you can scale them however is convenient).

Because the normal distribution is the conjugate prior for normal sampling, the posterior distribution is also a normal distribution, and is shown in red.

By Bayes theorem:

\[ \text{Pr}(\mu \, | \, \mathbf{y}, \sigma^2) \propto \text{Pr}(\mathbf{y} \, | \, \mu, \sigma^2) \text{Pr}(\mu) \]

\[ \text{N}(\mu_1, \tau_1^2) = \text{N}(\mu, \sigma^2) \text{N}(\mu_0, \tau_0^2) \]

where the posterior mean:

\[ \mu_1 = \frac{\frac{\mu_0}{\tau_0^2} + \frac{n \bar{y}}{\sigma^2}}{\frac{1}{\tau_0^2} + \frac{n}{\sigma^2}} \]

and the posterior variance:

\[ \tau_1^2 = (\frac{1}{\tau_0^2} + \frac{n}{\sigma^2})^{-1} \]

An interactive app that shows how the posterior distribution will change when the prior and the (scaled) data likelihood changes:

The source code of the app can be found here.

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