## Sunday, March 28, 2010

### Bayesian inference for the Gaussian

Given the prior probability
$p(\mu) = \mathcal{N}(\x_0,\sigma_0^2)$
and the likelihood
$p(x_1|\mu) = \mathcal{N}(\mu,\sigma_1^2)$,
the expectation of the posterior probability
$p(\mu|x_1)$
has a very simple and elegant form:
$(\alpha \x_0 + \beta x_1) / (\alpha + \beta)$
where
$\alpha = 1/(\sigma_0^2)$ and $\beta = 1/(\sigma_1^2)$
are the precisions.

Please refer to Bishop's PRML book section 2.3.6.