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.

## Sunday, March 28, 2010

### Bayesian inference for the Gaussian

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