Bayesian Statistics: An Introduction, 4th Edition

2.2 Normal prior and likelihood

2.2.1 Posterior from a normal prior and likelihood

We say that x is normal of mean θ and variance and write

when

Suppose that you have an unknown parameter θ for which your prior beliefs can be expressed in terms of a normal distribution, so that

and suppose also that you have an observation x which is normally distributed with mean equal to the parameter of interest, that is

where θ0, and are known. As mentioned in Section 1.3, there are often grounds for suspecting that an observation might be normally distributed, usually related to the Central Limit Theorem, so this assumption is not implausible. If these assumptions are valid

and hence

regarding