The continuous form

For probability density functions:

\[p(\theta \mid \mathbf{x}) = \frac{p(\mathbf{x} \mid \theta)\, p(\theta)}{p(\mathbf{x})} \propto p(\mathbf{x} \mid \theta)\, p(\theta)\]

• $p(\theta)$ — prior: beliefs before seeing data
• $p(\mathbf{x} \mid \theta)$ — likelihood: how probable the data is under $\theta$
• $p(\theta \mid \mathbf{x})$ — posterior: updated beliefs after seeing data

Gaussian conjugate update

If $x \mid \theta \sim \mathcal{N}(\theta, \sigma^2)$ and $\theta \sim \mathcal{N}(\mu_0, \tau^2)$, the posterior is:

\[\theta \mid x \sim \mathcal{N}\!\left(\frac{\tau^2 x + \sigma^2 \mu_0}{\sigma^2 + \tau^2},\; \frac{\sigma^2 \tau^2}{\sigma^2 + \tau^2}\right)\]

The posterior mean is a precision-weighted average of the prior mean and the data.