Conjugate Inference: Gaussian Likelihood with Known Variance

Domain: Here we consider only univariate normal distributions.
\[x\in\mathbb{R}\]

Parameters: \[\Theta = \{\mu\}\] where \(\mu\) is the mean of the Gaussian likelihood.

Likelihood: \[P(x|\Theta) = N(x|\mu,v)\] Note that \(v\) is the known variance of the Gaussian likelihood.

Prior: A normal distribution.
\[P(\Theta) = N(\mu|\mu_0,\sigma_0^2)\]

Posterior:
\[P(\Theta|D) = N(\mu|\mu_1,\sigma_1^2)\]
with
\begin{eqnarray}
\mu_1 &=& \sigma_1^2 \left( \frac{\mu_0}{\sigma_0^2} + \frac{S^{(1)}}{v} \right) \\
\sigma_1^2 &=& \left( \frac{1}{\sigma_0^2} + \frac{S^{(0)}}{v} \right)^{-1}
\end{eqnarray}
where
\begin{eqnarray}
S^{(0)} &=& |D| \\
S^{(1)} &=& \sum_{x\in D} x \\
S^{(2)} &=& \sum_{x\in D} x^2
\end{eqnarray}

Marginal likelihood:
\[P(D) =
\left(\frac{1}{\sqrt{2\pi v}}\right)^{S^{(0)}}
\frac{\sigma_1}{\sigma_0}
\exp\left( -\frac{1}{2}\left( \frac{\mu_0^2}{\sigma_0^2} - \frac{\mu_1^2}{\sigma_1^2} + \frac{S^{(2)}}{v} \right) \right)\]