Normal-normal conjugacy with known observation variance

For independent $X_i\sim N(\theta ,{\sigma }^2)$ and prior $\theta \sim N(\mu ,{\nu }^2)$, the posterior has variance and meanAs $n\to \infty$, $m_n-\overline{X}=O_p(n^{-1})$ and $v_n^{1/2}\sim \sigma /\sqrt{n}$, so fixed-level posterior credible intervals agree asymptotically with the corresponding known-variance confidence intervals.

For repeated independent $N(\theta ,{\sigma }^2)$ measurements with prior $\theta \sim N(\mu ,{\tau }^2)$ and known scales, normal-normal conjugacy gives $v_k=({\tau }^{-2}+k{\sigma }^{-2}{)}^{-1}$ and $m_k=v_k(\mu {\tau }^{-2}+{\sigma }^{-2}{\sum }_{j=1}^k{y}_j)$. The next measurement has posterior predictive distribution $N(m_k,v_k+{\sigma }^2)$. Its one-sided false-alarm level $\alpha$ yields the displayed limit. Predictive width tends to the irreducible measurement width, while the center adapts to the individual's data. To assess a held-out measurement, fit with earlier measurements only.

For independent normal observations conditional on $\Theta$, with mean $\Theta$ and variances $v/m_j$, and normal prior mean $\mu$ and variance $a$, write $W={\sum }_j{m}_j$. Multiplying likelihood and prior adds precisions $W/v+1/a$. Completing the square gives the displayed posterior distribution. Its posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, where $Z=aW/(v+aW)$ and ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Thus the Bayes estimator under squared error loss is exactly the Bühlmann–Straub credibility estimate, not merely an approximation to it. Its posterior variance is $av/(v+aW)$.