Known-variance Normal-Normal conjugate update

Let $\mathrm{\ell }(\mu ,y_{1:n})={\prod }_{i=1}^n(2\pi {\sigma }^2{)}^{-1\mathrm{/}2}\exp (-(y_i-\mu {)}^2\mathrm{/}(2{\sigma }^2))$. The coordinate maps on the parameter-data product are measurable by [F12], so [F8] makes $\mathrm{\ell }$ product-measurable. For each fixed $\mu$, [F10] and Tonelli [F9] show that $\mathrm{\ell }(\mu ,\cdot )$ integrates to one. On every product rectangle, the density measure $\mathrm{\ell }(\mu ,\cdot )d{\lambda }_n$ has the product of the $N(\mu ,{\sigma }^2)$ marginal probabilities by [F9] and [F10]; conditional independence gives the same rectangle values, so [F11] identifies it with the conditional sample law. Define $K(\mu ,D)={\int }_D\mathrm{\ell }(\mu ,y)d{\lambda }_n(y)$ for Borel $D$. By [F9] each section is a probability measure. The constant ${\lambda }_n$ kernel has the finite exhaustion $(-j,j{]}^n$ by [F10]; since $\mathrm{\ell }(\mu ,y){\mathbf{1}}_D(y)$ is product-measurable, [F12] makes $\mu \mapsto K(\mu ,D)$ measurable. Thus [F7] makes $K$ a probability kernel, and with the proper normal prior it gives the Bayesian model in [F3]. Now put $P={\tau }_0^{-2}+n{\sigma }^{-2}>0$, $q={\mu }_0{\tau }_0^{-2}+{\sigma }^{-2}{\sum }_i{y}_i$, and $r={\mu }_0^2{\tau }_0^{-2}+{\sigma }^{-2}{\sum }_i{y}_i^2$. By [F1] at positive scales $\sqrt{{\tau }_0^2}$ and $\sqrt{{\sigma }^2}$, whose existence is [F13], the product of the prior and sample densities is $C\exp \{-\frac{1}{2}(P{\mu }^2-2q\mu +r)\},C=(2\pi {\tau }_0^2{)}^{-1\mathrm{/}2}(2\pi {\sigma }^2{)}^{-n\mathrm{/}2}>0.$

Completing the square gives $P{\mu }^2-2q\mu +r=P(\mu -q\mathrm{/}P{)}^2+\mathrm{\Delta }$, where $\mathrm{\Delta }=r-q^2\mathrm{/}P\ge 0$ because the left side is a sum of nonnegative weighted squares and at $\mu =q\mathrm{/}P$ equals $\mathrm{\Delta }$. Hence the evidence is $C{e}^{-\mathrm{\Delta }\mathrm{/}2}\sqrt{2\pi \mathrm{/}P}\mathrm{.}$ It is finite and strictly positive; the Gaussian integral follows by scaling the normalized density in [F1]–[F2] at scale $\sqrt{1\mathrm{/}P}$, which exists by [F13].

Dividing the likelihood times prior in step 1.1 by the evidence computed in the preceding step gives the density $\sqrt{P\mathrm{/}(2\pi )}\exp \{-P(\mu -q\mathrm{/}P{)}^2\mathrm{/}2\},$ which is $N(q\mathrm{/}P,1\mathrm{/}P)$ by [F1] at scale $\sqrt{1\mathrm{/}P}$ from [F13]. Since $1\mathrm{/}P=V_n$ and $q\mathrm{/}P={\mu }_n$, this is the stated posterior. Finite-positive evidence is the valid domain by [F4] and [F5], and [F6] supplies the posterior-kernel identity; [F2] normalizes the resulting Gaussian density.