Finite Gaussian probability with a positive precision matrix

Assume the Axiom of Countable Choice: every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets has a function $f$ with $f(n)\in X_n$ for every $n$. For $N\ge 1$, a real symmetric positive-definite matrix $K\in {\mathbb{R}}^{N\times N}$ (meaning $z^{\mathsf{T}}Kz>0$ for every nonzero $z\in {\mathbb{R}}^N$) defines the finite Gaussian probability ${\gamma }_K$ on ${\mathbb{R}}^N$ by

$Z_K:={\int }_{{\mathbb{R}}^N}{e}^{-z^{\mathsf{T}}Kz\mathrm{/}2}d{\lambda }_N(z),{\gamma }_K(dz):=Z_K^{-1}{e}^{-z^{\mathsf{T}}Kz\mathrm{/}2}d{\lambda }_N(z)\mathrm{.}$

Under the stated Axiom of Countable Choice, $Z_K=(2\pi {)}^{N\mathrm{/}2}(\det K{)}^{-1\mathrm{/}2}$. For the coordinate identity $X(z)=z$ on ${\mathbb{R}}^N$, define its mean vector by ${\mu }_i:={\int }_{{\mathbb{R}}^N}{z}_i{\gamma }_K(dz)$ and its covariance matrix by ${\mathrm{\Sigma }}_{ij}:={\int }_{{\mathbb{R}}^N}(z_i-{\mu }_i)(z_j-{\mu }_j){\gamma }_K(dz)$ for $1\le i,j\le N$. These integrals are finite under the Gaussian theorem Gaussian covariance for positive quadratic forms and a boundary counterexample, which gives $\mu =0$ and $\mathrm{\Sigma }=K^{-1}$. Thus ${\gamma }_K$ is centered, meaning its mean vector is zero. Here $K$ is called the precision matrix because it appears in the quadratic exponent; for this centered law its covariance matrix is its inverse. For $N=0$, define ${\mathbb{R}}^0$ to be the singleton containing the empty vector and define ${\gamma }_{\mathrm{\varnothing }}$ to put mass one on that point; its partition factor and empty determinant are both $1$.

The variables $z_i$ and exponent $z^{\mathsf{T}}Kz\mathrm{/}2$ are dimensionless in this mathematical definition. In a physical quadratic model, if each component of $\mathrm{\Phi }$ has declared field unit ${\mathrm{\Phi }}_{\ast }$ and $H(\mathrm{\Phi })=\frac{1}{2}{\mathrm{\Phi }}^{\mathsf{T}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{\Phi }$ is energy in joules, then each entry of $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ has units $\mathrm{J}\mathrm{/}({\mathrm{\Phi }}_{\ast }{)}^2$. With $\beta =(k_{B}T{)}^{-1}$, where in SI $k_B$ is Boltzmann's constant with units $\mathrm{J}{\mathrm{K}}^{-1}$ and $T>0$ is absolute temperature in kelvins, $\beta$ has units ${\mathrm{J}}^{-1}$. For $z_i={\mathrm{\Phi }}_i\mathrm{/}{\mathrm{\Phi }}_{\ast }$, the dimensionless precision is $K=\beta {\mathrm{\Phi }}_{\ast }^2{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. This dimensional conversion does not assert an empirical model.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Gaussian covariance for positive quadratic forms and a boundary counterexample