Definition. Covariance and variance of Gaussian observables [fdip-002Z]
AGENTDRAFTED
Let \(\mu _Q\) be the Gaussian law on a finite real
inner-product space \((E,h)\), and write \(\omega _Q\) for its
expectation. For Borel \(X,Y:E\to \mathbb R\) with
\(X\), \(Y\) and \(XY\) absolutely integrable, define their
covariance by
\[\begin {aligned}
\operatorname {Cov}_Q(X,Y)
&=\omega _Q(XY)-\omega _Q(X)\omega _Q(Y)\\
&=\omega _Q\bigl ((X-\omega _Q(X))(Y-\omega _Q(Y))\bigr ).
\end {aligned}\]
Expansion and normalization prove the second equality; all 4 terms
are integrable by the stated hypotheses. This extends the finite-sum
covariance to integrals.
When \(X\) and \(X^2\) are integrable, its variance is
\[\begin {aligned}
\operatorname {Var}_Q(X)&=\operatorname {Cov}_Q(X,X)\\
&=\omega _Q\bigl ((X-\omega _Q(X))^2\bigr )\\
&=\omega _Q(X^2)-\omega _Q(X)^2\geq 0.
\end {aligned}\]
The inequality is positivity of the integral. In particular all these
expressions exist for polynomials by Proposition [fdip-002W], consistently with
the polynomial state.
For \(\ell _j(x)=h(j,x)\), set \(K(j,k)=\operatorname {Cov}_Q(\ell _j,\ell _k)\).
It is a finite symmetric bilinear form. There is a unique linear
covariance operator \(C_Q:E\to E\), relative to the specified
\(h\), such that \(h(j,C_Qk)=K(j,k)\).
For the counting field space, the coordinate means vanish by
Proposition [fdip-002Y], and the former coordinate notation is
\[\begin {aligned}
\operatorname {Cov}_Q(u,v)
&=\left \langle (\phi _u-\langle \phi _u\rangle _Q)
(\phi _v-\langle \phi _v\rangle _Q)\right \rangle _Q\\
&=\langle \phi _u\phi _v\rangle _Q.
\end {aligned}\]
The partition function, this two-point expectation, and the expectation
of a different observable are different quantities. The separate
inverse-precision theorem computes this covariance.
The definitions extend Notation [ftip-008H] directly; the integral identities
use the preceding moment and state results. Positivity uses the ordinary
real convention of [waldmann2004states, Definition 3.9, p. 11]
through Definition [fdip-0024], not an additional inverse-covariance assumption.
The four-cycle covariance calculation evaluates these
quantities for the running example.