Theorem. The law on a selected subspace [fdip-005B]
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  • November 1, 2026

For a selected background, view \(P\) as a map \(H\to X=\operatorname {im}P\) and let \(r=\dim X\). The law \(\nu _b=P_*\mu _Q\) is the centered Gaussian on \(X\) with intrinsic covariance and precision \[C_P=(PGP)|_X>0,\qquad S_P=C_P^{-1}:X\to X.\] Relative to the induced Euclidean volume \(\lambda _X\), its density is \[\frac {d\nu _b}{d\lambda _X}(x) =\frac {\sqrt {\det _X S_P}}{(2\pi )^{r/2}} e^{-\langle x,S_Px\rangle /2}.\] If \(B:\mathbb R^r\to H\) is an isometric inclusion with image \(X\), then \(BB^*=P\); its chosen coordinate matrices are \[B^*C_PB=B^*GB,\qquad [S_P]_B=(B^*GB)^{-1}.\] For another such basis \(B'=BU\), \(U\in O(r)\), the latter changes to \(U^*[S_P]_BU\). The intrinsic operator \(S_P\) is independent of this choice.

Proof. 

  • November 1, 2026

For nonzero \(x\in X\), \(\langle x,C_Px\rangle =\langle x,Gx\rangle >0\). Set \(C=B^*GB\), so \(C>0\). The positive square roots of \(Q\) and \(C\) exist by the real spectral theorem. The density change of variables \(z=Q^{1/2}\phi \) gives the standard Gaussian density \((2\pi )^{-|V|/2}e^{-|z|^2/2}\): its Jacobian and the normalizer cancel. Put \[T=C^{-1/2}B^*Q^{-1/2}.\] Then \(TT^*=C^{-1/2}(B^*GB)C^{-1/2}=I_r\), so its rows extend to an orthonormal basis of \(H\). The resulting square orthogonal matrix carries \(z\) to a standard Gaussian, since its Jacobian has absolute value one and it preserves the squared norm. Its first \(r\) coordinates are \(Tz\); integration over the other coordinates gives the standard \(r\)-dimensional density. Hence \[B^*\phi =C^{1/2}Tz\] has density \((2\pi )^{-r/2}(\det C)^{-1/2} e^{-y^TC^{-1}y/2}\) by an invertible change of variables. The isometry \(B\) carries Euclidean volume to \(\lambda _X\) and \(B(B^*\phi )=P\phi \). This proves the law and its density, rather than inferring a law from its covariance alone.

For \(x,x'\in X\), the covariance of the corresponding linear observations is \(\langle x,Gx'\rangle =\langle x,C_Px'\rangle \) by the inverse-precision formula. The inverse identity and \(B'=BU\) give the coordinate statements.

In the orthogonal split \(H=X\oplus Y\), write \[Q=\begin {pmatrix}A&K\\K^*&D\end {pmatrix},\qquad D>0.\] Then \[S_P=A-KD^{-1}K^*.\] Indeed, solving \(Q(x,y)=(u,0)\) gives \(y=-D^{-1}K^*x\) and \((A-KD^{-1}K^*)x=u\). Positivity follows by evaluating the quadratic form on \((x,-D^{-1}K^*x)\). Thus the \(X,X\) block of \(Q^{-1}\) is the inverse of this Schur complement, and it is exactly \(C_P\). For \(P=P_R\), this recovers the coordinate marginal. If \(PQ=QP\), then \(K=0\) and \(S_P=Q|_X\); symmetry of a subspace alone does not justify that equality.

[rasmussen2006gaussian, Appendix A.2, p. 200] gives formulas (A.4)-(A.6) for Gaussian densities and marginals.

[friedli2017statistical, Definition 8.3, p. 384] characterizes Gaussian vectors by linear observations.

[friedli2017statistical, Theorem 8.4, p. 385] gives their nonsingular density. The row-isometry argument above supplies the linear-image step, including its Euclidean-volume identification.

A partial formal correspondence is supplied by TauCeti's TauCeti.Probability.map_affine_multivariateGaussian. It allows finite real coordinate spaces, a positive semidefinite covariance and an arbitrary rectangular linear map. Set its mean and shift to zero, its covariance to \(G=Q^{-1}>0\), and its matrix to \(B^*\). After the precision-density identification, the result gives the coordinate image law with covariance \(B^*GB\). It does not require full row rank. Positivity of this compressed covariance, its inverse precision, the isometric identification with \(X\), and the graph data are the additional arguments above.