Proposition. An explicit conditional Gaussian kernel [fdip-003A]
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  • October 29, 2026

In the orthogonal positive setting of Gaussian elimination, let \(X\) be retained and \(Y\) eliminated. The components of \(Q\) are \(A,B,B^\dagger ,C\). Put \(T=C-B^\dagger A^{-1}B\) and \(c_A=(2\pi )^{\dim X/2}/\sqrt {\det A}\). The maps \(A,T\) are positive definite by exchanging the eliminated summand. For every \(y\in Y\), define a probability measure on \(X\) by \[\begin {aligned} &k_y(x)=c_A^{-1}\exp \bigl (-\tfrac 12\\ &\qquad \cdot h_X(x+A^{-1}By,A(x+A^{-1}By))\bigr ),\\ &\kappa (y,D)=\int _D k_y(x)\,d\lambda _X(x) \end {aligned}\] for Borel \(D\subseteq X\). These measures form a measurable conditional probability kernel for the retained field given the eliminated field. Its mean is \(-A^{-1}By\) and covariance operator is \(A^{-1}\).

Proof. 

  • October 29, 2026

Exchanged square completion gives \[\begin {aligned} &h_E(J(x,y),QJ(x,y))\\ &\quad =h_X(x+A^{-1}By,A(x+A^{-1}By))\\ &\qquad +h_Y(y,Ty). \end {aligned}\] The joint density relative to \(\lambda _X\otimes \lambda _Y\) is \(p_Q(x,y)=Z_Q^{-1}w_Q(J(x,y))\). Its \(Y\)-marginal density is \[\begin {aligned} r(y)&=\int _Xp_Q(x,y)\,d\lambda _X(x)\\ &=\frac {c_A}{Z_Q}e^{-h_Y(y,Ty)/2}>0. \end {aligned}\] It is continuous, finite and integrates to one by Tonelli.

Division gives \(k_y(x)=p_Q(x,y)/r(y)\). Translation invariance and positive normalization prove \(\int _Xk_y\,d\lambda _X=1\) for every \(y\), including zero-dimensional spaces. Thus \(\kappa (y,\cdot )\) is a probability measure.

For fixed Borel \(D\), the function \((y,x)\mapsto \mathbf 1_D(x)k_y(x)\) is nonnegative Borel. Measurability of parameter integrals for sigma-finite volume shows that \(y\mapsto \kappa (y,D)\) is Borel. This is the measurable-parameter part of [tao2011measure, Theorem 1.7.15, pp. 200-201, preliminary version], applied to the uncompleted Borel product.

For Borel \(D\subseteq X\) and \(H\subseteq Y\), Tonelli and \(k_yr=p_Q\) give \[\mu _Q(J(D\times H)) =\int _H\kappa (y,D)r(y)\,d\lambda _Y(y).\] The set on the left is Borel because \(J\) is a homeomorphism. This rectangle identity is the conditional-distribution identity with respect to the actual \(Y\)-marginal. It divides densities, not the probability of the event \(Y=y\); in positive dimension the latter singleton has probability zero. Our version is defined for every \(y\), whereas conditional distributions in general are identified only almost everywhere.

Under \(\kappa (y,\cdot )\), the translated variable \(u=x+A^{-1}By\) has centered precision \(A\). Its mean is zero and its covariance is \(A^{-1}\) by Theorem [fdip-000B]. Translating back proves the stated mean and covariance.

The retained marginal instead has mean zero and covariance \(S^{-1}\) for \(S=A-BC^{-1}B^\dagger \). At \(y=0\), the conditional precision is \(A\), not \(S\). If \(B=0\), then \(S=A\) and the conditional mean is zero for every \(y\), so the laws coincide. The four-cycle exhibits their different off-diagonal covariances when the coupling is nonzero.

To compare covariance parameters, put \(\Sigma =Q^{-1}\) and write its components as \(\Sigma _{XX},\Sigma _{XY},\Sigma _{YY}\). The opposite-pivot inverse formulas give \[\begin {aligned} \Sigma _{YY}&=T^{-1},\\ \Sigma _{XY}&=-A^{-1}BT^{-1},\\ \Sigma _{XX}&=A^{-1}+A^{-1}BT^{-1}B^\dagger A^{-1}. \end {aligned}\] Consequently \[\begin {aligned} \Sigma _{XY}\Sigma _{YY}^{-1}&=-A^{-1}B,\\ \Sigma _{XX}-\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}&=A^{-1}. \end {aligned}\] The covariance Schur complement for conditioning differs from the precision Schur complement for marginalization.

[rasmussen2006gaussian, Appendix A.2, equation (A.6), p. 200] gives these conditional Gaussian parameters. The density construction above additionally checks normalization, parameter measurability and the conditional integral identity.

TauCeti's TauCeti.Probability.condDistrib_multivariateGaussian identifies a conditional distribution with its Gaussian kernel almost everywhere for the conditioning marginal, assuming a Gaussian joint law, positive-semidefinite covariance and a positive-definite conditioning covariance block. Here \(\Sigma =Q^{-1}>0\) and \(\Sigma _{YY}=T^{-1}>0\); the displayed identities supply the parameter conversion. The totalized Gaussian constructor or a Markov-kernel instance alone would not identify the conditional distribution.