Example. One-vertex marginals with many extensions [fdip-006A]
✍️source

November 1, 2026

Compatibility of local Gaussian data alone does not imply uniqueness. On the cover \(\{0\},\{1\}\) of two vertices, prescribe a standard normal law at each vertex. Every covariance \[G_\rho =\begin {pmatrix}1&\rho \\\rho &1\end {pmatrix}, \qquad -1<\rho <1,\] is a positive-definite global extension. The product law is the member \(\rho =0\). Uniqueness will require an additional conditional-independence condition.

Example. One-vertex marginals with many extensions [fdip-006A]
✍️source

Local Gaussian laws and gluing [fdip-0063]

Definition. Local Gaussian data and marginal restriction [fdip-0064]

Example. One-vertex marginals with many extensions [fdip-006A]
✍️source

Theorem. Precision zeros and Gaussian conditional independence [fdip-0065]

Proof.

Theorem. Gluing two Gaussian laws across a separator [fdip-0066]

Proof.

Example. Separator reconstruction on the four-cycle [fdip-006B]

Example. Compatible edge laws with no global extension [fdip-0067]

Proof.

Theorem. Chordal Gaussian gluing along a junction tree [fdip-0068]

Proof.

Remark. What the gluing theorem establishes about descent [fdip-0069]

Theorem. Chordal Gaussian gluing along a junction tree [fdip-0068]

Proof.

Definition. Local Gaussian data and marginal restriction [fdip-0064]

Remark. What the gluing theorem establishes about descent [fdip-0069]

Definition. Local Gaussian data and marginal restriction [fdip-0064]