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

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Fix a finite vertex set \(V\). For every subset \(T\subseteq V\), let \(\mathcal G(T)\) be the set of centered real Gaussian probability laws on \(\mathbb R^T\) with positive-definite covariance. Put \(\mathcal G(\varnothing )=\{1\}\), the unit mass on the zero-dimensional space. For \(S\subseteq T\), coordinate projection \(\pi _{S,T}:\mathbb R^T\to \mathbb R^S\) defines marginal restriction \[\operatorname {res}_{S,T}\mu =(\pi _{S,T})_*\mu .\] If \(\mu =N(0,G)\), its restriction is \(N(0,G_{SS})\); a principal submatrix of \(G>0\) is positive definite. Identity and composition of projections make \(\mathcal G\) a presheaf on the poset of all subsets of \(V\).

A cover of \(T\) is a finite family \((T_i)\) with \(\bigcup _iT_i=T\). A family \(\mu _i\in \mathcal G(T_i)\) is compatible local Gaussian data if its two marginals on each \(T_i\cap T_j\) agree. A global extension is a law \(\mu \in \mathcal G(T)\) with these restrictions. In covariance coordinates, compatibility means equality of the two principal submatrices on each overlap.

The domain contains all subsets and all such positive Gaussian laws. It is not restricted to the Markov laws of one fixed graph: marginalizing vertices can create new precision entries. For a law with precision \(Q=G^{-1}\), the Schur formula gives the precision of its \(S\)-marginal as \[\Lambda _S=(G_{SS})^{-1} =Q_{SS}-Q_{S,S^c}Q_{S^c,S^c}^{-1}Q_{S^c,S},\] where \(S^c=T\setminus S\). This is generally different from \(Q_{SS}\). Empty blocks follow the zero-dimensional convention.

The two-singleton example shows why overlap compatibility alone does not ensure uniqueness.