# Sufficient statistics for adjacency matrix of a graphical model

Assume a graph with $$d$$ nodes with adjacency matrix $$\boldsymbol{A}\in\{0,1\}^{d\times d}$$ is given and continuous data $$\boldsymbol{X}\in\mathbb{R}^{M\times d}$$ is observed from a graphical model given the graph.

If the graphical model is a Gaussian graphical model, the covariance matrix of $$\boldsymbol{X}$$ is a sufficient statistics for the precision matrix, and thus also for the graph strucutre.

Is there a way to classify the graphical models, for which there exists a smooth transformation $$f:\mathbb{R}^{M\times d}\mapsto \mathbb{R}^{M\times d}$$ of the data, such that the covariance matrix $$\propto f(\boldsymbol{X})^T f(\boldsymbol{X})$$ is a sufficient statistics for the graph strucutre?

For structural equation models, is it true that these are models with error term from the exponential family? Also, for copula Gaussian models, where data can be transformed to Gaussian distribution by a probability integral transform and applying inverse normal CDF. Is there any literature about this topic / concept?

• I am not sure I follow. The covariance matrix is not even a statistic. Do you mean the empirical covariance matrix? But if so I cannot understand how the third paragraph follows from the first two.
– 温泽海
Commented Aug 2 at 20:53
• sorry for being imprecise. Yes I mean the empirical covariance matrix $\propto X^T X$. Here: en.wikipedia.org/wiki/Precision_(statistics) the zeros in the precision matrix (inverse covariance matrix) correspond to conditionally independent variables. This is equal to the skeleton of a DAG, also used by Glasso e.g. Commented Aug 3 at 11:05