Statistical Learning with Sparsity

GRAPHICAL MODELS WITH HIDDEN VARIABLES 261

9.5 Graphical Models with Hidden Variables

Chandrasekaran et al. (2012) propose a method for undirected graphical mod-

els, in which some of the variables are unobserved (or “latent”). Suppose for

example that we are modelling stock prices in a certain sector, and they are

heavily dependent on the price of energy, the latter being unmeasured in our

data. Then the concentration matrix of the stock prices will not look sparse

in our data, but may instead be sparse if we could condition on energy price.

Let the covariance matrix of all variables—observed and unobserved—be

Σ. The sub-block of Σ corresponding to the observed variables is Σ

O

. Let

K = Σ

−1

be the concentration matrix for the set of observed and hidden ...