Sorry, you do not have access to this eBook
A subscription is required to access the full text content of this book.
Graphical Gaussian models are one of the main tools for the analysis of high-dimensional data with applications in a variety of disciplines. A graphical Gaussian model for the random vector Z ∈ R r is a Gaussian model where the dependencies between the components of Z are represented by means of a graph. The parameter of a centered Gaussian model is its covariance Σ or equivalently its concentration or precision matrix K = Σ - 1 ∈ P r where P r is the cone of r × r positive definite matrices. Because the dependencies between the components of Z are represented by means of a graph G, the expression of Σ or K depends upon G. In a Bayesian framework, we express uncertainty on the parameters by putting prior distributions f 2 ( K | G ) and f 3 ( G ) on K given G and G respectively. Given a sample D = ( Z 1 , … , Z n ) from the Gaussian distribution with precision matrix K, the joint distribution for ( D , K , G ) is f ( z 1 , … , z n , K , G ) = f 1 ( z 1 , … , z n | K , G ) f 2 ( K | G ) f 3 ( G ) from which we derive the posterior distribution of the model, that is G, as p ( G ∣ D ) ∝ f 3 ( G ) ∫ f 1 ( z 1 , … , z n | K , G ) f 2 ( K | G ) d K .
A subscription is required to access the full text content of this book.
Other ways to access this content: