Algebraic Representation of Gaussian Markov Combinations

Markov combinations for structural meta-analysis problems provide a way of constructing a statistical model that takes into account two or more marginal distributions by imposing condi- tional independence constraints between the variables that are not jointly observed. This paper considers Gaussian distributions and discusses how the covariance and concentration matrices of the dierent combinations can be found via matrix operations. In essence all these Markov combinations correspond to nding a positive denite completion of the covariance matrix over the set of random variables of interest and respecting the constraints imposed by each Markov combination. The paper further shows the potential of investigating the properties of the com- binations via algebraic statistics tools. An illustrative application will motivate the importance of solving problems of this type.