On a wider class of prior distributions for graphical models
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Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of 𝒢_n, the set of all graphs in n vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of 𝒢_n. In this paper, we study subsets that are vector subspaces with the cycle space 𝒞_n as main example. We propose a novel prior on 𝒞_n based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implemented a Markov chain Monte Carlo algorithm and show that (i) posterior edge inclusion estimates compared to the standard technique are comparable despite searching a smaller graph space and (ii) the vector space perspective enables straightforward MCMC algorithms.
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