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On a wider class of prior distributions for graphical models

Part of: Graph theory

Published online by Cambridge University Press:  08 June 2023

Abhinav Natarajan ,
Willem van den Boom Open the ORCID record for Willem van den Boom [Opens in a new window] ,
Kristoforus Bryant Odang  and
Maria de Iorio
Abhinav Natarajan*
Affiliation:
University of Oxford
Willem van den Boom*
Affiliation:
National University of Singapore
Kristoforus Bryant Odang*
Affiliation:
National University of Singapore
Maria de Iorio*
Affiliation:
National University of Singapore; Singapore Institute for Clinical Sciences, A*STAR; University College London
*
*Postal address: Mathematical Institute, University of Oxford, Wellington Square, Oxford OX1 2JD, UK. Email address: natarajan@maths.ox.ac.uk
**Postal address: Yong Loo Lin School of Medicine, National University of Singapore, 10 Medical Dr, Singapore 117597.
**Postal address: Yong Loo Lin School of Medicine, National University of Singapore, 10 Medical Dr, Singapore 117597.
*****Postal address: Yong Loo Lin School of Medicine, National University of Singapore, 10 Medical Dr, Singapore 117597. Email address: mdi@nus.edu.sg
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Abstract

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 $\mathcal{G}_n$, 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 $\mathcal{G}_n$. In this paper we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as the main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.

Keywords

Gaussian graphical modelBayesian statisticsnetwork inferencecycle spaceMarkov chain Monte Carlovector space

MSC classification

Secondary: 05C80: Random graphs 05C90: Applications

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 230 - 243
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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