Abstract
One of the main approaches to performing computation in Bayesian networks (BNs) is clique tree clustering and propagation. The clique tree approach consists of propagation in a clique tree compiled from a BN, and while it was introduced in the 1980s, there is still a lack of understanding of how clique tree computation time depends on variations in BN size and structure. In this article, we improve this understanding by developing an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from BNs, specifically: (i) the ratio of the number of a BN's non-root nodes to the number of root nodes, and (ii) the expected number of moral edges in their moral graphs. Analytically, we partition the set of cliques in a clique tree into different sets, and introduce a growth curve for the total size of each set. For the special case of bipartite BNs, there are two sets and two growth curves, a mixed clique growth curve and a root clique growth curve. In experiments, where random bipartite BNs generated using the BPART algorithm are studied, we systematically increase the out-degree of the root nodes in bipartite Bayesian networks, by increasing the number of leaf nodes. Surprisingly, root clique growth is well-approximated by Gompertz growth curves, an S-shaped family of curves that has previously been used to describe growth processes in biology, medicine, and neuroscience. We believe that this research improves the understanding of the scaling behavior of clique tree clustering for a certain class of Bayesian networks; presents an aid for trade-off studies of clique tree clustering using growth curves; and ultimately provides a foundation for benchmarking and developing improved BN inference and machine learning algorithms.
Highlights
Bayesian networks (BNs) play a central role in a wide range of automated reasoning applications, including in diagnosis, sensor validation, probabilistic risk analysis, information fusion, and decoding of error-correcting codes [64, 6, 59, 38, 37, 60, 43, 58]
A key research question, which we investigate in this article, is how clique tree size relates to parameters that can be computed for a BN in polynomial time, such as the following parameters: V = jV j, the number of root nodes in a BN, with V 1
In the clique tree approach, which we emphasize in this article, BN inference consists of propagation in a clique tree that is compiled from a Bayesian network
Summary
Bayesian networks (BNs) play a central role in a wide range of automated reasoning applications, including in diagnosis, sensor validation, probabilistic risk analysis, information fusion, and decoding of error-correcting codes [64, 6, 59, 38, 37, 60, 43, 58]. We develop a more precise understanding of this easy-hard-harder pattern This is done by formulating macroscopic and approximate models of clique tree growth by means of restricted growth curves, which we illustrate by using bipartite BNs created by the BPART algorithm [45]. There is a need to develop similar laws for clique tree clustering’s performance, and in this article we obtain such laws in the form of Gompertz growth curves for BPART BNs [45]. While they admittedly have a strong empirical basis, these.
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