Thinning a Triangulation of a Bayesian Network or Undirected Graph to Create a Minimal Triangulation

Abstract

In one procedure for finding the maximal prime decomposition of a Bayesian network or undirected graphical model, the first step is to create a minimal triangulation of the network, and a common and straightforward way to do this is to create a triangulation that is not necessarily minimal and then thin this triangulation by removing excess edges. We show that the algorithm for thinning proposed in several previous publications is incorrect. A different version of this algorithm is available in the R package gRbase, but its correctness has not previously been proved. We prove that this version is correct and provide a simpler version, also with a proof. We compare the speed of the two corrected algorithms in three ways and find that asymptotically their speeds are the same, neither algorithm is consistently faster than the other, and in a computer experiment the algorithm used by gRbase is faster when the original graph is large, dense, and undirected, but usually slightly slower when it is directed.

Analysing Sensitive Data from Dynamically-Generated Overlapping Contingency Tables

“…Under the overlapping structure we consider, the graph generated by stitching saturated marginal models together is already decomposable, so no triangulation is needed. However, for stitching non-saturated marginal models together we suggest beginning with the triangulation equivalent to the saturated models, then using thinning (removing edges added during triangulation) to construct a minimal triangulation (Jones and Didelez 2017). In some circumstances, edge removal (removing existing model edges) may also be necessary to guarantee a model that is both decomposable and graphical -in order to ensure that the necessary sufficient statistics are available from the marginal models.…”