Trail-Weighted Influence Scores for Approximate Outcome Prediction in Bayesian Networks

We study trail-weighted pruning of evidence for Bayesian-network prediction. A traversal enumerates the active trails from each evidence node to the outcome, scores each trail by the product of its edge strengths, and drops evidence whose score falls below a threshold. For collider-free networks (chains and forks) over binary variables, we show that the trail score equals the exact correlation between a trail's endpoints, that a chain and a fork with equal edge strengths have identical joint distributions, and that the pruned posterior is within a computable sum of per-trail terms of the exact posterior when each dropped trail lies in its own branch of the outcome node. We prove this bound, check it exhaustively on all small trees, and show that it fails when dropped trails share a branch. In simulations on random trees, thresholding trail scores gives 1.4 to 13 times lower error than random pruning of the same size. We also apply the score, as an unproven heuristic, to seven bnlearn networks with colliders and loops. Keeping the top half of the active evidence by trail score gives lower error than random pruning on all seven, but it is not clearly better than a mutual-information ranking, and scoring costs more than the inference it saves on six of the seven, so we show no speedup. Models trained on trail, graph and information features and tested on networks they never saw do not beat the trail score or mutual information, but trail features carry most of what they use and clearly outperform generic graph features. A bound for general networks is left to future work.