Friday, 7 October 2016

The Bayesian Networks mutual exclusivity problem

Several years ago when we started serious modelling of legal arguments using Bayesian networks we hit a problem that we felt would be easily solved. We had a set of mutually exclusive events such as "X murdered Y, Z murdered Y, Y was not murdered" that we needed to model as separate variables because they had separate causal pathways and evidence.

It turned  out that existing BN modelling techniques cannot capture the correct intuitive reasoning when a set of mutually exclusive events need to be modelled as separate nodes instead of states of a single node. The standard proposed ’solution’, which introduces a simple constraint node that enforces mutual exclusivity, fails to preserve the prior probabilities of the events and is therefore flawed.

In 2012 myself (and the co-authors listed below) produced an initial novel and simple solution to this problem that works in a reasonable set of circumstances, but it proved to be difficult to get people to understand why the problem was an important one that needed to be solved. After many changes and iterations this work has finally been published and, as a 'gold access paper' it is free for anybody to download in full (see link below).

During the current Programme "Probability and Statistics in Forensic Science" that I am helping to run at the Isaac Newton Institute for Mathematical Sciences, Cambridge, 18 July - 21 Dec 2016, it has become clear that the mutual exclusivity problem is critical in any legal case where there are diverse prosecution and defence narratives. Although our solution does not work in all cases (and indeed we are working on more comprehsive approaches) we feel it is an important start.

Norman Fenton, Martin Neil, David Lagnado, William Marsh, Barbaros Yet, Anthony Constantinou, "How to model mutually exclusive events based on independent causal pathways in Bayesian network models", Knowledge-Based Systems, Available online 17 September 2016
http://dx.doi.org/10.1016/j.knosys.2016.09.012

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