In any given week a terrorist organisation may or may not carry out an attack. There are several independent cells in this organisation for which it may be possible in any week to determine heightened activity. If it is known that there is no heightened activity in any of the cells, then an attack is unlikely. However, for any cell if it is known there is heightened activity then there is a chance an attack will take place. The more cells known to have heightened activity the more likely an attack is.In the case where there are three terrorist cells, it seems to reasonable to assume the BN structure here:
To define the probability table for the node "Attack carried out" we have to define probability values for each possible combination of the states of the parent nodes, i.e., for all the entries of the following table.
That is 16 values (although, since the columns must sum to one we only really have to define 8).
When data are sparse - as in examples like this - we must rely on judgment from domain experts to elicit these values. Even for a very small example like this, such elicitation is known to be highly error-prone. When there are more parents (imagine there are 20 different terrorist cells) or more states other than "False" and "True", then it becomes practically infeasible. Numerous methods have been proposed to simplify the problem of eliciting such probability tables. One of the most popular methods - “noisy-OR”- approximates the required relationship in many real-world situations like the above example. BN tools like AgenaRisk implement the noisy-OR function making it easy to define even very large probability tables. However, it turns out that in situations where the child node (in the example this is the node "Attack carried out") is observed to be "False", the noisy-OR function fails to properly capture the real world implications. It is this weakness that is both clarified and resolved in the following two new papers.
- Noguchi, T., Fenton, N. E., & Neil, M. (2018). "Addressing the Practical Limitations of Noisy-OR using Conditional Inter-causal Anti-Correlation with Ranked Nodes". IEEE Transactions on Knowledge and Data Engineering DOI: 10.1109/TKDE.2018.2873314 (This is the pre-publication version)
- Fenton, N. E., Noguchi, T. & Neil, M, (2018). "An extension to the noisy-OR function to resolve the “explaining away” deficiency for practical Bayesian network problems", IEEE Transactions on Knowledge and Data Engineering, under review
Hence the first paper provides a 'complete solution' but requires software like AgenaRisk for its implementation, while the second paper provides a simple approximate solution.
Acknowledgements: The research was supported by the European Research Council under project, ERC-2013-AdG339182 (BAYES_KNOWLEDGE); the Leverhulme Trust under Grant RPG-2016-118 CAUSAL-DYNAMICS; Intelligence Advanced Research Projects Activity (IARPA), to the BARD project (Bayesian Reasoning via Delphi) of the CREATE programme under Contract [2017-16122000003]. and Agena Ltd for software support. We also acknowledge the helpful recommendations and comments of Judea Pearl, and the valuable contributions of David Lagnado (UCL) and Nicole Cruz (Birkbeck).