The Three Prisoners Problem Revisited: issues in the use of Bayesian networks for security applications

Bayesian networks are a type of causal network used
for probabilistic reasoning, which have found wide application in
biomedical environments and machine vision.We have considered
their application in the realm of security, where behaviour that
is deliberately intended to deceive has to be considered. As a
first step to the the analysis of this behaviour we have analysed
problems in which an one agent provides truthfull, but evasive,
information to the other agents.
The three prisoners problem, and its simpler relation the
Monty Hall problem, are classic examples of statistical analysis
giving rise to counter intuitive results. In this paper the source
of the counter intuitive results is identified as an agent that only
releases partial data about the true state of the system. Furthermore
the data that is communicated is a function of the identity
of the agent requesting the data. Under these circumstances two
significant results are demonstrated; first different questioning
agents, will arrive at different probability estimates for the
same problem. Secondly, although if all the data is requested
the estimated probability will converge, the convergence may
be nonmonotonic. This means that some questions, truthfully
answered, will lead to a less precise probability measurement.
Keywords — Bayesian reasoning; three prisoner problem;