  In forensics it is a classical problem to determine, when a suspect $S$
shares a property $\Gamma$ with a criminal $C$, the probability that $S=C$. In
this paper we give a detailed account of this problem in various degrees of
generality. We start with the classical case where the probability of having
$\Gamma$, as well as the a priori probability of being the criminal, is the
same for all individuals. We then generalize the solution to deal with
heterogeneous populations, biased search procedures for the suspect,
$\Gamma$-correlations, uncertainty about the subpopulation of the criminal and
the suspect, and uncertainty about the $\Gamma$-frequencies. We also consider
the effect of the way the search for $S$ is conducted, in particular when this
is done by a database search. A returning theme is that we show that
conditioning is of importance when one wants to quantify the "weight" of the
evidence by a likelihood ratio. Apart from these mathematical issues, we also
discuss the practical problems in applying these issues to the legal process.
The posterior probabilities of $C=S$ are typically the same for all reasonable
choices of the hypotheses, but this is not the whole story. The legal process
might force one to dismiss certain hypotheses, for instance when the relevant
likelihood ratio depends on prior probabilities. We discuss this and related
issues as well. As such, the paper is relevant both from a theoretical and from
an applied point of view.
