77,230 research outputs found
Assessing forensic evidence by computing belief functions
We first discuss certain problems with the classical probabilistic approach
for assessing forensic evidence, in particular its inability to distinguish
between lack of belief and disbelief, and its inability to model complete
ignorance within a given population. We then discuss Shafer belief functions, a
generalization of probability distributions, which can deal with both these
objections. We use a calculus of belief functions which does not use the much
criticized Dempster rule of combination, but only the very natural
Dempster-Shafer conditioning. We then apply this calculus to some classical
forensic problems like the various island problems and the problem of parental
identification. If we impose no prior knowledge apart from assuming that the
culprit or parent belongs to a given population (something which is possible in
our setting), then our answers differ from the classical ones when uniform or
other priors are imposed. We can actually retrieve the classical answers by
imposing the relevant priors, so our setup can and should be interpreted as a
generalization of the classical methodology, allowing more flexibility. We show
how our calculus can be used to develop an analogue of Bayes' rule, with belief
functions instead of classical probabilities. We also discuss consequences of
our theory for legal practice.Comment: arXiv admin note: text overlap with arXiv:1512.01249. Accepted for
publication in Law, Probability and Ris
Picturing classical and quantum Bayesian inference
We introduce a graphical framework for Bayesian inference that is
sufficiently general to accommodate not just the standard case but also recent
proposals for a theory of quantum Bayesian inference wherein one considers
density operators rather than probability distributions as representative of
degrees of belief. The diagrammatic framework is stated in the graphical
language of symmetric monoidal categories and of compact structures and
Frobenius structures therein, in which Bayesian inversion boils down to
transposition with respect to an appropriate compact structure. We characterize
classical Bayesian inference in terms of a graphical property and demonstrate
that our approach eliminates some purely conventional elements that appear in
common representations thereof, such as whether degrees of belief are
represented by probabilities or entropic quantities. We also introduce a
quantum-like calculus wherein the Frobenius structure is noncommutative and
show that it can accommodate Leifer's calculus of `conditional density
operators'. The notion of conditional independence is also generalized to our
graphical setting and we make some preliminary connections to the theory of
Bayesian networks. Finally, we demonstrate how to construct a graphical
Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture
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