(and of the use of Bayesian inference in forensics)

- A multilevel introduction to probabilistic reasoning -

**G. D'Agostini**
*Università ``La Sapienza'' and INFN, Roma, Italia*

(giulio.dagostini@roma1.infn.it,
`http://www.roma1.infn.it/~dagos`)

**Note**: the automatic translation LaTeX -> html (performed by *latex2html*)
is quite poor.

The original document (in several versions) can be found in
arXiv:1003.2086v2.

Triggered by a recent interesting New Scientist article
on the too frequent incorrect use of probabilistic
evidence in courts, I introduce the basic concepts of
probabilistic inference with a toy model, and
discuss several important issues that need to be
understood in order
to extend the basic reasoning to real life cases.
In particular, I emphasize
the often neglected point
that degrees of beliefs
are updated not by `bare facts' alone, but by all
available information pertaining to them,
including how they have been acquired.
In this light I show that,
contrary to what claimed in that article,
there was no ``probabilistic pitfall''
in the Columbo's episode pointed as example
of ``bad mathematics'' yielding ``rough justice''.
Instead, such a criticism could have a `negative
reaction' to the article itself and to the
use of Bayesian reasoning in courts,
as well as in all other places
in which probabilities need to be assessed and
decisions need to be made. Anyway, besides
introductory/recreational aspects, the paper touches important
questions, like: role and evaluation of priors;
subjective evaluation of Bayes factors;
role and limits of intuition;
`weights of evidence' and `intensities of beliefs'
(following Peirce) and `judgments leaning'
(here introduced), including their uncertainties
and combinations;
role of relative frequencies to assess and express
beliefs; pitfalls due to `standard' statistical education;
weight of evidences mediated by testimonies.
A small introduction to Bayesian networks, based
on the same toy model (complicated by the
possibility of incorrect testimonies) and implemented
using Hugin software, is also
provided, to stress the importance of formal, computer aided
probabilistic reasoning.

(I.J. Good's

- Introduction
- One in thirteen - Bayesian reasoning illustrated
with a toy model
- Bayes theorem and Bayes factor
- Role of priors
- Adding pieces of evidence
- How the independent arguments sum up in our judgement - logarithmic updating and its interpretation
- Recap of the section

- Weight of priors and weight of evidence in real life
- Assessing subjective degrees of beliefs - virtual bets
- Beliefs versus frequencies
- Subjective evaluation of Bayes factors
- Combining uncertain priors and uncertain weights of evidence
- Agatha Christie's ``three pieces of evidence''
- Critical values for guilt/innocence - Assessing beliefs versus making decisions

- Columbo's priors versus jury's priors

- The weight of evidence of the
full sequence of actions

- Comments and conclusions
- Bibliography
- The rules of probability

- Belief versus frequency

- Intuitions versus formal, possibly computer aided, reasoning
- Bare facts and complete state of information
- Some remarks on the use of logarithmic updating of the odds
- AIDS test
- Which generator?
- Likelihood and maximum likelihood methods
- Evidences mediated by a testimony
- A simple Bayesian network
- About this document ...