Editing Bayesian inference (section)

===In the courtroom===
{{Main|Jurimetrics#Bayesian analysis of evidence}}
Bayesian inference can be used by jurors to coherently accumulate the evidence for and against a defendant, and to see whether, in totality, it meets their personal threshold for "[[beyond a reasonable doubt]]".<ref>Dawid, A.&nbsp;P. and Mortera,&nbsp;J. (1996) "Coherent Analysis of Forensic Identification Evidence". ''[[Journal of the Royal Statistical Society]]'', Series&nbsp;B, 58, 425–443.</ref><ref>
Foreman, L.&nbsp;A.; Smith, A.&nbsp;F.&nbsp;M., and Evett, I.&nbsp;W. (1997). "Bayesian analysis of deoxyribonucleic acid profiling data in forensic identification applications (with discussion)". ''Journal of the Royal Statistical Society'', Series&nbsp;A, 160, 429–469.</ref><ref>Robertson, B. and Vignaux, G.&nbsp;A. (1995) ''Interpreting Evidence: Evaluating Forensic Science in the Courtroom''. John Wiley and Sons. Chichester. {{ISBN|978-0-471-96026-3}}.</ref> Bayes' theorem is applied successively to all evidence presented, with the posterior from one stage becoming the prior for the next. The benefit of a Bayesian approach is that it gives the juror an unbiased, rational mechanism for combining evidence. It may be appropriate to explain Bayes' theorem to jurors in [[Bayes' rule|odds form]], as [[betting odds]] are more widely understood than probabilities. Alternatively, a [[Gambling and information theory|logarithmic approach]], replacing multiplication with addition, might be easier for a jury to handle.

[[Image:Ebits2c.png|thumb|right|Adding up evidence]]

If the existence of the crime is not in doubt, only the identity of the culprit, it has been suggested that the prior should be uniform over the qualifying population.<ref>Dawid, A. P. (2001) [http://128.40.111.250/evidence/content/dawid-paper.pdf Bayes' Theorem and Weighing Evidence by Juries]. {{Webarchive|url=https://web.archive.org/web/20150701112146/http://128.40.111.250/evidence/content/dawid-paper.pdf |date=2015-07-01. }}</ref> For example, if 1,000 people could have committed the crime, the prior probability of guilt would be 1/1000.

The use of Bayes' theorem by jurors is controversial. In the United Kingdom, a defence [[expert witness]] explained Bayes' theorem to the jury in ''[[Regina versus Denis John Adams|R v Adams]]''. The jury convicted, but the case went to appeal on the basis that no means of accumulating evidence had been provided for jurors who did not wish to use Bayes' theorem. The Court of Appeal upheld the conviction, but it also gave the opinion that "To introduce Bayes' Theorem, or any similar method, into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task."

Gardner-Medwin<ref>Gardner-Medwin, A. (2005) "What Probability Should the Jury Address?". ''[[Significance (journal)|Significance]]'', 2 (1), March 2005.</ref> argues that the criterion on which a verdict in a criminal trial should be based is ''not'' the probability of guilt, but rather the ''probability of the evidence, given that the defendant is innocent'' (akin to a [[frequentist]] [[p-value]]). He argues that if the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. This will depend on the incidence of the crime, which is an unusual piece of evidence to consider in a criminal trial. Consider the following three propositions:
: ''A'' – the known facts and testimony could have arisen if the defendant is guilty.
: ''B'' – the known facts and testimony could have arisen if the defendant is innocent.
: ''C'' – the defendant is guilty.

Gardner-Medwin argues that the jury should believe both ''A'' and not-''B'' in order to convict. ''A'' and not-''B'' implies the truth of ''C'', but the reverse is not true. It is possible that ''B'' and ''C'' are both true, but in this case he argues that a jury should acquit, even though they know that they will be letting some guilty people go free. See also [[Lindley's paradox]].