interpreting probability as degrees of belief or, as it is sometimes defined
subjective probability.
In a court of law, questions of guilt or innocence are sometimes decided by
the ‘balance of probabilities’. Strictly speaking this criterion only applies to civil
cases but we shall imagine a scenario where it applies to criminal cases as well.
The frequentist statistician has a problem ascribing any meaning to the
probability of a prisoner being guilty of a crime. Not so the Bayesian who does
not mind taking feelings on board. How does this work? If we are to use the
balance-of-probabilities method of judging guilt and innocence we now see how
the probabilities can be juggled. Here is a possible scenario.
A juror has just heard a case in court and decided that the probability of the
accused being guilty is about 1 in 100. During deliberations in the jury room the
jury is called back into court to hear further evidence from the prosecution. A
weapon has been found in the prisoner’s house and the leading prosecution
counsel claims that the probability of finding it is as high as 0.95 if the prisoner is
guilty, but if he is innocent the probability of finding the weapon would be only
0.1. The probability of finding a weapon in the prisoner’s house is therefore
much higher if the prisoner is guilty than if they are innocent. The question
before the juror is how should they modify their opinion of the prisoner in the
light of this new information? Using our notation again, G is the event that the
prisoner is guilty and E is the event that new evidence is obtained. The juror has
made an initial assessment that prob(G) = 1/100 or 0.01. This probability is
called the prior probability. The reassessment probability prob(G | E) is the
revised probability of guilt given the new evidence E, and this is called the
posterior probability. Bayes’s formula in the form
shows the idea of the prior probability being updated to the posterior
probability prob(G | E). Like working out prob(S) in the medical example, we can
work out prob(E) and we find
This will present a quandary for the juror because the initial assessment of a