In 2004, The United States government executed Cameron Todd Willingham via lethal injection for the crime of murdering his young children by setting fire to his house.
In 2009, David Grann wrote an extended examination of the evidence in the Willingham case for The New Yorker, which has called into question Willingham's guilt. One of the prosecutors in the Willingham case, John Jackson, wrote a response summarizing the evidence from his current perspective. I am not summarizing the evidence here so as to not give the impression of selectively choosing the evidence.
A prior probability estimate for Willingham's guilt (certainly not a close to optimal prior probability) is the probability that a fire resulting in the fatalities of children was intentionally set. The US Fire Administration puts this probability at 13%. The prior probability could be made more accurate by breaking down that 13% of intentionally set fires into different demographic sets, or looking at correlations with other things such as life insurance data.
My question for Less Wrong: Just how innocent is Cameron Todd Willingham? Intuitively, it seems to me that the evidence for Willingham's innocence is of higher magnitude than the evidence for Amanda Knox's innocence. But the prior probability of Willingham being guilty given his children died in a fire in his home is higher than the probability that Amanda Knox committed murder given that a murder occurred in Knox's house.
Challenge question: What does an idealized form of Bayesian Justice look like? I suspect as a start that it would result in a smaller percentage of defendants being found guilty at trial. This article has some examples of the failures to apply Bayesian statistics in existing justice systems.
The bias toward false positives is probably especially strong in criminal cases. The archetypal criminal offense is such that it unambiguously happened (not quite like the Willingham case), and in the ancestral human environment there were far fewer people around who could have done it. That makes the priors for everyone higher, which means that for whatever level of probability you're asking for it takes less additional evidence to get there. That a person is acting strangely might well be enough -- especially since you'd have enough familiarity with that person to establish a valid baseline, which doesn't and can't happen in any modern trial system.
Now add in the effects of other cognitive biases: we tend to magnify the importance of evidence against people we don't like and excessively discount evidence against people we do. That's strictly noise when dealing with modern criminal defendants, but ancestral humans actually knew the people in question, and had better reason for liking or disliking them. That might count as weak evidence by itself, and a perfect Bayesian would count it while also giving due consideration to the other evidence. But these weren't just suspects, but your personal allies or rivals. Misweighing evidence could be a convenient way of strengthening your position in the tribe, and having a cognitive bias let you do that in all good conscience. We can't just turn that off when we're dealing with strangers, especially when the media creates a bogus familiarity.