Well, the number could hardly be made explicit, for political reasons ("you mean it's acceptable to have x wrongful convictions per year?? We shouldn't tolerate any at all!").
In any case, let me not be interpreted as arguing that the legal system was designed by people with a deep understanding of Bayesianism. I say only that we, as Bayesians, are not prevented from working rationally within it.
This is the third time on LW that I've seen the percentage of certainty for convictions conflated with the percentage of wrongful convictions (I suspect it's just quick writing or perhaps my overwillingness to see that implication on this particular post). They're not identical.
Suppose we had a quantation standard of 99% certainty and juries were entirely rational actors, understanding of the thin slice 1% is, and given unskewed evidence. The percentage of wrongful convictions would be well under 1% at trial; juries would convict on cases from 99% certainty to c. 100% certainty. The actual percentage of wrongful convictions would depend on the skew of the cases in that range.
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.