There are problems with an adversarial system-- attorneys are rewarded for using weak but plausible arguments for their clients, and as stated above, there are also inequities in the quality of attorneys that different people can afford. Further, attorneys are so expensive that poorer people get an attorney supplied by the court who may not bother to try, or be so ill-paid that they can't afford to put a good case together.
There are also problems with an inquisitorial system. Iterated prisoner's dilemma implies that the investigator may be on the side of the court more than on the side of the accused. In theory, juries are supposed to prevent that, but US practice has been to give juries less and less information and to tell them that they have less and less freedom to use their whole minds.
I assume that a Bayesian court would permit-- possibly even encourage-- jurors to do additional research during the trial.
Some of the problems can be ameliorated by combining the systems-- three voices, speaking for the accused, the defense, and a third that's hopefully a neutral investigator.
Permitting the jurors to ask questions (afaik, a relatively recent addition to US practice and not universal here-- I don't know about other countries) is a significant contribution to getting more minds on the problem of getting all the relevant information and arguments on the table.
Iterated prisoner's dilemma implies that the investigator may be on the side of the court more than on the side of the accused.
Please call them "prosecutors" since in the U.S., "investigators" are people without law degrees hired by the defense or prosecution to interview potential witnesses, etc.
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.