I think the coolest idea for justice systems that I've heard of (dunno its origin) is to have decisions done though prediction markets.
There's a couple ways I can see to keep it grounded to reality. The 'jury' (which would include people that think they can make money doing it) could not be given all of the evidence all the time, and their predictions would be checked (and their bets payed off) on the cases where they were withholding strong evidence.
There could also be two markets, and they have to predict the other juries verdict (possibly given more evidence).
Hell, even play money prediction markets do pretty well and you don't have to worry about reliably confirming actual guilt for payoffs. Just getting people to think with the "my money!" part of their brains helps.
Of course, there are all sorts of details to figure out, but it shouldn't be hard to beat the consensus of 12 people with no stake in the matter that were selected for being too dumb to get out of jury duty.
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.