So obviously if everyone was a perfect Bayesian agent a jury would be fine. I actually think imagining how things would work with perfect Bayesians is boring. I was thinking more realistically, as in, how would I design a new justice system tomorrow (or in 5 years when we're really good at this) to minimize irrationality as much as possible. No perfect Bayesians just smart people you can teach things to.
The advantage of judges over juries is that we could teach judges to be rationalists as part of their job. Also, the adversarial system strikes me as bias inducing. The wealthy get better representation. I think even if we want to keep adversaries in place we should at least give whoever is making the decision more investigative and active in their role. Instead of just listening to arguments they should be asking questions and thinking things through in real time. More like the Supreme Court hears cases than the way a regular jury does. Listening to two pieces of propaganda and then deciding doesn't seem like the ideal way of settling questions of fact. It might be a decent starting point, though.
Making that biasing evidence admissible is definitely a good call assuming properly trained judges.
The advantage of judges over juries is that we could teach judges to be rationalists as part of their job.
I still think you are missing a primary reason for having a jury system. The role of the judge in a jury system is to be an expert on the law and to explain to juries how it applies to the current case, to ensure the trial is conducted under the rules of the system and to pass sentencing.
The decision of guilt or innocence is delegated to the jury in an attempt to avoid conflicts of interest. Because of the requirement for the judge to be an expert ...
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.