I was in an exchange here a while back about this question. Komponisto and I basically agree that a trial, whether the guilt is decided by a judge or jury, should work like this:

• For a conviction, they (jury or judge) must find that the odds of guilt meet some specific, high threshold, such as 100:1 odds.

• They should be allowed to incorporate any relevant Bayesian evidence (such as "probability that a random person committed this crime")

• But there should be a list of exceptions to the above for cases where cognitive biases significantly distort the evidence (i.e. their sex life, their religion, hearsay, etc.) or where the society has made a conscious decision that something shouldn't matter or be presented, irrespective of how informative it is (i.e. race/gender of the accused, illegally gathered evidence, stuff endangering national security, etc.).

• Retain the adversarial system, but require each side to present (and the jury be trained in) a causal/Bayes network indicating what they think happened and how strong each piece of evidence is within that net.

• The jury should be required to construct their own diagram (possibly borrowing from one side) and determine the weights of each piece of evidence with it, and then use it to compute the posterior of the defendant's guilt, which can then be compared against the guilt threshold.

This last one is important because it will clarify the presence of any poor inference on the part of the jury. While the jury can just fudge the numbers to match their "gut feeling", they cannot do so without assigning unreasonable evidence weight somewhere in the diagram, allowing the public to see if the jury wasn't thinking right. ("Whoa, you gave eight freaking bits of evidence toward guilt in a homicide case because the defendant "dresses like a slut"???)

Beyond that, where I may or may not agree with komponisto, I think the justice system should also have these properties:

• Be continuous rather than binary: For at least some cases, it shouldn't be all-or-nothing, but the compensation paid to the alleged victim should vary with the probability of guilt the jury finds.

• Also, punishments should be amplified by the inverse of the recovery rate (basically, the rate at which that crime is solved), with the extra compensation going to a victims' fund to pay those for whom no suspect is ever caught.

• The issue of jail should be less coupled to the issue of compensation and guilt. Whether someone is jailed should be determined by how dangerous they are expected to be, as judged by some insurer, whether or not they are found guilty. Whether they should pay compensation or be punished should be determined by causal culpability, whether or not the accused is generally dangerous.

This decoupling is most important for cases like drunk driving or accidental homicide. If it's something that the accused is unlikely to do again, they shouldn't be jailed, but they should have to pay compensation anyway to those they hurt. In cases of drunk driving that don't hurt anyone, the evidence that the person drove recklessly while drunk should be fed to their liability insurer, who can then update the risk assessment of the insured.