Why are conditional probabilities so rarely used in court, and sometimes even prohibited? Here’s one more good reason: prosecution bias.
Suppose that a piece of evidence X is correlated with guilt. The prosecutor might say, “Conditional on evidence X, the likelihood ratio for guilt versus innoncence is Y, update your priors accordingly.” Even if the prosecutor is correct in his statistics his claim is dubious.
Because the prosecutor sees the evidence for all suspects before deciding which ones to bring to trial. And the jurors know this. So the fact that evidence like X exists against this defendant is already partially reflected in the fact that it was this guy they brought charges against and not someone else.
If jurors were truly Bayesian (a necessary presumption if we are to consider using probabiilties in court at all) then they would already have accounted for this and updated their priors accordingly before even learning that evidence X exists. When they are actually told it would necessarily move their priors less than what the statistics imply, perhaps hardly at all, maybe even in the opposite direction.