But it seems quite bizarre that somebody might be convicted...on a favoured Bayesian prior.

How would you describe how an ideal jury should perform its task? Not how real ones work, but an ideal one.

Is there really any particular reason why a Bayesian shouldn't have a significantly higher prior probability that members of certain ethnic or religious groups commit certain crimes (whatever the reasons for that may be), based on government statistics?

1. Conviction should not be based merely on probability of guilt; consider for example where society has for good reason excluded rational evidence from being legal evidence, such as with the 5th Amendment in the United States.

2. The relevant comparison will be between the likelihood of arrest and trial for an innocent person of group X and a guilty person of group X, not between the rates of likelihood of conviction for random individuals of group X and Y, respectively.

If the defendant is a woman, it is not relevant that "women commit (or are convicted of) crimes less often then men". What is relevant is how likely a female defendant is to be guilty. This may be less or more than a male defendant, but I don't consciously have a different prior for defendants based on gender.

For example in a case of terrorism, we may prefer a likelihood of guilt (derived in whatever manner) to be sufficient cause to convict.

The likelihood of guilt we convict at should differ among crimes, just as punishment differs among crimes. This is good discrimination, discriminating between importantly different cases on the basis of important differences among them.

Convicting people differently based on the type of evidence that gives the same probability of their having committed the crime is generally baseless discrimination. If two people Al and Bob each may have committed the crime of public urination, and A did it with probability of X% considering the legal evidence, and B did it with probability of Y% considering the legal evidence, and X>Y, I don't know if one, both, or neither should be convicted. But I do know that if Al is not convicted, then Bob shouldn't be either.

There are a few exceptions for which good public policy depends on type of evidence, generally involving excluding it entirely for public policy reasons. For example, societies restrict how police may gather evidence and then restrict what they may do with inappropriately gathered evidence to make police comply with the rules.