The trouble with P(evidence given Bob is irrational) isn’t that it’s inherently impossible to find, it’s that you haven’t given quite enough information to have a satisfying answer. There are multiple ways that irrational people could work—they could have a truly random set of beliefs, in which case you can just check for consistency. They could have a 50⁄50 chance of answering yes or no to any self-posed question, in which case they might think they were consistent. If we restrict the ways to be irrational to be based on just those two possibilities (although there are others, these seem most likely form your description), even if we don’t know what the real mixture is we can just take the average to get P(E|I)=0.25.
That 0.25 leaves off a literally negligible exponential term, but I should note that the exponential term becomes all there is when talking about Boltzmann brains, and so becomes important.
The trouble with P(evidence given Bob is irrational) isn’t that it’s inherently impossible to find, it’s that you haven’t given quite enough information to have a satisfying answer. There are multiple ways that irrational people could work—they could have a truly random set of beliefs, in which case you can just check for consistency. They could have a 50⁄50 chance of answering yes or no to any self-posed question, in which case they might think they were consistent. If we restrict the ways to be irrational to be based on just those two possibilities (although there are others, these seem most likely form your description), even if we don’t know what the real mixture is we can just take the average to get P(E|I)=0.25.
That 0.25 leaves off a literally negligible exponential term, but I should note that the exponential term becomes all there is when talking about Boltzmann brains, and so becomes important.
EDIT: whoops, double-post.