In Bayesian terms, Pr(X) just skyrocketed with that added detail.
Well, whether you subscript the background information or declare it among the givens, you're courting confusion to compare two different conditional probabilities, but using identical symbols for both. An editor wouldn't let you suppress the notation in such a case, and it isn't what Hans was talking about anyway. Maybe he should be talking about it, but one hitch at a time.
In Bayesian terms,
none of the probabilities change during a problem, because they're all conditional or prior (unchanged by evidence, by definition). The probabilities represent somebody's beliefs. The probability that a green marble is drawn given that the bag contains only green marbles is 1. Always, in every problem, throughout the problem.
What changes during your problem is that the "given all green" probability becomes salient, because the truth of its "given" part becomes known. The only thing that changes in a pure Bayes problem is what you know about the situation, and so which probabilities are salient.
That, of course, is conceptual or ideal. Realisitcally, nobody has actual beliefs about the specific effect of learning every possible thing that might be learned. But if I do learn something unexpected, then I can usually "pretend" that I had the needed conditional belief all along.
Hans is talking about priors. Pr(A and B given C) cannot be greater than P(A given C) and cannot be greater than P(B given C), for all A and for all B and for all non-contradictory C. That is a feature of all belief representations, not just probability.
If you present me with an opaque bag of marbles,
~ I believe a green marble might be drawn from it,
~ I believe it might contain only green marbles,
but my confidence that both are true is strictly less than my confidence in the first, and equal to my confidence in the second. In no case can my confidence in both be greater than in either one alone.
What is more relevant to the evaluation of stories, however, is that my confidence that a story is true given that someone told me the story and included a detail can be greater than my confidence in the story given that someone told me the story and didn't mention the detail.
Pr(drawing a green marble given you told me there is a bag and that it's all green) may be greater than Pr(drawing a green marble given you told me there is a bag). Not necessarily, since I may not believe you, but of course it is possible that I would.
Every day, people can and do increase their listeners' confidence by including details, by doing it without contradicting themselves, introducing other anomalies, or including things far-fetched all by themselves. Just visit any trial courtroom and watch the witnesses testify.
But while you're there, notice that even the most successful witness never persuades opposing counsel. Conclude that priors do matter. Spoken or written testimony, unsupported by other evidence, is almost always thin. A sprinkling of local color is unlikely to persuade many people who aren't already persuaded anyway. It just isn't doomed on logical or belief representational grounds