In general, I would think that the more information is in a theory, the more specific it is, and the more specific it is, the smaller is the proportion of possible worlds which happen to comply with it.
Regarding how much emphasis we should place on it: I woud say "a lot" but there are complications. Theories aren't used in isolation, but tend to provide a kind of informally put together world view, and then there is the issue of degree of matching.
Which theory has more information?
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