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"S is an increasing sequence" is a less specific hypothesis than "S consists of all prime numbers whose decimal representations end in 1, in increasing order". But "The only constraint governing the generation of S was that it had to be an increasing sequence" is not a less specific hypothesis than "The only constraint governing the generation of S was that it had to consist of primes ending in 1, in increasing order".
If given a question of the form "guess the rule governing such-and-such a sequence", I would expect the intended answer to be one that uniquely identifies the sequence. So I'd give "the numbers are increasing" a much lower probability than "the numbers are the primes ending in 1, in increasing order". (Recall, again, that the propositions whose probabilities we're evaluating aren't the things in quotation marks there; they're "the rule is: the numbers are increasing" and "the rule is: the numbers are the primes (etc.)".
Moving back to your question about analytic functions: Yes, more specific hypotheses may be more useful when true, and that might be a good reason to put effort into testing them rather than less specific, less useful hypotheses. But (as I think you appreciate) that doesn't make any difference to the probabilities.
The subject concerned with the interplay between probabilities, preferences and actions is called decision theory; you might or might not find it worth looking up.
I think there's some philosophical literature on questions like "what makes a good explanation?" (where a high probability for the alleged explanation is certainly a virtue, but not the only one); that seems directly relevant to your questions, but I'm afraid I'm not the right person to tell you who to read or what the best books or papers are. I'll hazard a guess that well over 90% of philosophical work on the topic has close to zero (or even negative) value, but I'm making that guess on general principles rather than as a result of surveying the literature in this area. You might start with the Stanford Encyclopedia of Philosophy but I've no more than glanced at that article.