[dfan]

Returning to your original question -

For NL at least, if you had an infinitely large database of hands on a player who played the same preflop hand range throughout, you could determine this hand range precisely and simply.

Say, for example, you wanted to know what percentage of the time a player plays A9 UTG. Simply look at all hands where this player was UTG, one of the blinds was forced all-in due to low chips, there is no preflop raise and the flop was AA9. Every time the player has A9 in this situation he will show it down (assuming he doesn't fold flopped boats). If the player plays A9 100% of the time then on 1.2% of such instances (blind all-in, no raise, flop AA9) the player will show down his AAA99 boat. If he plays A9 UTG 1/2 the time, he will show down AAA99 0.6% of the time, and so on.

Once you determine the percentage of time he enters utg with A9, you could also figure out how often he folds it to a raise preflop. Again look at AA9 flops after a preflop raise. If he shows down AAA99 1/2 as often as when there was no preflop raise then obviously he folds 1/2 the time to this raise.

You could do this for the rest of his hand range.

In limit, you could use a similar approach of looking at flops that would give him a boat. The problem is that often everyone will fold after he bets so no showdown. So I'm not sure how you could get the same deterministic backtracking to work. There is probably a clever way to do so though, I just can't think of it off the top of my head.

Wait I just thought of such method. You can compute how often he will be dealt A9 UTG and another player will be dealt AA. I think we can assume that whenever this occurs and the flop is AA9 it will always get shown down. So just look at that situation and see how often the player shows down his bad beat 2nd best boat.