[sweetjazz]

ILP, there is plenty of information available: the strength of your hand for that board, the general tendency of players in that game to overplay weaker hands vs. their tendency to wait for the turn to check/raise with their very strong hands. One reason I (and probably most of us) can't answer this question that well is that we don't have access to a lot of the peripheral info that was available to BK.

Often, if it is close I take a cautious approach and just showdown here without a further raise. Sometimes, I use very marginal clues to put in one more aggressive action. It could be something as obvious as seeing check/raise second pair on the turn before to seeing him fastplay a flopped two pair earlier.

I think your standard for information is way too high here. Yes, it would be nice if we had played 1000 hands versus this guy and watched each one carefully, and then it would be obvious what to do. But it wouldn't take a whole lot of information for me to suspect this player would play almost all Kx hands like this, and if that were the case I would definitely be looking to put in another raise on most rivers.