Hmm it seems like perhaps this is a problem involving weighting the chance your opponent calls 3 streets of value with a hand you beat, and the danger of them sucking out with the extra card you grant by checking the flop IP.
Obviously if they will never call three streets of value with any worse hand and had no chance to improve, it's trivially better to check behind. You get the same outcome (2 streets of value), plus additional bets fired at you by PP on the turn, and a sightly likelier chance of getting your bets called on the turn and river than on the flop+turn. You also potentially lose less in the hand when villain has flopped a set. You may still lose the same if the villain goes for another c/r, but I can't see a sane villain taking that line.
However the higher the chance for villain to improve and the higher chance a hand can give you 3 streets of value, the more betting the flop looks good. Seems like just a number crunching game with the villain's stats. And if you don't have stats, knowing the most likely style of an unknown at the level/site you're playing.
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