Assume 100bb effective stacks in a standard NLHE blind format.

A tight/aggressive villain (say 15/10/3.0) in early position limps, it's folded to you, and you raise to 5x from late position with AKo. The remaining players fold, and the limper calls your raise.

The pot is now 11.5bb and you both have 95bb behind. The flop comes A83 rainbow, and villain checks to you.

Now if I've got this right, a good portion of villain's range will be pocket pairs, but also other unpaired hands that missed the flop completely. So basically, villain either has 2nd/3rd pair, a set, or a high card hand.

If we bet here, the high card hands are folding, and a good part of the time 2nd/3rd pair are folding depending on our image. Obviously if we bet, sets are going to drag us along and milk us a bit. So, it seems like the only value we get out of a bet are when we get called down by a pocket pair.

What about if we check? Villain is drawing to 2 outs a lot of the time (maybe 3 if he limp called Axs), so we allow him to draw for free, but we pick up some value from the times he tries to take the pot on the turn. Plus, the times he flopped a set, he's able to get in one less street of betting on us.

So under what conditions is it better to check behind and under what conditions is it better to just fire on the flop?