This is more of a general scenario that I'd like to start some discussion around. I was at a table last Friday that I was reading like an open book - it was a zone I had never been in before, and a few times I was able to put people on a flush draw with 85-90% certainty. The question, then is this:

You hold AX in position, raise, and get a few callers. Flop comes down fairly wet, something like Tc9c7s. Checks to you, you bet, get 1 caller. Turn hits no draws (e.g. 7h), checks to you, you bet, opponent calls. Based on timing, etc., you have him on a draw. River blanks (3s), opponent checks.

Under what conditions do you bet? When do you check behind?

Why would you wanna get checkraised on the river with a hand that has good showdown value and you believe is good? Not very many worse hands will ever call you on boards that you mentioned.

Well, for one, you might not want to chop with Ac5c. Or you might want to bluff 7c4c off of the best hand. So, what factors come together to make this a bet, or a check?

Player dependent but in general if the non flush cards are undercards to the flush cards (or I am otherwise sure they didn't pick up top pair or 2nd pair) and my read is strong that they they were on the FD I will bet to fold out a weak pair they may have picked up along the way otherwise rely on my A high showdown value.

Very few LLSNL players are c/r ing rivers light. Hell most don't do it with less than the nuts or darned close. If they c/r your read was way off and you fold. The few (and they are few) players capeable of c/r bluffing rivers are the ones I check behind and I identify these guys ASAP because I also do not bet/fold for thin value against these guys.

To answer your question, you have to estimate what % of time does villain have a FD here and what % of time does he have a weak pair.
As a simple example, if there is an equal chance he has a FD or a pair, checking behind is 0EV. Therefore, in this situation, you are waay better off betting the river to gain a +EV edge.
In this example, if there is even a 5% chance villain folds a pair, then betting is clearly +EV, while checking behind is neutral EV

To get more complicated, if villain range is skewed more towards pair than a FD, then checking back is clearly -EV
So, you have to factor in the size of your river bet, to gain a +EV edge. A weak bet has a higher chance of being called by villain's middling pair, and may end up being -EV. So as the size of the bet increases, the %chance villain will call down light goes down (not necessarily in a linear fashion)

Cliffs: checking back is least +EV. betting 1/3-1/2pot as an estimate is more +EV