Value bet (Draw high)
 

Probably not that intereting of a hand but....

5/10 Draw High at PR.
UTG and UTG+1 fold. I raise KK99 on the button. Only BB calls. BB is solid and doesn't call with pairs unimproved. I suspect BB is defending with at least QQ here. BB draws 3. I draw one and get a Q. BB checks. The Q I drew makes it more likely that BB was defending with AA and will only call with a better hand. Should I still value bet this with the chances that BB has QQ-up or less?

Re: Value bet (Draw high)
 

If he'll only defend with QQ or better and won't call with 1 pair after the flop, you don't have a value bet here even if you caught an ace.

Re: Value bet (Draw high)
 

Since standard poker theory says that you should only valuebet if you think you will be called by a worse hand than yours the answer is no.

Re: Value bet (Draw high)
 

If the BB typically check/calls after the draw when he improves then I agree with the others and think you should check behind. However, if the BB frequently bets out when he improves, then I think you should bet when he checks. Betting in this situation is an image play and not a value bet. Frequently betting in situations like this where your opponent is almost guaranteed to fold helps to establish a reckless image for yourself. Do it a couple times and your opponents will be a lot more tempted to call your legitimate value bets with weak hands.

Re: Value bet (Draw high)
 

Knowing what you know the answer is...no.