The sudden big bet can definitely be someone hitting their set. I ran into that last night. But be wary of putting people on a specific hand. If we put people on the one hand that beats us we will end up misjudging and misplaying a lot of hands where we're actually good.

If your read is he's tight or you trust him you could certainly just call and take a note. Poker is all about making adjustments and following your reads.

Against the average player I would definitely raise and get it in expecting to see 98s, AJ, KQcc way more often than 44.

Plus your hand is under-repped due to the turn check which I really like. You got there, he's interested, this is a very +ev situation.