Would it be possible to get some probability of cheating per player based on how often they play the best moves according to engines in online and live matches?
One thing that Magnus mentioned about a year ago (I think in relation to his oblique insinuation that Hans Niemann cheated) is that you don't have to cheat throughout the game.
He posited that if he checked an engine only once per game (at some critical juncture), he would be unbeatable (by any other non-cheating super GM).
If you cheat like that, it would be considerably more difficult to detect.
Caruana recently suggested that ~50% of players had cheated in cash online tournaments. He wasn't particularly clear about what he meant by "50%" but I suspect he thinks that many lesser GMs have done what Magnus suggested: they occasionally "dip their toe" into the cheating realm by checking at some critical point against a more formidable opponent.
They don't even need to do this every game. And, human nature being what it is, I bet they can convince themselves that it's not really cheating (e.g., "I was just double checking my favorite move....")
Not even to that extent.
They just need to be told which single move mid-game to think harder on, and they'll often find it.
So all they need is a metaphoric tap on the shoulder.
That's right! I think Carlsen even said something to that effect.
"This is a critical move." is all he might need to know....