A much better analysis imo would be trying to find the probability of someone at his ELO finding surprising moves. EG I played a 1900 online recently who happened to completely turn around a game by setting up a forced mate in 6, with several branching moves a few moves down which all happened to result in mate because of incredibly lucky piece positions. I can't calculate the probability of someone at a relatively low level like that finding such a move, but I bet it's very low. This is the type of analysis which I'm guessing Magnus is using to assess Niemann as a cheater.
That's not true. Pick an engine, set to a few hundred points above your strength, then try to beat it using only one or two moves from an engine. You will lose nearly every game, because so many of your other moves will be so below the other player that the 1-2 good moves cannot make it up.
This is demonstrated quite often by the games where GMs are "helped" by others in multi-player games, and it shows that help against a much better opponent takes far more than 1-2 moves.
Among really close players it helps. But not once you get a few 100 ELO points apart.
Which brings us back to a sort of basic question - if this guy is cheating, do we think he's doing it fairly competently or not? If he's not cheating, the statistics will show normal play, if he is cheating and he's fairly competent the statistics will show normal play. So what has this analysis done? It's proved that he's not totally incompetent, which we already knew because he's pretty well established as a good chess player even he isn't truly a 2700ELO.
If there are not statistical differences, then there are no performance differences. Cheating by definition should imply performance differences.
If he is cheating, then at some point in the future, if that method becomes detectable and he has to stop, then his play will suddenly suffer, which would be more evidence.
Claiming that statistics cannot answer this question with statistics is not true. It may be hard, or the current sample too small, but claiming stats is not usable is a misunderstanding of statistics.
>This is like the statistical analyses that show election rigging by highlighting a statistically improbable distribution of results
This only works on the public, and is not what professional statisticians that analyze elections do.
And even here, if the event is rare enough, say 1 part in quadrillions, and the analysis is correct, then yes, we would certainly conclude there was rigging.
All human knowledge is statistical. Things we claim to be true are only statistically true to large odds, so even for election rigging, if the stats reach some level of certainty, then it is completely valid proof that would hold in court.
The pop idiocy of election rigging claims has never risen to that level.
>it is completely avoidable if you cheat competently
No, it is not. It may only lower the signal to noise ratio, but there is still detectable differences. If you continually improve the statistics and are forced to lower the signal, eventually the signal would be so low as to not affect the system, which in this case is chess games.
Physics, for example, can tease events out of on the order of 1 part in trillions and demonstrate signal. Plenty of other things do the same.