Say your opponent likes to make unusual plays, for instance, betting really small, overbetting, or donking in spots you don’t expect. How do you tell if your opponent is just being a donkey or if they’re found a glaring weakness in your play that they’re exploiting?
The only way you really know is to take a bunch of those hands to showdown and see what they mean. Sometimes they will do a particular play only with a specific type hand. For example, they might overbet the pot on the flop when they have top pair.
If they are mixing it up though, there’s no way to know.
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- Observe the player’s interactions with other players. Are these typical plays he’s making or just vs you? If the former, then it is a playing style. If the latter, there has probably been an adjustment to your play that you should investigate.
- When your opponent is making these unconventional moves against you, what preceded them and what followed? How are you reacting to each of these moves once they happen? When we play an exploitative style, there are natural countermoves against us. Look at these hands to see if this is what’s happening or if there is another explanation?
You should be able to ferret out what’s going on through the 2 methods above. If not, I’d be happy to help investigate further.
Sure there is - you call in the accountants 
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What you described is useful, but doesn’t fit my definition of “knowing.” If you lower the standard to “suspect” or “statistically likely,” then sure, your methods will get you there, or close to there, or make you think you’re there. You are looking at indeterminate probabilistic wave functions, I’m looking at determinate eigenstates!
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I’m looking to beat up on players in a game of incomplete information. We will likely never have a sufficient sample of hands seen at showdown to “know” anything. We may be able to infer what’s going on to a high degree of probability from observations without knowing villains hole cards. Hey, that sounds like a game I enjoy playing. I forget what its called at the moment 
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The game you are playing is “torture SPG with magical accountant hocus-pocus.”
If they are using these plays as part of a mixed strategy across their entire range, how useful are these inferences?
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It would lead to more questions than answers. Chief among them would be “Why are they testing Pluribus here?” I very much doubt that’s what’s going on. Even the top 1% of cash game players in the world have noticeable frequency errors. That occurs even when they are not intentionally trying to exploit their opponents.
When I hear hoofbeats, I think horses 1st, not zebras. This is either someone with a strange style, someone clicking buttons or someone who has found an exploit.
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Leading to more questions than answers is exactly why you should be doing it. I can say with 100% certainty that at least some players are using these plays this way.
That depends on the context. If I happen to be in Africa, I think zebras first.
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Speaking of context…
Are these plays being made isolated in position or OOP or what? Are you seeing them, for example, when you open from the button and they are in the BB? How many limpers? What size opens? What you may infer as as an exploit against you could also be a positional or situational play that has nothing to do with exploiting you specifically.