You are at least partly correct. My statement was correct, but it was intended only as an explanation of some of the mechanics of the applied math and applied

You are at least partly correct. My statement was correct, but it was intended only as an explanation of some of the mechanics of the applied math and applied probability of getting a doable hypothesis test. The role of the test in a scientific theory was deeper than my statement! In the main, novel hypothesis test I invented and published, there was more than just what I outlined and you criticized, more along the lines you mentioned: That "more" was something like "if the next observation is a long way from anything we've seen before, then maybe it is an anomaly?". The actual work of the math in the paper showed, intuitively, what "a long way" meant and, in particular, for the math, how actually to calculate, along the lines of my statement, the hypothesis test false alarm rate. So, in short, in nearest neighbors anomaly detection, with meager assumptions, we actually can calculate false alarm rate and, thus, get an hypothesis test. And, then, we are not limited strictly to the single nearest but can do some weighted sum over several of the relatively near. Right, such things appear to be necessarily close to multi-variate probability density estimation. But such estimation commonly requires so much data that it is not a good, direct way to calculate or estimate probabilities; curiously we can do well with the calculations without actually getting a local density estimate although in a sense we are close to doing that. The other remarks here are even better, e.g., get to how to do science, maybe at the LHC or some such. Good and beyond what I tried to say.