What Is And What We Know Of It, Probabilistically Speaking

Ontology is the study of what is and what is not. Epistemology is the study of our knowledge of what is and what is not. Though there are obvious points of overlap and dependencies, it would be a mistake or fallacy to confuse that a thing exists with whether this or that person knows the thing existed. This mistake or fallacy accounts for the frequentist interpretation of probability which mislabels the existence of things (relative frequencies) as the probability of things.

This mistake makes sense because sometimes the relative frequency matches the probability. But it does not always do so. For instance, if in a bag there are n objects just one of which is labeled X and just one will be pulled out, the relative frequency of objects labeled X is 1/n which matches the probability that this object is drawn. The existence of the object matches our knowledge of it.

But if our premises are that “All Martians wear hats and George is a Martian” relative to the proposition “George wears a hat”, then there is no relative frequency—there is no ontology (to speak loosely and in the form of computer scientists); there are no Martians wearing hats and therefore no Martians named George. But there is a probability (which equals 1). “All” in the premises may be changed to “Some” or “Just 3 of n” or whatever and the conclusion that no relative frequency but a probability exists remains the same. And (of course) no counterfactual proposition has a relative frequency, but most (all?) can have a probability.

Suppose you don’t know, you have a “blank epistemology”, of the number of objects labeled X in the bag, but you at least know that each object has the possibility of being X. This is akin to suspecting a “loaded” coin, or in “trials” where there will be a defined success (the X) and failure, or whether an elementary particle in this field and measured in that way will show a certain property¹, or to any situation where the concern is one thing whose presence is contingent. It is easy to show that there is a still a probability of “drawing” an X (which is 1/2). But while there exists (at least in the here-and-now fixed bag) a relative frequency, it is unknown and therefore cannot be equated with the probability.

The relative frequency in a (say) drug “trial” is trickier. The number of elements (the “sample size”) will be finite. Of course, the relative frequency will eventually be something, say m/n successes, but at no point until we have reached the end will we know what the relative frequency is. Yet at each point the probability is still calculable (in the same way as discovering the initial 1/2), and of course eventually becomes the relative frequency—relative to the proposition “This element in the experiment was X” and knowing only there were m successes and n possibilities. But then the probability is also 1 relative to the same proposition but this time including all the knowledge from the trial (because we know whether each individual was a success or failure).

Now suppose we want to extrapolate what we have learned from the trial—of which everything is known, thus any proposition relative to this knowledge will have extreme probabilities (0 or 1) or any probability (propositions which have no logical relation to the trial but which are contingent will have the interval from 0 to 1). If we want to say something about the next n people before they take our drug, again there will eventually be a relative frequency but it is now unknown. Yet the probability is known (in the same manner as before). And again, once we reach the end of the n, we know everything there is and our probabilities are once again extreme or any probability (the interval (0,1)).

The next abstraction is to assume the trial’s (or even initial) results will be with respect to an infinite population: n goes to the limit, a mathematically desirable state. But nothing changes. We are still able to discover a probability at any point before the “long run” expires. We will, of course, wait for forever and a day before a relative frequency (of the entire set) exists. Once the Trump of Doom sounds and time ends, we will have everything we need to know and the probability and relative frequency will match. But nobody (at that point) will care.

How do we know this? The strong² “law” of large numbers states that (or, in other words, it can be proved beyond all doubt that):

which is to say the probability that the growing sample’s relative frequency (the average) equals the “expected” value of an observation given some evidence V is 1, but only at the limit. Notice we have used the limit twice, one time boldly and the other hidden in , the expected value. Calculating the expected value thus assumes the probability is known (deduced via V). In other words, the law is right and always has been, and those who use as a justification for calling the relative frequency the probability in finite slices of infinite samples have got it backwards.

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¹There is a healthy debate whether quantum theory is epistemological or ontological, or a mixture of the two. See inter alia the work of Anton Zeilinger (here or here). Zeilinger has scientist hair, so you know you can trust him.

²The difference between the strong and weak laws for this discussion are negligible.