The Frequency Interpretation Of Probability Is False: Some Reasons Contra von Mises

The Frequency Interpretation Of Probability Is False: Some Reasons Contra von Mises

What is the chance that “Donald Trump wins the 2024 Presidential election”? Impossible to say, says Richard von Mises, because the event is singular and must be embedded in an infinite collection for us to make any sense of it. Indeed, he says assigning probabilities to single events is “utter nonsense.” Even though you and I, and many bookies, can understand it.

All right, do this. Go to the store and buy a set of dice. Pop one out and toss it. What is the chance that “A six lands uppermost.” Impossible to say, says the increasingly grumpy von Mises, even though we might be able to envision of these collective thingess. Such that you may be tempted to say 1 in 6, but vos Mises says the die must first “have been the subject of sufficiently long series of experiments to demonstrate this fact.”

Which is going to cost casinos a lot of money, because by the time they get done testing a set of dice to ensure the probabilities accord with our simple model, they’ll be worn out and have to be tossed—I’m such a racket—away.

What is a collective? An infinite set of events all of which are the same but all of which are “randomly” different. Events must be embedded, in a mathematical sense, in these infinite collectives, such that—and here comes some mathematical pain—each infinite subsequence inside the sequence converges to the same limit as the collective itself.

Now if you have studied analysis, this will make perfect sense. If you haven’t, all it means, for us, is that after an infinite amount of time, the ratio of times the events happen to the total goes to some definite limit. That definite limit becomes the probability of any event.

What about events before the infinite gets here? You heard the man: we can’t say what the probability is for any of them. We must first wait until they all get here. Then we can say.

The sequence must be infinite, otherwise the math doesn’t work out. Not 100, nor 1,000, nor even a million. Infinite. If the sequence is only 1,000, it may as well be only 100, and it if only 100, it may as well only be 1, and we are back to singular events which von Mises says flummox him.

In the paper “The limits of numerical probability: Frank H Knight and Ludwig von Mises and the frequency interpretation” by Hans-Hermann Hoppe (thanks to Anon for the tip), von Mises himself is quoted about what frequency means (ellipsis original):

Imagine, for instance, a road along which milestones are placed, large ones for whole miles and smaller ones for tenths of a mile. If we walk long enough along this road, calculating the relative frequencies of large stones, the value found in this way will lie around 1/10. . . . The deviations from the value 0.1 will become smaller and smaller as the number of stones passed increases; in other words, the relative frequency tends towards the limiting value 0.1.

No it won’t, and no they won’t. They tend toward whatever the actual value is, because there will be an end of the road. It does not go on forever, and cannot. It is a terrible example of a collective because it is necessarily finite, and according to von Mises’s own rules, finite events cannot have probabilities assigned to them.

Of course, or at least so far, all events in Reality are finite. So no probabilities have been known yet. But wait.

Can you imagine a collective for the stones? You cannot. You can imagine lots of roads, but if you start building them you’ll quickly fill up the earth. Not infinite. So you start paving outer space, which is plenty big. Alas, not big enough. You will run out of space before you even get close to infinite. It’s much bigger than you thought.

This being so, von Mises, and all other frequency bros, cheat. “Hey, 1,000 is close enough to infinity that I’m calling it infinity. Now I can do probabilities.” Yet if 1,000 is close enough, why not 100? And if 100, why not 1?

Nobody, and I mean but nobody, follows frequency theory as it is on the books. They are all satisfied by cheating. And must cheat.

Here’s another way they cheat. Return to the Trump question. Let’s embed that into a collective. Which one? All men running against other men in leadership elections? All men running against other men in constitutional republicans that have fallen into the hell of democracy? Any election of any kind? Elections in which all those 18 years old vote?

You can do on like that for, well, forever, never coming to a definite collective which gives an event which is identical to all other events in the collective, except that each are “randomly” different.

You can pick just one definition and bluff that it’s the only one that makes sense. You’ll surely get away with it. But then you have to explain just what you mean by “identical but randomly different.” Be out of the room when this question arises.

These are only a couple of many criticisms of frequentism, which is false. I have more in Uncertainty.

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  1. JerryR

    The most amazing thing about this OP is the conclusion that infinity in time and space is an impossibility.

    As far as math is concerned, no fractions, rational or irrational numbers. It’s all in your head. Extremely useful because it built the modern world but like frequency probability, doesn’t exist in real world

    And for all you non believers out there, infinity of time and space is an essential part of your beliefs. But it cannot exist. So where do you go?

  2. Rudolph Harrier

    It’s no wonder that students graduate from statistics classes immediately use statistics to lie, when the classes themselves told them to use one definition and then said it was okay to ignore it later. The same issue lies behind how people effectively use p-values, confidence intervals, etc.: they will dutifully memorize a definition to get points on the exam, and then proceed to use the concept as if it said something completely contrary to what the definition says, because that is how their instructor acted too. The lesson learned is that if you say the right incantation before you start, you can proceed with any lie that you want.

    Some people expected statisticians to hold strong for truth against obviously false beliefs like transgenderism, but if they paid attention to who statisticians actually act they should have expected them to fold immediately.

  3. Glenn Ammons

    Should it be Richard or Ludwig in the first paragraph?

  4. cdquarles

    If I am remembering correctly, and I am familiar with the economist Ludwig, there are two being discussed. Ludwig famously panned applying statistics to economics. His mathematician brother is the other one; with whom I am far less familiar such that I’m not sure what his first name was.

  5. whoa.. no. Or, at least. I don’t think so. specifically “collective” is a bad translation – I imagine neither mises(es?) nor the translater/interpreter studied group theory. Second, what the more famous mises was on about was the idea that our estimate of the probability that the next stone will be large is largely a fn of our knowledge about how the stones are distributed – with small n, that depends on where we started counting but with sufficiently large n is does so less and less.. until we hit all n and get perfect knowledge = get p(stone is large) = 1 or 0 right for all stones.

    Ultimately I think your arguments against frequentism amount to discomfort with inductive proof – I’d scribble an example in the box here, but have to go…. 🙂 (freally, dentist appt)

  6. Chad W Jessup

    Trump winning the 2024 election? That really depends on successful Democrat Party voting fraud procedures.

  7. Cary D Cotterman



  8. Rhetocrates

    I love that example of the road, because it’s prima faciae absurd. If you keep count of the number of large stones and the number of total stones on that road, the percentage (which is what it is, not frequency, which implies speed) of large stone won’t ‘tend toward’ 0.1, and the deviations from that value won’t become smaller and smaller.

    The percentage of large stones simply is 0.1, and your measurement of the whole (finite) population reveals this.

  9. (~wannabee~) theorists oftentimes have a “trick” ready to exploit all kinds of deception by omission: Example: set up the milestones in a confusing circle, but don’t reveal anything to the students. [that’s not! you, Briggs]

  10. The True Nolan

    What is the chance that Trump wins the 2024 election? There are a lot of variables involved in that. Unfortunately, “the accurate number of legitimate votes cast for him”, is probably not one of those factors.

  11. To me, what this discussion is missing is the idea of the error of the mean.

    In the real world, we don’t need an infinite series. We only need a series long enough that the uncertainty of the mean is small enough for practical purposes.

    (My wonderful high school teacher, Mr. Hedji, explained “practical purposes” as follows:

    “Suppose we line up the boys on one wall and the girls on the other wall. Every time I ring the bell, they move halfway towards the other group. Will the groups ever meet?

    Of course not … but at some point they’ll be close enough for all practical purposes …”)

    So we don’t need an infinite number of trials. The theoretical average of repeated throws of a single die is 3.5. I just ran a sample of 100,000 trials on the computer. The answer was 3.50063 … close enough for all practical purposes.

    Matt, thanks as always for another in your endlessly intriguing posts,


  12. Briggs


    Indeed you are thinking like a Bayesian, where large is good enough. And is, also, in the logical probability.

    But for frequentism you need infinity. This is what the theory itself says. Nobody likes this, so nobody remembers it, and all dismiss it.

    See also my coin flip video, linked in the banner atop the site.

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