Cite articles as: Author, “Title,” Libertarian Papers [volume #], [article number] (year). Example: Jan Narveson, “Present Payments, Past Wrongs: Correcting Loose Talk about Nozick and Rectification,” Libertarian Papers 1, 1 (2009).
26. “On the Possibility of Assigning Probabilities to Singular Cases, or: Probability Is Subjective Too!”
by Mark R. Crovelli
Abstract: Both Ludwig von Mises and Richard von Mises claimed that numerical probability could not be legitimately applied to singular cases. This paper challenges this aspect of the von Mises brothers’ theory of probability. It is argued that their denial that numerical probability could be applied to singular cases was based solely upon Richard von Mises’ exceptionally restrictive definition of probability. This paper challenges Richard von Mises’ definition of probability by arguing that the definition of probability necessarily depends upon whether the world is governed by time-invariant causal laws. It is argued that if the world is governed by time-invariant causal laws, a subjective definition of probability must be adopted. It is further argued that both the nature of human action and the relative frequency method for calculating numerical probabilities both presuppose that the world is indeed governed by time-invariant causal laws. It is finally argued that the subjective definition of probability undercuts the von Mises claim that numerical probability cannot legitimately be applied to singular, non-replicable cases.
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I enoyed the paper. My best example for showing that probability is in the eye of the beholder is a simple thought experiment: Bob flips a coin and looks at the result, but hides it from Mike. Then both Bob and Mike state the probability of heads. Bob is either going to state 0% or 100% while Mike will state 50%. Thus, probability is subjective, merely a measure of the observer’s ignorance (along with his understanding of the process and the things he does know about it).
I want to point out a third type of universe that you didn’t seem to touch on, which could use more discussion:
A universe that is mostly deterministic with some uncaused random events would be indistinguishable to a human observer from a purely deterministic one, as he wouldn’t be able to tell whether his uncertainty was due to lack of knowledge or an uncaused random process. As long as enough events were deterministic, he would still be able to act productively and form useful models.
When applying the relative frequency method, one simply needs the conditions to vary in typical ways. Whether this variance is due to the complexity of the environment (pseudo-randomness) or true randomness is irrelevant, so the fact that the method would work or be useful is not proof that all events are deterministic, merely that there is sufficient determinism.
In such a universe, an observer’s probability estimate would be based both on his ignorance, and uncaused randomness. If he has complete knowledge, then such a probability would be objective, but if his ignorance was sufficient, it would outweigh the inherent uncaused randomness and make the probability effectively subjective.
So the critical issue is then the amount of uncaused randomness. If it were sufficient enough to be a major factor in probability estimates, then we would surely notice it, unless there were an intelligent being behind the scenes ensuring it occurred only when we weren’t looking. From this, I conclude that if there is uncaused randomness, then it’s only present in areas that we are generally ignorant and thus can’t tell whether the unexpected outcome was due to ignorance or uncaused randomness.
Also, a typographical error: both occurrences of “discreet” on page 14 should be changed to “discrete”.
I can’t agree with the author’s example of the outcome of a boxing match as a singular event to which it makes sense to attach a probability. Suppose A is fighting B tonight, and my bookie gives odds of .61 for A. The author claims that bookie’s are right an astounding portion of the time. My question is – just what counts as my bookie being right or wrong? Would we count him as “right” just in case A wins? If that’s the case, then he could have given any number greater than .5 and he’d be just as right. On the other hand, what if B wins? Why would this mean my bookie is wrong? In sum, just what does the author mean in claiming that there is a numerical probability attached to this fight which can be meaningful?
It seems to me that the natural understanding of my bookie’s odds is to say something like “with a 95% confidence, we believe that if 100 fights just like the one tonight were fought, A would win 61 times.” This is perfectly reasonable, but there’s no way that the one fight that actually takes place can be taken to check such a claim. The claim refers, as Richard von Mises says, to a collective.
Here’s another collective we can put boxing probability into, albeit with a bit more work: form the set of ordered pairs of outcomes and odds. For instance, tonight’s fight (assuming A wins) would be recorded as where the second number is the given probability of winning of whichever combatant is predicted to win, and the first word tells if that combatant wins or loses. Tomorrow, if C wins an upset over D, and the odds of D winning were supposed to be .8, we’ll get . There are plenty of other equivalent ways of defining the ordered pairs to include the same information. Now from this collective, it is possible to calculate the frequency with which the favorite wins, and to mathematically compare it to the probability, giving a confidence interval for this particular bookie’s predictions. Another way of saying that this confidence interval is high, of course, is to say what the author says – that the bookie makes money. So, the fact that the bookie makes money doesn’t establish that he successfully applies probability to singular cases – the very fact that he makes money over time is referring to his performance at guessing many fights, not one.
Professor Katz,
Perhaps I can explain the point here more concretely with an example taken from basketball, since there is an important basketball game tonight. According to the brothers von Mises, it would be absolutely absurd and meaningless to assign a numerical probability to a singular basketball game such as tonight’s game. According to them, we must only calculate numerical probabilities for collectives. But, if I open up any newspaper in America right now, (and I am indeed looking inside my local newspaper), I can find the “line” for the game, which happens to be that Orlando will win tonight’s game by three points. For anyone committed to the von Mises definition of probability, such a number is absolutely ludicrous, but anyone who has ever watched the “lines” for sporting events would be hard-pressed to explain how often such probabilities are accurate.
How could frequentists like the brothers von Mises explain this really obvious empirical reality?
Thanks for your response. First, I must note that I am a high school teacher, not a professor. Now, on basketball – we have a newspaper claiming that Orlando will beat some other team by 3 points. But this certainly is not a probability, just a prediction. What has this to do with probability, other than the separate claim that I can attach a probability to the event of this prediction being correct?
If we want to talk probability, let’s define the experiment and the sample space. It seems to me that there’s two ways to do that. First, we can say that the experiment is simply the difference in scores (say the signed difference Orlando-the other team, so that the sign will show who won.) The sample space, then, is the integers. In this manner, the newspaper is saying that they think the outcome will be three. What they have not done, though, is attached a numerical probability to this outcome. So we’re still in the realm, it seems, that can be covered by verbal description, not in numerical probability.
Another is to say that the experiment is comparing the outcome to the newspaper’s prediction, and the sample space consists of two points – “same” and “different.” Which of these is more likely? I don’t know, but once again, no one is claiming to have a numerical probability.
In neither case have we attached numerical probabilities to any elements in the sample space. However, it may be useful to do so. We can take your local newspaper and ask “what is the probability that they are correct?” This corresponds to the probability of a 3 in the first experiment, or a “same” in the second. Given this question, how would you go about finding the answer?
It seems that you’ve already given the answer above – such reports tend to be correct quite often, and this fact gives you confidence in the number you’re looking at now. In fact, we can even look at how often they’re right and take the limit of the number of times they’re right over position in the sequence, or substitute for “right” falling within some range of the correct answer…but now we’re doing frequentist probability. Barring that, how do you attach a number to the likelihood that the prediction is correct?
Mr. Katz,
I think you’re trying to approach this question from a technical point of view rather than a logical point of view, and I think that is a mistake. What we are after, (or, rather, what my paper is seeking to identify) is an epistemologically defensible definition for probability. In order to accomplish this task, what we need to do is first of all figure out why man tries to attach numerical probabilities to events or phenomena. As I have argued, and Ludwig von Mises argued, man seeks to employ probabilistic methods because he is uncertain about some event or phenomenon. A vital question is thus raised; namely, why is he uncertain? It is because the events and phenomena in the world are themselves completely random and erratic, or simply because man is not omniscient? For Austrians, this leads us to the unavoidable conclusion that man is uncertain for the latter reason– he is not omniscient, even though everything in the world has a prior and certain cause. And this means that probability is simply a subjective measure of man’s uncertainty about the causes in the world.
This seems to me to be the only defensible way to approach this definitional question.
Dr. Crovelli,
I agree with you on these points. In particular, I hold that probability is an epistemic question. In fact, classes I’ve generally argued for a slightly more radical position than yours in that direction – I hold that regardless of whether or not there is real indeterminism in the world (and I think there’s a good case that there might be, both at the quantum level and because chaos and subtle order) probability is still an epistemic question. I didn’t argue with you on those points because I agree with you on them, although I’d go about establishing it somewhat differently. I’m not, for instance, convinced that the fact that causal determinism is logically necessitated by human action implies that the world is causally deterministic, but I do think it implies that we treat it as such, which is good enough for me.
So, yes, man seeks to assign numerical probabilities to various events, classes, and so on, because of this epistemic uncertainty. But this is also the reason that man assigns verbal probabilities to various events. To use the example I heard raised (I wish I could remember who by) a few years back, if the President proposes nationalization of all industry, it makes sense to say things like “Hans Hoppe is more likely to oppose this than is Paul Krugman” or “Hans Hoppe is more likely than not going to oppose this” or something similar – verbal probabilities – but we would not say or understand “there is a probability of .95 that Hans Hoppe will oppose this.” If someone said this, and I was being charitable, I’d rephrase it in my head to mean “Hans Hoppe is very likely to oppose this” and not try to take it literally. I choose this example because, based on your paper, I think you’ll agree that we can’t apply numerical probability to singular events involving human choice of this sort.
Certainly, though “man seeks to” do something does not imply that man succeeds in so doing. It would be great to attach numerical probability, in a meaningful way, to singular events, but that doesn’t make it possible.
Your examples, I think, were meant to motivate the idea that there are meaningful numerical probabilities for singular events. I cannot agree that they have done so. I think that this claim is not necessary for your more substantial claim, with which I agree. That is, the idea that probability is epistemic need not imply that we can assign meaningful numerical probabilities to singular events. These examples, I think, better serve as cases for illustrating just this fact.
[...] trials of these subsets arriving at the established frequencies that define the probabilities. Crovelli (2009) argues that this is a mistaken approach, and that a subjective assessment of individual trials [...]
I just started reading your article and feel compelled to comment on the boxing analogy. I beleive the profit from gamblng is not made from a calculation of probability of a single event. The profit from gambling is made by the actions of the class of investors. The profit from probability correlates to the class of investors and not the singular event. The utility of a numerical probability measurement of a single event of win or loss is close to meaningless. For example, even 60/40 odds when restricted to a single event has little utility for profit making. Would you bet your life on a single event even if you accurately calculated the probability to be 90% in your favor? On the contrary you might bet your life if I gave you the opportunity to bet on the average of 1 million events? That is what casinos do, the more investors the more likely the probabilty calculation will work in their favor. Does this profit depend solely on whether one boxer wins or loses a boxing match? No, the profit is made regardless of the outcome.
Hi Robb,
I think you are misunderstanding my point with regard to the boxing analogy. The point is that bookies and casinos generate odds that are remarkably accurate for singular cases. Whether they generate these odds through a sportsbook, or by employing people capable of picking winners, or whatever, the point is that the odds are accurate. Just take a look at the odds on, say, basketball in your local paper. These are remarkably accurate numerical predictions about singular events, are they not? If so, then how can the von Mises brothers say that numerical probabilities cannot be calculated for singular events?