Is chance in the map or the territory?

by Ian Pollock

www2.isye.gatech.edu/

[Related (and much more in-depth): Fifteen Arguments Against Finite Frequentism; Fifteen Arguments Against Hypothetical Frequentism; Frequentist versus Subjective View of Uncertainty.]

Stop me if you’ve heard this before: suppose I flip a coin, right now. I am not giving you any other information. What odds (or probability, if you prefer) do you assign that it will come up heads?

If you would happily say “Even” or “1 to 1” or “Fifty-fifty” or “probability 50%” — and you’re clear on WHY you would say this — then this post is not aimed at you, although it may pleasantly confirm your preexisting opinions as a Bayesian on probability. Bayesians, broadly, consider probability to be a measure of their state of knowledge about some proposition, so that different people with different knowledge may correctly quote different probabilities for the same proposition.

If you would say something along the lines of “The question is meaningless; probability only has meaning as the many-trials limit of frequency in a random experiment,” or perhaps “50%, but only given that a fair coin and fair flipping procedure is being used,” this post is aimed at you. I intend to try to talk you out of your Frequentist view; the view that probability exists out there and is an objective property of certain physical systems, which we humans, merely fallibly, measure.

My broader aim is therefore to argue that “chance” is always and everywhere subjective — a result of the limitations of minds — rather than objective in the sense of actually existing in the outside world.

Random Experiments

What, exactly, is a random experiment?

The canonical example from every textbook is a coin flip that uses a fair coin and has a fair flipping procedure. “Fair coin” means, in effect, that the coin is not weighted or tampered with in such a way as to make it tend to land, say, tails. In this particular case, we can say a coin is fair if it is approximately cylindrical and has approximately uniform density.

How about a fair flipping procedure? Well, suppose that I were to flip a coin such that it made only one rotation, then landed in my hand again. That would be an unfair flipping procedure. A fair flipping procedure is not like that, in the sense that it’s … unpredictable? Sure, let’s go with that. (Feel free to try to formalize that idea in a non question-begging way, if you wish.)

Given these conditions, frequentists are usually comfortable talking about the probability of heads as being synonymous with the long-run frequency of heads, or sometimes the limit, as the number of trials approaches infinity, of the ratio of trials that come up heads to all trials. They are definitely not comfortable with talking about the probability of a single event — for example, the probability that Eugene will be late for work today. Will Feller said: “There is no place in our system for speculations concerning the probability that the sun will rise tomorrow. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines ‘out of infinitely many worlds one is selected at random...’ Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.”

The first, rather practical problem with this is that it excludes altogether many interesting questions to which the word “probability” would seem prima facie to apply. For example, I might wish to know the likelihood of a certain accident’s occurrance in an industrial process — an accident that has not occurred before. It seems that we are asking a real question when we ask how likely this is, and it seems we can reason about this likelihood mathematically. Why refuse to countenance that as a question of probability?

The second, much deeper problem is as follows (going back to coin flipping as an example): the fairness (i.e., unpredictability) of the flipping procedure is subjective — it depends on the state of knowledge of the person assigning probabilities. Some magicians, for example, are able to exert pretty good control over the outcome of a coin toss with a fairly large number of rotations, if they so choose. Let us suppose, for the sake of argument, that the substance of their trick has something to do with whether the coin starts out heads or tails before the flip. If so, then somebody who knows the magicians’ trick may be able to predict the outcome of a coin flip I am performing with decent accuracy — perhaps not 100%, but maybe 55 or 60%. Suppose that a person versed in such tricks is watching me perform what I think is a fair flipping procedure. That person actually knows, with better than chance accuracy, the outcome of each flip. Is it still a “fair flipping procedure?”

This problem is made even clearer by indulging in a little bit of thought experimentation. In principle, no matter how complicated I make the flipping procedure, a godlike Laplacian Calculator who sees every particle in the universe and can compute their past, present and future trajectories will always be able to predict the outcome of every coin flip with probability ~1. To such an entity, a “fair flipping procedure” is ridiculous — just compute the trajectories and you know the outcome!

Generalizing away from the coin flipping example, we can see that so-called “random experiments” are always less random for some agents than for others (and at a bare minimum, they are not random at all for the Laplacian Calculator), which undermines the supposedly objective basis of frequentism.

The apparent saviour: Argumentum ad quantum

“Ah!” you say, laughing now. “But you, my friend, are assuming determinism is true, whereas Quantum Mechanics has proven that determinism is false — objective chance exists! So your Laplacian Calculator objection fails altogether, being impossible, and the coin example fails if quantum randomness is involved, which it might be.”

Let’s review the reasons why Quantum Mechanics is held to imply the existence of objective chance (i.e., chance that does not depend on states of knowledge).

There are a variety of experiments that can be performed in order to bring out this intuition, but the simplest by far is simply to take a piece of radioactive material — say, uranium — and point a Geiger counter at it. The Geiger counter works because passing radiation makes an ionized (hence, electrically conductive) path in a gas between two electrodes that are at different voltages from each other. Electrical current flows through that ionized path, into a speaker, and makes a clicking sound, signaling the passage of an ionizing particle.

The ionizing radiation itself is caused by the radioactive decay of elements with unstable nuclei — for example, uranium. Uranium-238 has a half-life of 4.5 billion years, which means that in a given sample of U-238, about half of the atoms will have decayed after 4.5 billion years. However — and here is where the supposed objective randomness comes in — one can never say for any given atom when it will decay: now, or next Tuesday, or in 10 billion years. All one can say is that one would give odds of 1:1 (50% probability) that it will decay in the next 4.5 billion years (unfortunately, collecting on that bet is somewhat impractical).

For reasons that I do not have space to get into, the idea that the U-238 atom has some sort of physical “hidden variable” that would determine the time of its decay, if only we could measure this variable, is pretty much ruled out (by something called Bell’s theorem). So prima facie, it appears that nature itself is characterized by objective, irreducible randomness — therefore, chance is objective, therefore frequentism is saved!

There are at least two problems with this argument.

One is that the interpretation of the quantum mechanical experiments outlined above as exhibiting “objective chance” is highly controversial, although it is true that the above is the orthodox interpretation of QM (mainly due to historical accident, it must be said). Yudkowsky has done an excellent job of arguing for the fully deterministic Many Worlds Interpretation of QM, and so has Gary Drescher in “Good and Real,” so I am not going to try to recapitulate it. In essence, all you need in order to reject the standard interpretation above (usually called the Copenhagen interpretation) is to properly apply Ockham’s razor and guard against mind-brain dualism.

The most important problem with the argument that QM rescues frequentism, however, is that even given the existence of objective chance for the reasons outlined above, experiments that are actually characterized by such quantum randomness in anything higher than the tenth decimal place, are incredibly rare.* In other words, even if we accept the quantum argument, this only rescues a very few experiments — essentially the experiments done by quantum physicists themselves — as being objectively random.

It’s worth clarifying what this means. It does not mean that QM doesn’t apply to everyday situations; on the contrary, QM (or whatever complete theory physicists eventually come up with) is supposed to apply without exception, always & everywhere. No, the issue is rather that for macro-scale systems, Quantum Mechanics is well-approximated by classical physics — for example, fully deterministic Newtonian mechanics and fully deterministic electromagnetic theory (that’s why those theories were developed, after all!). Macro-scale systems, in other words, are almost all effectively deterministic, even given the existence and (small) influence of quantum indeterminacy. This definitely applies to something as flatly Newtonian as a coin toss, or a roulette wheel, or a spinning die — let alone statistics in the social sciences.**

So, unless frequentists wish to have their interpretation of probability apply ONLY to the experiments of quantum mechanics — and even that only arguably — they had better revise their philosophical views of the nature of probability.

Chaos theory?

Another common way to try to sidestep the problems of frequentism is to say that many physical systems are truly unpredictable, according to chaos theory. Take, for example, the weather — surely, this is a chaotic system! If a butterfly flaps its wings there, why, we might have a tornado here.

The trouble is that chaos theory openly acknowledges its status as a phenomenon of states of knowledge rather than ontology. Strictly speaking, chaos refers to the fact that many in-principle deterministic systems are, in practice, unpredictable, because a tiny change in initial conditions — perhaps too small a change to measure — makes the difference between two or more highly distinct outcomes. The important point is that higher resolving power (better knowledge) reduces the chaotic nature of these phenomena (i.e., reduces the number of possibilities), while never eliminating chaos completely. Hence, although chaos theory chastens us by revealing that the prediction of many deterministic physical systems is a fool’s errand, our view of chance remains essentially the same — chance is still subjective in the sense of depending sensitively on our state of knowledge. The weather is a chaotic system, but it is less chaotic to a good weatherman than to an accountant!

Some additional thoughts

Let’s return to the concept of objective chance. What does this phrase even MEAN, exactly?

Going back to the example of radioactive decay, how does the radioactive atom “decide” to decay at one time as opposed to another? We have already agreed that it does not have any physical guts which determine that it will decay at 15:34 on May 13, 2012. So are we saying that its decay is uncaused? Very well, but why is it un-caused to decay on May 13 as opposed to May 14? Your insistence that the decay is “objectively random” doesn’t seem to remove my confusion on this score.

The basic problem is that there is just no way of defining or thinking about “randomness” without reference to some entity that is trying to predict things. Please go ahead and attempt to do so! Imagine a universe with no conscious beings, then try to define “randomness” in terms that do not reference “knowing” or “predicting” or “calculating” or whatever.

Don’t get me wrong, I’m okay with the idea of unpredictability even in principle. Goodness knows, humans have epistemic limits, and some are doubtless insuperable forever. The problem is that unpredictability, randomness, chaos — whatever you want to call it, however unavoidable it is — is a map-level phenomenon, not a territory-level phenomenon.

So sure, you can tell me that there are some things that exist, but are un-mappable even in principle. But please don’t tell me that un-mappability itself is out there in the territory — that’s just flat out insane! You’re turning your own ignorance into ontology!

... And yet, that is exactly what the standard interpretations of QM say. But one person’s modus ponens is another’s modus tollens. Some would say “Standard interpretations of QM imply objective chance, therefore objective chance exists.” But it’s also possible to say “Standard interpretations of QM imply objective chance, but objective chance is gibberish, therefore the standard QM interpretations are incorrect.” ***

Hey, Einstein agreed with me, so I MUST be right!

To summarize:

> Frequentism would seem to be completely undermined by the fact that uncertainty is always subjective (i.e., less uncertain for agents with more knowledge).

> Quantum mechanics appears to offer an ‘out,’ by apparently endorsing objective chance against determinism.

> However, such an interpretation of QM is highly controversial.

> Also, even if that interpretation were accepted, the universe would still be deterministic enough to eliminate most of the classic questions of probability from a frequentist framework, making frequentism correct but almost useless.

> When considered carefully, the entire concept of ‘objective chance’ is highly suspicious, since (from a philosophical point of view) it turns epistemic limits into novel ontology, and (from a scientific point of view) it makes basic physics (e.g., radioactive decay) dependent on terminology (“unpredictable,” etc.) that is unavoidably mental in character.

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Footnotes:

* Okay, I admit it. I don’t actually know which decimal place. But it’s definitely not the first, second, or third.

** Likewise for free will. Even if objective chance would rescue contracausal free will (it wouldn’t), and even if objective chance actually existed (don’t take that bet!), the universe just would not be objectively chancy enough to make everyday decisions non-deterministic. Good thing free will isn’t undermined by determinism after all!

*** I wish I could give you the answer to the radioactive decay riddle now, but it’s not going to fit in one post — not even close.