Probability

JM Keynes wrote on fundamental uncertainty that for some events in the future (such as whether or not there would be another European war or the interest rates 20 years from), we simply do not know what will happen. This is to say that there is no probability distribution at all - just complete uncertainty. Is this a coherent statement? It seems that there is always a probability for any given scenario (even if it the variables are extremely complicated). Chaos theory also seems to tell us that in a deterministic world there are some events that are too complex to predict. Are these not just a result of a lack of data or, perhaps, mathematical technique?

Comments

It depends on what you think

It depends on what you think probability is, but even then the answer is probably (heh!) no.

Nothing in the mathematical theory of probability requires that all events have probabilities. Probability theory simply imposes coherence conditions on any probability assignments there may be. And the mathematical theory of probability doesn't tell us what probability is but only what its formal properties are.

Some believe that there are objective probabilities—that if we specify our probability question appropriately, then there may be an answer to the question independent of what anyone thinks. For example: someone might think that if a quantum system has been prepared in a certain way, then the probability that a measurement interaction will have a certain result is, say, 1/3 regardless of what anyone thinks. This may or may not be right, though it still leaves us in the dark about what exactly this probability is. Is it a propensity or tendency of some sort? Is it a disguised way of talking about hypothetical long-run frequencies? There's no agreement about this. But notice: even if there are objective probabilities, it doesn't follow that every possible event has an objective probability. The probabilities of quantum mechanics are arguably a consequence of the mathematical structure of quantum states and the contingent facts about what states particular quantum systems end up in. But not every event we might be interested in corresponds to what's called an effect in quantum theory, effects being the things with which probabilities are associated. We might be interested in the probability that the winner of the 2020 Presidential election will be a Democrat. Neither quantum theory nor any other theory in physics will have much of anything to say about this.

So far: if we understand Keynes's claim as a claim about objective probability, then there's no good reason to doubt it. There's no good argument for thinking that all possible events have objective probabilities.

But there's another point: on one important view, there's no such thing as "objective probability." On this view, probabilities are ways of quantifying degrees of belief. Probabilities understood this way are sometimes called subjective probabilities. This has two consequences for your question. First, there doesn't have to be any one answer to a probability question. If you and I disagree about a probability question, there's no objective fact that makes one of us right, though if our probabilities don't fit the formal rules, we could both be wrong. Second, if the subjective account of probability is the best one, then there will be plenty of possible events that no one has assigned any probability to. If that's so, then there is no probability for those events, because no one has any degree of belief about them.

Finally, some remarks on chaos theory. The problem has nothing to do with mathematical technique. The relevant fact about chaos is this: suppose a system is chaotic. Suppose x and y are two possible, perfectly precise initial conditions for the system, and suppose that x and y are as similar as you like—as close together in the mathematical "state space" as you like. If the theory is deterministic (which chaos theory presupposes), then for any later time t, the equations tell us what state x' the state x would evolve into at t, and also what state y' the state y would evolve into at t. But if the system is chaotic, then even though x and y are as close together as you like, x' and y' can be as far apart as you like. We just have to pick t appropriately. A roulette wheel illustrates the point. Even if we know the initial conditions with high accuracy, even a small error will essentially leave us unable to predict the result.

This makes it sound like the problem has to do with poor data, but that misses the point. For non-chaotic systems, small errors in the initial data don't lead to large errors in prediction. For chaotic systems, they do. But knowing the initial state with absolute precision is typically not possible, and even if we did know it, the equations for the state at a later time typically don't have analytic solutions. That means the math itself (not just out techniques) forces us to use approximations, and that means the predictions about what will happen will be infected by the approximation.

In any case, Keynes' claim stands. There's no reason why every possible event must have a probability.

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