What is a probabilistic physical theory?

Taking another shot at what the fundamental question is: a normative theory tells us something about how agents ought to behave, whereas a descriptive theory tells us something about what is; physical theories seem to be descriptive rather than normative, but when they're merely probabilistic, how can probabilities tell us anything about what is?

The idea that a descriptive theory tells us about "what really is" is rooted in the correspondence theory of truth, and deeper in a generally Aristotelian metaphysics and logic which takes as a self-evident first-principle the Law of Excluded Middle (LEM), that "of one subject we must either affirm or deny any one predicate". Even if a probabilistic theory enables us to affirm the open sets of probability 1, and to deny the open sets of probability 0, the question remains: how can a probabilistic theory "tell us" anything more about what really is? What does "a probability of 0.4" correspond to in reality?

If we accept LEM wholesale in both metaphysics (the domain of what is) and logic (for my purposes, the normative characterization of rational speech), then our descriptive theories are absolutely limited to deterministic ones. For any metaphysical proposition P about reality, either P actually is or P actually is not; "P actually is" is a logical proposition Q, and a rational speaker must either affirm Q or deny Q, and he speaks truth iff his answer agrees with what actually is. To accommodate nondeterministic theories, one must give way either in the metaphysics or the logic.

This is so pragmatically crippling that even Aristotle recognized it, and for propositions like "there will be a sea-battle tomorrow", he seems to carve out an exception (although what exactly Aristotle meant in this particular passage is the subject of embarrassingly much philosophical debate). My interpretation is that he makes an exception on the logical side only, i.e. that a rational speaker may not be required to affirm or deny tomorrow's sea-battle, even though metaphysically there is an actual fact of the matter one way or the other. If the rational speaker does choose either to affirm or to deny tomorrow's sea-battle, then the truth of his claim is determined by its correspondence with the actual fact (which presumably will become known soon). My guess is that you'd be sympathetic to this direction, and that you're willing to go further and get on board with probabilistic logic, but then your question is: how could a probabilistic claim like "with probability 0.4, there will be a sea-battle tomorrow" conceivably have any truth-making correspondence with actual facts?

A similar problem would arise for nondeterminism if someone said "it is indeterminate whether there will be a sea-battle tomorrow": how could that claim correspond, or fail to correspond, to an actual fact? However, we can adopt a nondeterministic theory and simply refuse to answer, and then we make no claim to judge true or false, and the crisis is averted. If we adopt a probabilistic theory and try the same trick, refusing to answer about $A$ when its probability is $0<P\left(A\right)<1$, then we can say exactly as much as the mere nondeterminist who knows only our distribution's support—in other words, not very much (especially if we thoroughly observe Cromwell's Rule). We have to be able to speak in indeterminate cases to get more from probabilistic theories than merely nondeterministic theories.

The metaphysical solution (for the easier case of nondeterminism) is Kripke's idea of branching time, where possible worlds are reified as ontologically real, and the claim "it is indeterminate whether there's a sea-battle tomorrow" is true iff there really is a possible future world where there is a sea-battle tomorrow and another possible future world where there isn't. Kripke's possible-world semantics can be naturally extended to the case where there is a probability measure over possible successor worlds, and "with probability 0.4, there will be a sea-battle tomorrow" is made true by the set of {possible future worlds in which a sea battle takes place tomorrow} in fact having measure exactly 2/3 that of the set of {other possible future worlds}. But there are good epistemological reasons to dislike this metaphysical move. First, the supposed truthmakers are, as you point out, epiphenomenal—they are in counterfactual worlds, not observable even in principle, so they fail Einstein's criterion for reality. Second, some people can be better-informed about uncertain events than others, even if both of their forecasts are false in this metaphysical sense—as would almost surely always be the case if, metatheoretically, the "actual" probabilities are continuous quantities. The latter issue can be mitigated by the use of credal sets, a trick I learned from Definability of Truth by Christiano, Yudkowsky, et al.; we can say a credal set is made true by the actual probability lying within it. But still, one credal set can be closer to true than another.

The epistemological solution, which I prefer, is to transcend the paradigm that rational claims such as those about probabilities must be made true or false by their correspondence with some facts about reality. Instead of being made true or false, claims accrue a quantitative score based on how surprised they are by actual facts (as they appear in the actual world, not counterfactual worlds). With the rule $S\left(A\right)=\log P\left(A\right)$, if you get the facts exactly right, you score zero points, and if you deny something which turns out to be a fact, you score $-\infty$ points. In place of the normative goal of rational speech to say claims that are true, and the normative goal of rational thought to add more true claims to your knowledge base, the normative goals are to say and believe claims that are less wrong. Bayesian updating, and the principle of invariant measures, and the principle of maximum entropy (which relies on having some kind of prior [LW(p) · GW(p)], by the way), are all strategies for scoring better by these normative lights. This is also compatible with Friston's free energy principle, in that it takes as a postulate that all life seeks to minimize surprise (in the form of $-\log p\left(A|\theta \right)$). Note, I don't (currently) endorse such sweeping claims as Friston's, but at least within the domain of epistemology, this seems right to me.

This doesn't mean that probabilistic theories are normative themselves, on the object-level. For example, the theory that Brownian motion (the physical phenomenon seen in microscopes) can be explained probabilistically by a Wiener process is not a normative theory about how virtuous beings ought to respond when asked questions about Brownian motion. Of course, the Wiener process is instead a descriptive theory about Brownian motion. But, the metatheory that explains how a Wiener process can be a descriptive theory of something, and how to couple your state of belief in it to observations, and how to couple your speech acts to your state of belief—that is a normative metatheory.

It might seem like something is lost here, that in the Aristotelian picture with deterministic theories we didn't need a fiddly normative metatheory. We had what looked like a descriptive metatheory: to believe or say of what is that it is, is truth. But I think actually this is normative. For example, in a heated moment, Aristotle says that someone who refuses to make any determinate claims "is no better off than a vegetable". But really, any theory of truth is normative; to say what counts as true is to say what one ought to believe. I think the intuition behind correspondence theories of truth (that truth must be determined by actual, accessible-in-principle truth-makers) is really a meta-normative intuition, namely that good norms should be adjudicable in principle. And that the intuition behind bivalent theories of truth (that claims must be either True or False) is also a meta-normative intuition, that good norms should draw bright lines leaving no doubt about which side an act is on. The meta-norm about adjudication can be satisfied by scoring rules, but in the case of epistemology (unlike jurisprudence), the bright-line meta-norm just isn't worth the cost, which is that it makes talk of probabilities meaningless unless they are zero or one.

As I see it, probability is essentially just a measure of our ignorance, or the ignorance of any model that's used to make predictions. An event with a probability of 0.5 implies that in half of all situations where I have information indistinguishable from the information I have now, this event will occur; in the other half of all such indistinguishable situations, it won't happen.

For example, all I know is that I have a coin with two sides of equal weight that I plan to flip carelessly through the air until it lands on a flat surface. I'm not tracking how all the action potentials in the neurons of my motor cortex, cerebellum, and spinal cord will affect the precise twitches of individual muscle fibers as I execute the flip, nor the precise orientation of the coin prior to the flip, nor the position of every bone and muscle in my body, nor the minute air currents that might interact differently with the textures on the heads versus tails side, nor any variations in the texture of the landing surface, nor that sniper across the street who's secretly planning to shoot the coin once it's in the air, nor etc., etc., etc. Under the simplified model, where that's all you know, it really will land heads half the time and tails half the time across all possible instantiations of the situation where you can't tell any difference in the relevant initial conditions. In the reality of a deterministic universe, however, the coin (of any particular Everett branch of the multiverse) will either land heads-up or it won't, with no in-between state that could be called "probability".

Similarly, temperature also measures our ignorance, or rather lack of control, of the trajectories of a large number of particles. There are countless microstates that produce identical macrostates. We don't know which microstate is currently happening, how fast and in what direction each atom is moving. We just know that the molecules in the fluid in the calorimeter are bouncing around fast enough to cause the mercury atoms in the thermometer to bounce against each other hard enough to cause the mercury to expand out to the 300K mark. But there are vigintillions of distinct ways this could be accomplished at the subatomic level, which are nevertheless indistinguishable to us at the macroscopic level. You could shoot cold water through a large pipe at 100 mph and we would still call it cold, even though the average kinetic energy of the water molecules is now equivalent to a significantly higher temperature. This is because we have control over the largest component of their motion, because we can describe it with a simple model.

To a God-level being that actually does track the universal wave function and knows (and has the ability to control) the trajectories of every particle everywhere, there is no such thing as temperature, no such thing as probability. Particles just have whatever positions and momenta they have, and events either happen or they don't (neglecting extra nuances from QM). For those of us bound by thermodynamics, however, these same systems of particles and events are far less predictable. We can't see all the lowest-level details, much less model them with the same precision as reality itself, much less control them with God-level orchestration. Thus, probability, temperature, etc. become necessary tools for predicting and controlling reality at the level of rational agents embedded in the physical universe, with all the ignorance and impotence that comes along with it.

1. You spend a few paragraphs puzzling about how a probabilistic theory could be falsified. As you say, observing an event in a null set or a meagre set does not do the trick. But observing an event which is disjoint from the support of the theory's measure does falsify it. Support is a very deep concept; see this category-theoretic treatise that builds up to it.
2. However, I think the more fundamental question here isn't "how can I discard most of the information in a probabilistic theory so that it fits into Popperian falsificationism?", but rather "why should I accept Bayesian epistemology when it doesn't seem to fit into Popperian falsificationism?" For that, I refer you to Andrew Gelman's nuanced views and Sprenger and Hartmann's Bayesian Philosophy of Science.

What do we mean when we say that we have a probabilistic theory of some phenomenon?

If you have a probabilistic theory of a phenomenon, you have a probability distribution whose domain, or sample space, is the set of all possible observations of that phenomenon.

A probabilistic theory can be considered as a function that maps random numbers to outcomes. It tells us to model the universe as a random number generator piped through that function. A deterministic theory is a special case of a probabilistic theory that ignores its random number inputs, and yields the same output every time.

Here's an example: We can use the probabilistic theory of quantum mechanics to predict the outcome of a double slit experiment. If we feed a random number to the theory it will predict a photon to hit in a particular location on the screen. If we feed in another random number, it will predict another hit somewhere else on the screen. Feed in lots of random numbers, and we'll get a probability distribution of photon hits. Believing in the probabilistic theory of quantum mechanics means we expect to see the same distribution of photon hits in real life.

Suppose we have a probabilistic theory, and observe an outcome that is unlikely according to our theory. There are two explanations: The first is that the random generator happened to generate an unlikely number which produced that outcome. The second is that our theory is wrong. We'd expect some number of unlikely events by chance. If we see too many outcomes that our theory predicts should be unlikely, then we should start to suspect that the theory is wrong. And if someone comes along with a deterministic theory that can actually predict the random numbers, then we should start using that theory instead. Yudkowsky's essay "A Technical Explanation of Technical Explanation" covers this pretty well, I'd recommend giving it a read.

The takeaway is that quantum mechanics isn't a decision theory of how humans should act. It's a particular (very difficult to compute) function that maps random numbers to outcomes. We believe with very high probability that quantum mechanics is correct, so if quantum mechanics tells us a certain event has probability 0.5, we should believe it has probability 0.50001 or 0.49999 or something.

Also, in real life, we can never make real-number measurements, so we don't have to worry about the issue of observing events of probability 0 when sampling from a continuous space. All real measurements in physics have error bars. A typical sample from the interval [0,1] would be an irrational, transcendental, uncomputable number. Which means it would have infinitely many digits, and no compressed description of those digits. The only way to properly observe the number would be to read the entire number, digit by digit. Which is a task no finite being could ever complete.