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Erkenntnis

, Volume 83, Issue 6, pp 1195–1212 | Cite as

No Interpretation of Probability

  • Wolfgang SchwarzEmail author
Open Access
Article

Abstract

I argue that none of the usual interpretations of probability provide an adequate interpretation of probabilistic theories in science. Assuming that the aim of such theories is to capture noisy relationships in the world, I suggest that we do not have to give them classical truth-conditional content at all: their probabilities can remain uninterpreted. Indirectly, this account turns out to explain what is right about the frequency interpretation, the best-systems interpretation, and the epistemic interpretation.

1 Interpreting Science

Probabilities play a central role in many theories all across science, from quantum physics to statistical mechanics to chemical kinetics, to systems biology, evolutionary theory, and neoclassical economics. If we want to take seriously what science tells us about the world, we have to ask what the probability statements in these theories mean.

Consider Boltzmann-style statistical mechanics. Here the objects of study are isolated physical systems consisting of a large number N of particles. The possible micro-states of such a system (with fixed energy) correspond to a region \(\Gamma \) in a 6N-dimensional state space, each point of which specifies the precise location and momentum of every particle. According to the ‘basic postulate’ of statistical mechanics, the probability with which the system’s state lies in a subregion S of \(\Gamma \) is equal to the ratio \(\mu (S)/\mu (\Gamma )\), where \(\mu \) is a measure of volume (the Liouville measure) associated with the space.

The postulate in some sense identifies the probability P(S), that the system’s state lies in region S, with the quantity \(\mu (S)/\mu (\Gamma )\). But the identification is not a stipulative definition: ‘\(P(S) = x\)’ is not just shorthand for ‘\(\mu (S)/\mu (\Gamma ) = x\)’. That would turn the probabilistic predictions of statistical mechanics into trivial analytic truths. But statistical mechanics is an empirical theory, explaining real-life phenomena such as the melting of ice cubes and the diffusion of soy milk in coffee, and these explanations turn on the identification of P(S) with \(\mu (S)/\mu (\Gamma )\). In fact, it has been argued that in order to yield the right empirical predictions, the measure \(\mu \) should be replaced by another measure \(\mu '\) that gives lower weight to systems moving from a high entropy past to a low entropy future (see e.g. Albert 2000: ch. 4).

So the identification of P(S) with \(\mu (S)/\mu (\Gamma )\) or \(\mu '(S)/\mu '(\Gamma )\) is not just a definition. It seems to have empirical consequences. What are these consequences? What does statistical mechanics say about the world when it says that that \(P(S) = \mu (S)/\mu (\Gamma )\)?

Some hold that there is a fundamental probabilistic quantity of chance built into the very fabric of physical reality, accounting for the propensity of physical systems to evolve in one way rather than another. Tritium atoms, for example, have a propensity for decaying into helium. On the present view, this propensity is understood as an intrinsic physical property, not unlike mass or charge. If tritium atoms have this intrinsic propensity, then the basic laws of physics must be indeterministic: if the laws dictate that a given atom will decay in 8.7 years, then the decay would no longer be a matter of chance (see e.g. Schaffer 2007).

There is an extensive debate in philosophy over whether the idea of primitive chance is ultimately coherent and whether the probabilities in quantum physics can be interpreted as primitive chances. We do not need to enter this debate, for it is quite clear—and widely agreed—that the probabilities of statistical mechanics are not primitive chances. My cup of coffee can hardly be said to have a propensity to be, right now, in one micro-state rather than another. Indeed, classical statistical mechanics assumes a deterministic micro-dynamics. It certainly does not presuppose that the fundamental laws are stochastic.1

One might suggest that while the probabilities of statistical mechanics do not pick out primitive quantum-physical propensities, they pick out another primitive probabilistic quantity—call it ‘statistical mechanical chance’. But there is no good reason to believe in such a quantity. Among other things, the quantity would seem to be epiphenomenal. The future state of a physical system is completely determined (to the extent that it is) by its present micro-state and the fundamental dynamical laws. It isn’t sensitive to the values of any other fundamental quantity.2 Moreover, it is plausible that the laws of statistical mechanics supervene on the fundamental structure of the world, in the sense that a world with the very same distribution of micro-properties and the same micro-laws couldn’t have different statistical mechanical probabilities. Again this suggests that these probabilities are not fundamental.

So even those who believe in primitive chance still need an interpretation of the probabilities in statistical mechanics and other high-level scientific models—an interpretation that is neutral on the existence of primitive chance.

There is no shortage of proposals on the market. An old classic is the frequency interpretation, which identifies probabilities with actual or counterfactual frequencies. The proposal suffers from a range of serious and well-known problems (see e.g. Hájek 1997, 2009). For example, scientific practice clearly allows probabilities to deviate from actual frequencies, especially if the relevant conditions are rare. Turning to counterfactual frequencies only seems to makes things worse, since it is hard to give a coherent definition of counterfactual frequencies that preserves their status as a worthy object of empirical study.

A more sophisticated relative of the frequency interpretation is the best-systems interpretation, first developed by Lewis (1986: 128, 1994). Here the probability of an outcome is defined as the probability assigned to the outcome by whatever empirical theory best combines the virtues of simplicity, strength, and fit, where ‘fit’ measures the extent to which the theory assigns high probability to actual events. Lewis originally restricted the analysis to dynamical probabilities in fundamental physics, but it has often been observed that the analysis can be extended to statistical mechanics and other areas of science (see e.g. Loewer 2004, 2007; Schrenk 2007; Hoefer 2007; Cohen and Callender 2009).

Another popular proposal is the epistemic interpretation, which treats probability statements as statements about rational belief. Roughly speaking, to say that an event has a 50% probability is here taken to mean that it would be rational to be 50% confident that the event will occur. The epistemic interpretation has been especially influential in statistical mechanics (see e.g. Jaynes 1957; Uffink 2011).

I will argue that none of these proposals is plausible as an interpretation of probabilistic theories in science. They all make the same mistake: they assume that probabilistic theories are in the business of making straightforward, categorical claims about the world. I will suggest that this is not the right way to understand probabilistic theories. The probability claims in scientific theories are not meant to be true or false, and thus do not need an interpretation. The idea may sound radical and revisionary, but it turns out to be quite ecumenical. We will see that the frequency account, the best-systems account, and the epistemic account all have a natural place in the resulting picture.

2 Troubles

Let me begin with a somewhat curious (and so far overlooked) problem for the best-systems account. The problem is easiest to see in the application for which the account was originally developed: stochastic dynamical theories in fundamental physics.

The leading version of quantum physics that postulates a stochastic dynamics is the ‘GRW’ theory first proposed by Ghirardi, Rimini, and Weber (1986). GRW assigns probabilities to certain transitions between states of physical systems. The details don’t matter for our purpose. For concreteness, let’s imagine GRW directly postulates an 0.5 probability for tritium atoms to decay within 12 years; for short, it says that \(P(Decay )=0.5\). What does that mean?

The best-systems account defines physical probability (what Lewis calls ‘chance’) indirectly via scientific theories. Imagine a list of all possible physical theories, understood as deductively closed systems of sentences in a suitable language. Given full information about the world, each such system can be evaluated for correctness, simplicity, strength, and other theoretical virtues. If a theory involves probabilities, we can also evaluate the extent to which the theory assigns high probability to actual events. Suppose some probabilistic theory T comes out best, on balance, by those criteria. The best-systems account now defines the true probability of an event as whatever probability this ‘best system’ T assigns to the event.

Here is how Lewis puts it:

Consider deductive systems that pertain not only to what happens in history, but also to what the chances are of various outcomes in various situations [...]. Require these systems to be true in what they say about history. We cannot yet require them to be true in what they say about chance, because we have yet to say what chance means; our systems are as yet not fully interpreted. [...] [S]ome systems will be simpler than others. [...] [S]ome will be stronger than others: some will say either what will happen or what the chances will be when situations of a certain kind arise, whereas others will fall silent both about the outcomes and about the chances. And further, some will fit the actual course of history better than others. That is, the chance of that course of history will be higher according to some systems than according to others. [...] The virtues of simplicity, strength, and fit trade off. The best system is the system that gets the best balance of all three. [...] [T]he laws are those regularities that are theorems of the best system. But now some of the laws are probabilistic. So now we can analyse chance: the chances are what the probabilistic laws of the best system say they are. (Lewis 1994: 233f).

On Lewis’s account, what is the truth-conditional content of a statement such as \(P(Decay )=0.5\)? What does the statement say about the world? To be sure, it says that the probability of decay (i.e., of a tritium atom decaying within 12 years) is 0.5. Everyone agrees about that. What we want to know is what that means: does it mean that a fundamental physical measure of chance assigns value 0.5 to the decay event? Or does it mean that the relative frequency of tritium atoms decaying within 12 years is 0.5? According to the best-systems account, it means neither of these things. Rather, \(P(Decay )=0.5\) seems to mean the following:
(*)

Whichever physical theory best combines the virtues of simplicity, strength, fit, etc. assigns probability 0.5 to tritium atoms decaying within 12 years.

The truth-conditional equivalence is straightforward. By the best-systems account, the objective chance of an event is defined as whatever probability the best system assigns to the event. So if the best system assigns probability 0.5 to an event—as (*) says—then the event’s chance must be 0.5. Conversely, if the chance is 0.5, then 0.5 must be the value the best system assigns to the event. So, if probability statements in physical theories are interpreted along the lines of the best-systems account, then our imagined law \(P(Decay )=0.5\) is analytically equivalent to (*).

The problem is that (*) is not the kind of proposition I would expect to find in the basic laws of physics.3

Why not? One reason is that I expect the basic laws of physics to specify relations between fundamental physical quantities. (That is, I expect them to be truth-conditionally equivalent to a sentence stating relations between fundamental quantities.) This is what’s wrong, for example, with the Copenhagen interpretation of quantum mechanics, according to which the basic laws attribute a special role to measurements: measurement is a gerrymandered, anthropocentric, and not at all fundamental physical kind. The same is true for probability as interpreted by the best-systems account. The theoretical virtues that go into the definition of a best system are not part of fundamental physical reality. Indeed, proponents of the best-systems account often emphasize the anthropocentric character of the interpretation, the fact that it reflects our contingent epistemic perspective. In any case, there are many ways of spelling out the virtues, and of balancing them against each other; it is hard to believe that one of these ways is somehow objectively privileged. On the best-systems interpretation, the precise content of the GRW laws would therefore depend on arbitrary choices in the ranking of theories.

The problem here is not that on the best-systems account, what counts as a (probabilistic or non-probabilistic) physical law might depend on somewhat arbitrary and anthropocentric facts. That is true, and widely accepted by advocates of the best-systems account. The present problem is that the content of probabilistic laws now involves gerrymandered, anthropocentric notions.

A second point that worries me about fundamental laws like (*) is that I expect the basic laws of physics to be explanatory bedrock, in some intuitive sense. Why do opposite charges attract? Perhaps there is no deeper scientific explanation. That’s just how things are.4 By contrast, if the basic laws say that \(P(Decay )=0.5\) and what this means is that the best system assigns probability 0.5 to Decay, then that is clearly not a basic fact. It is explained by patterns of occurrent events in the history of the world together with the relevant standards for evaluating theories.

Again, the problem should not be conflated with a superficially similar but different problem: that the best-systems account makes the laws depend on occurrent facts, while many people intuit that the dependence goes the other way. We must distinguish the claim that p is a law from the simpler claim that p. On the best-systems account, that it is a law that opposite charges attract is made true by patterns in occurrent events. If something is a law, then on the best-systems account there is always a non-trivial explanation of why it is a law. But the simpler claim that opposite charges attract is not a statement about laws; it may well be explanatory bedrock. The problem is that this can no longer be said for probabilistic claims such as \(P(Decay )=0.5\). If that claim is analysed as (*) then its truth clearly has an explanation.

This brings me to a third worry: the best-systems account threatens to collapse the important difference just mentioned between the claim that something is merely true and the claim that it is nomologically necessary. Newton’s second law, for example, says that \(F=ma\), not that it is nomologically necessary that \(F=ma\). Yet if we interpret \(P(Decay ) = 0.5\) as (*), then \(P(Decay ) = 0.5\) can’t be true without also being part of the best system and hence a law (on the best-systems account of laws).

All these problems arise because the best-systems account has implications for the (truth-conditional) content of probabilistic laws. Analogous problems do not arise for the best-systems account of non-probabilistic laws because physical theories generally do not involve the term ‘law’; whatever we say about the meaning of ‘law’ therefore can’t have implausible consequences about the content of physical laws. But probabilistic laws in science evidently do contain the term ‘probability’, or ‘P’. Our present concern is not what makes such laws laws. It is more basic: What do such laws say about the world? What does GRW say when it assigns such-and-such probability to transitions between physical states? The best-systems account suggests an answer: it suggests that the probability statements in GRW mean that whichever theory best combines such-and-such virtues assigns such-and-such probability to the relevant transitions. But that, I have argued, is implausible.

Analogous problems do arise for the epistemic interpretation of physical probability. Suppose, as before, that \(P(Decay )=0.5\) is (an instance of) a fundamental physical law. On a simple-minded Bayesian interpretation, the law states that some not-further-specified individual assigns subjective degree of belief 0.5 to the decay event. That is clearly absurd. The laws of nuclear decay are not statements about what some person happens to believe; they can be true even if no-one has the relevant degrees of belief. More sophisticated epistemic accounts interpret probability statements as statements about what it would be rational to believe. So \(P(Decay )=0.5\) is analysed as something like (\(\dagger \)).
(\(\dagger \))

Rational agents should assign degree of belief 0.5 to Decay.

But that is still absurd. For one thing, surely normative psychological notions do not figure in the fundamental laws of physics! As above, the relevant propositions also do not seem to be explanatory bedrock. If it is rational to have degree of belief 0.5 in certain events, and this is an epistemically contingent fact about the world (as physical laws are supposed to be), then surely there must be an explanation of why that degree of belief would be adequate. In addition, one can clearly entertain the hypothesis that there are no epistemic norms at all, or none beyond probabilistic coherence—a view prominent philosophers have endorsed—without concluding that the laws of nuclear decay are false. In other words, it would be absurd to argue that physics has established the existence of non-trivial epistemic norms. So the probabilities in physical theories cannot be straightforwardly interpreted as epistemic probabilities.

I have assumed that the epistemic account and the best-systems account are to be understood as genuine analyses (or explications) of probability statements: as directly and transparently spelling out their truth-conditions. For example, I have assumed that on the epistemic account, anyone who accepts a probabilistic theory is thereby committed to the existence of substantive epistemic norms, just as anyone who accepts that there are vixens is thereby committed to the existence of foxes; anyone who claims to accept the theory but reject the norms is either confused or misunderstands the theory.

In response, one might suggest that the accounts in question are not meant to provide analyses of that kind. Perhaps they only ‘fix the reference’: they identify physical probability by a certain role, without revealing the nature of the quantity that occupies the role. This kind of story is familiar and plausible for other theoretical terms. Perhaps our concept of inertial mass can be analyzed in terms that we don’t expect to find in the laws of fundamental physics, identifying inertial mass by its role in our experience of the world—roughly, as the property responsible for the fact that we find some things harder to accelerate than others. The role is realized by a fundamental physical quantity (as it turns out, by the very same quantity that also plays the role associated with the distinct concept of gravitational mass). The content of Newton’s second law is arguably a proposition directly about that quantity. Unfortunately, that story can hardly be adapted to probability terms. Advocates of the best-systems account or the epistemic account generally do not believe in fundamental probabilistic quantities, and even if they did, the story would at most apply to probabilities in quantum physics. We could alternatively take the referent to be a non-fundamental quantity such as \(\mu (S)/\mu (\Gamma )\), but that would turn the probability statements in the relevant theories into empirically empty tautologies.

So the problem remains: popular reductionist accounts of physical probability have implausible consequences for the content of probabilistic theories. So much the worse, you might say, for reductionist accounts of physical probability! If we believe in primitive chance, we can simply read \(P(Decay )=0.5\) as a statement about chance, without analyzing it in terms of anything else. But this interpretation, too, faces serious problems—for example, when it comes to explaining the link between physical probability and rational degree of belief (see Lewis 1994). In any case, the primitive chance account is at best applicable to a very narrow range of scientific theories. It says nothing about the probabilities in Bohmian mechanics, statistical mechanics, chemical kinetics or systems biology.

What about a mixed approach then: GRW talks about primitive chance, while the other theories talk about best-systems probabilities or epistemic probabilities? I agree that we should not take for granted that a unified interpretation can be given for probability statements in all areas of science. But most of the problems I just raised for the best-systems interpretation and the epistemic interpretation are not specific to GRW; analogous problems arise for probabilistic models in statistical mechanics or genetics. For example, the best-systems account would still collapse the distinction between p and it is a law that p, and the epistemic account would take statistical mechanics to have established non-trivial normative truths. In addition, the mixed approach would face all the problems of postulating primitive chances.

I do not claim to have refuted any—let alone all—candidate interpretation of probabilistic theories. Most advocates of the best-systems account have come to accept that the account has counterintuitive consequences, so they might accept the problems I have raised as further bullets that have to be bitten. Nonetheless, I hope I have said enough to motivate trying something new.

3 Theories Without Truth

I began with a question: what do probabilistic theories in science say about the world? What would a world have to be like for it to be true that tritium atoms have a 50% probability of decaying within 12 years? I want to suggest that we should reject the question. Probability statements in scientific theories do not express a special kind of fact. They are not meant to be true or false.

The idea is that we broaden our conception of scientific theories. On the traditional realist conception, scientific theories aim to register important truths about the world: interesting and robust patterns in the observable phenomena and in whatever lies behind these phenomena. The task is straightforward if the relevant patterns are crisp: all Fs are Gs; whenever a system is in state \(S_1\) it will later be in \(S_2\); whenever a phenotype has frequency x in one generation then it has frequency y in the next generation. But what if the world is more complicated? What if two quantities F and G are strongly and robustly correlated, but the value of G on any given occasion is not completely determined by the value of F, nor is there a simple formula for how G is determined by F together with other salient features of the situation? We could simply refrain from saying anything about the connection between the quantities. But then we might fail to capture an important fact about the world. What is a scientist supposed to do if she notes (or suspects) an interesting, robust, but noisy relationship between two quantities? How can she express such a relationship in a scientific theory?

This is where probability enters the picture. Let’s allow our scientist to specify a probabilistic relationship between F and G, perhaps by adding a noise term to an algebraic equation. The result is a probabilistic model or theory. The point of the model is to capture the noisy, stochastic relationship between F and G. It is not to capture a crisp relationship between F, G, and third quantity P. This is why we could not find a sensible answer when we asked what that quantity might be: primitive propensity, best-systems probability, rational credence, or what have you. All these answers misunderstand the point of probabilistic models.

When a scientist puts forward a probabilistic model, she commits herself to the assumption that the model fares well, on balance, in terms of simplicity, strength, fit and other relevant virtues. But this is not the content of her model. Her model doesn’t say of itself that it maximizes theoretical virtues, or that it captures noisy relationships in the world. In order to serve its purpose, it is enough that the model contains a probability function. The function does not need an interpretation.

Consider a toy example. Our object of study is a series of events with two kinds of outcome, call them ‘heads’ and ‘tails’. (If you want, imagine that this is all there is in the universe. At any rate, nothing else falls in scope of our inquiry.) There are a million outcomes in total, arriving in seemingly random order with heads having a stable relative frequency of around 0.8. How could we model this noisy pattern? We could simply list all individual outcomes in the order in which they arrive. But such a list would be unwieldy, and it wouldn’t reveal any patterns in the data. For many purposes, it might therefore be useful to put forward a probabilistic model. Specifically, we could put forward a model that assigns probability 0.8 to heads on each toss, independent of the other outcomes. The model’s probability for heads and tails then closely matches their relative frequency, but the probabilities are not meant to stand for relative frequencies. Indeed, by treating the events as independent the model assigns positive probability to many sequences of heads and tails (such as 1000 tails in a row) that never occur in the series at all. Nor are the probabilities meant to stand for fundamental propensities. The events in question may or may not be generated by an underlying deterministic mechanism; the usefulness of our model doesn’t depend on that. Nor are the probabilities meant to stand for rational credence, or anything else. The point of our probabilistic model is, as I said, to capture a noisy pattern in the world.

To a first approximation, we can spell out what that means by following the best-systems analysis—but without assigning an interpretation to the probabilities. Imagine all possible ways of assigning probabilities to the members of our series. These are our ‘theories’. Some of them are simpler than others. A theory that assigns probability 1 to every actual outcome and thus effectively lists the entire series is not very simple; a theory that treats the outcomes as independent is (other things equal) simpler than a theory that doesn’t. And so on. We can also compare our theories in terms of strength. A theory that assigns probabilities only to individual outcomes is (other things equal) weaker than a theory that also assigns probabilities to sequences of outcomes. And we can compare our theories in terms of probabilistic fit. For theories that assign a probability to the entire sequence, we can follow Lewis and use that probability as a measure of fit. Finally, then, what it means for a theory to capture the patterns in our sequence is that the theory fares comparatively well, on balance, in terms of simplicity, strength, and fit.

If the aim of scientific theories is to capture possibly noisy patterns in the world, we don’t need to interpret the theories’ probabilities. In fact, interpreting the probabilities would get in the way of this aim, since it is unclear how a theory which states a crisp relationship between three quantities F, G, and P is supposed to capture a non-crisp pattern in the relationship between F and G alone.

Admittedly, the present view of scientific theories may be unfamiliar and therefore somewhat counterintuitive. We are used to thinking that respectable scientific theories explicitly represent the world as being a certain way, for example (as I said above) by stating relations between fundamental quantities. On the present account, this is not quite true for probabilistic statements in scientific theories. If a theory ‘states’ a probabilistic connection between fundamental quantities, it doesn’t really state anything, insofar as it does not make a categorical, outright claim about the world.

As an analogy, it may help to imagine scientific theories as agents (‘experts’). On the traditional conception of theories, the expert only has binary beliefs: she believes that all Fs are Gs, that whenever quantity A has value x, then B has value y, and so on. Now we also allow partial beliefs. The expert can be more or less confident that something is G given that it is F, or that B has value y if A has value x. The expert can be 80% confident that the first outcome in a series is heads. Such partial beliefs are not outright beliefs with a special probabilistic content. To believe something to a given degree is not to have a full belief about a physical quantity, or about one’s own state of mind. As a consequence, a system of partial beliefs is in the first place not true or false, but more or less close to the truth. A good expert generally assigns high degree of belief to true propositions and low degree of belief to false ones. A range of ‘accuracy measures’ have been proposed to render this kind of distance to the truth precise (see e.g. Joyce 1998). Such measures can be applied not only to probability functions that represent degrees of belief but also to uninterpreted probability functions in scientific theories, where they offer a natural approach to measuring fit. Like a good expert, a good theory should generally assign high probability to true propositions and low probability to false ones.5

One might think that an ideal expert assigns degree of belief 1 to every truth and degree of belief 0 to every falsehood. Accordingly, an ideal theory would have no need to involve probabilities. But a complete theory of all truths is not only beyond our reach, it is also not what we seek in scientific theories. Science is looking for patterns in the total history of the world, for simple yet powerful principles that allow predicting a wide variety of facts. If these patterns are suitably noisy, even an ideal theory will be probabilistic.

So that’s my proposal. The probabilities in scientific theories do not have an interpretation. As a consequence, probabilistic theories cannot be true or false, except in their non-probabilistic parts. They can still be more or less simple, more or less unified, and more or less close to the truth, as measured by the difference between the (uninterpreted) probabilities and the actual events in the world. That is all we need. The point of probabilistic models in science is to provide a simple and informative systematization of noisy patterns in the world, and they can do that without being true (or false).

4 Capturing Patterns

Let me say a little more on how we may understand the goal of ‘capturing noisy patterns’. Above I explicated this notion by following the best-systems account: I suggested that a probabilistic theory captures a noisy pattern in certain events just in case it scores best, on balance, in terms of simplicity, strength, and fit among all possible theories of the relevant events. It is crucial for my proposal that a theory can do that without having truth-conditional content.

Intuitively, one might think that a theory’s strength is to be measured by how many possibilities it rules out. This would seem to require that the theory has truth-conditional content. But while the suggested notion of strength may be useful for certain applications (assuming one can find a sensible way of counting possibilities), there are independent reasons why it is not adequate in the context of either the best-systems account or the present proposal. In particular, we here need a measure of strength that is relative to a history of events.

To illustrate, consider the ‘problem of accidental regularities’. Many truths of the form all Fs are Gs—including many simple truths of that form—are clearly not laws. On the best-systems account, a regularity is a law only if (together with other members of the best system) it provides valuable information about the world. But whether all Fs are Gs provides valuable information depends on how many Fs and Gs there are. If there are many Fs, all of which are G (and many non-Fs that are not G), then it is useful to know that all Fs are Gs; the statement may then be part of the best summary of regularities in the world. Not so if there are few Fs, or no Fs at all. The statement is equally true, and equally simple, in either case, but it is more informative in the first. All else equal, the relative strength of all Fs are Gs in a given world should therefore be greater the more Fs and the fewer Gs there are in the world. The situation is the same, and the same criterion could be used, for probabilistic statements: the probability of an F being G is x.

Note that the best-systems account, too, assumes that one can evaluate theories for simplicity, strength, and fit without assigning an interpretation to the probabilities (see the above quote from Lewis (1994: 234)). Lewis suggests to measure simplicity by syntactic complexity, strength by the variety of circumstances and outcomes for which a theory specifies probabilities, and fit as the probability a theory assigns to the entire history of the world. Lewis does not defend these criteria. The observation that strength should be world-relative indicates that his criterion for strength is inadequate. His criterion for fit also runs into well-known problems in cases where theories assign either no probability or probability zero to the entire history of the world. Elga (2004) suggests an alternative characterization of fit in terms of typicality; in Schwarz (2014) I suggest yet another measure of fit which aggregates the differences between actual frequencies and theory-expected frequencies.

In general, it is fair to say that nobody has yet put forward fully satisfactory and precise criteria for simplicity, strength, and fit, and for how these are meant to trade off against each other. For the present proposal, this is less of an embarrassment than it is for the best-systems account. In the account I have put forward, the standards of simplicity, strength, and fit are only used to clarify the scientific aim of capturing patterns. This aim does not have to be absolutely precise and objective. We can allow that what scientists value in their theories is to some extent imprecise and varies from discipline to discipline, from school to school, or even from person to person.

I will not go into more details about how one might spell out the relevant notions of simplicity, strength, and fit. I do, however, want to highlight a further kind of goal that is often ignored in the literature on best systems.

Real scientific theories typically aim for more than a compact statistical summary of relevant events. They try to shed light not only on how the events are distributed, but also on why they are distributed the way they are. Accordingly, the probabilities in scientific theories are generally motivated by underlying explanatory assumptions, often about how the relevant events come about. The binomial probabilities in the Wright–Fisher model of neutral evolution, for example, are not based on inductive generalization from observed frequencies. Rather, they are motivated and explained by internal assumptions of the model.

A popular and powerful tool for motivating probabilities is the ‘method of arbitrary functions’ (see e.g. von Plato 1983). Paradigm applications of the method are gambling devices such as roulette wheels or dice. These devices are built in such a way that any reasonably smooth probability distribution over initial conditions is mapped by the dynamics of the system to approximately the same distribution over outcomes. The characteristic patterns in the observed outcomes can therefore be explained by the absence of very unusual patterns in the input conditions. Several authors have recently suggested that considerations along these lines can also justify the probabilities in statistical mechanics and other scientific theories (see e.g. Strevens 2003; Myrvold 2012).

What’s important for our present topic is not so much how this or that probabilistic model can be justified, but the more general fact that we expect the probabilities in a model to have some such underlying justification. Among other things, this explains why we tend to hold fixed the adequacy of our models under counterfactual suppositions: on the supposition that a fair coin were tossed a million times, we expect the relative frequency of heads to be approximately 1/2.

The method of arbitrary functions, the ergodic theorem, and other popular ways of justifying probabilistic models explain why a model can be expected to have good probabilistic fit, but they do not provide an interpretation of the model’s probabilities. Consequently, these explanations are often supplemented with an epistemic or frequentist interpretation of probability—leading to the usual problems for these interpretations. On the present approach, no supplementation is called for.

5 Theories, Predictions, Beliefs

At first glance, my proposal seems to create a host of problems. If probability statements don’t have truth-conditional content, how can they be believed, disbelieved or conjectured? How can they be confirmed or disconfirmed by observation? How do we interpret complex sentences that embed statements about probability?

In response, I should first stress that my proposal does not concern the interpretation of probability statements in ordinary language. My topic is the interpretation of scientific models or theories. Arguably, such models are best understood not as linguistic constructions at all. If they are expressed in language, that language always includes special-purpose technical vocabulary. On my proposal, probability terms should be treated as technical terms, and they should not be given an interpretation. I will say a little more on the interpretation of ‘probability’ in ordinary English below, but that is not the focus of my proposal.

So the problem with complex sentences only arises for complex sentences within a given scientific theory. That is, what if instead of assigning an outright probability to an event A, a theory merely states that the probability of A is either x or y? Or what if a theory says that if H, then \(P(A) = x\) (see footnote 3 above)?

Now, on the present account, probabilistic theories do not have a classical truth-conditional interpretation. They only need to be evaluated for simplicity, strength, probabilistic fit and other theoretical virtues. So we need to ask, for example, how to measure a theory’s fit with respect to actual events in the world if the theory merely specifies that the probability of A is either x or y. This might be an interesting question to ponder, but it is not a terribly urgent question, since real theories rarely take that form. (To the extent that there is a problem here, it is equally a problem for the best-systems account, which also assumes that one can evaluate theories for probabilistic fit without yet assigning a meaning to the probability terms.)

The issue of confirmation and belief is more serious. One response it to go full-on projectivist and say that beliefs about objective probability also do not straightforwardly represent the world as being one way or another (see e.g. Skyrms 1980, 1984; Jeffrey 1983: ch. 12; Spohn 2010). I have sympathies for this move, but let me defend a less radical response.

Suppose a scientist proposes or endorses a probabilistic theory T. On the account I suggested, she thereby commits herself to the hypothesis that T provides a good systematization of relevant patterns in the world. So the scientist commits herself to the truth not of T itself, but of a derivative proposition \(\Box T\): that T fares well in terms of simplicity, strength, fit, and other theoretical virtues.6 Unlike T, \(\Box T\) is an ordinary (albeit vague) proposition. It can be true or false. It can be believed, disbelieved, conjectured, and denied. It can be confirmed and disconfirmed by empirical observations.

So what appear to be propositional attitudes towards a probabilistic theory T are really attitudes towards an associated proposition \(\Box T\)—roughly, the proposition that T provides the best systematization of relevant patterns in the world.

The ‘relevant patterns’ are not just patterns in the phenomena. To be sure, a scientist might only half-heartedly and instrumentally ‘accept’ a theory, confident that it captures interesting patterns in past and future observations, but agnostic about whether the entities it postulates are real and whether they display the hidden patterns described by the theory. In contrast, to really endorse (say) GRW quantum mechanics, you have to believe (roughly) that the true state of an isolated physical system is accurately and completely characterized by its wavefunction, that the state mostly evolves in accordance with the Schrödinger equation, but that this evolution is occasionally punctured by collapse events whose frequency and outcome displays statistical regularities to which the probabilities in GRW are a good approximation. This (roughly) is the content of \(\Box {\text {GRW}}\). It goes far beyond the hypothesis that GRW is a useful tool for predicting measurement outcomes.7

In general, \(\Box T\) is closely related to propositions about disorder and relative frequency. Return to the toy example from Sect. 3. Let T be the so-far unnamed theory which assigns probability 0.8 to heads on each toss, treating the tosses as independent. T itself can’t be true or false, but \(\Box T\) can. What does \(\Box T\) entail about the sequence of outcomes? It obviously depends on the precise meaning of the box. For concreteness, let’s assume that \(\Box T\) states that T is the best systematization of the sequence as measured by Lewis’s (1994) criteria of simplicity, strength, and fit—setting aside the worries raised in the previous section. \(\Box T\) then entails that about 80% of the tosses actually come up heads. For suppose the actual frequency is only 70%. Then T provides a significantly worse systematization of the sequence than a rival theory \(T'\) that assigns probability 0.7 to heads on each toss. T and \(T'\) are equal in terms of simplicity and strength, but \(T'\) has much greater fit: the probability of 70% heads is approximately \(8.7 \times 10^{-4}\) according to \(T'\), but \(8.4 \times 10^{-12237}\) according to T. \(\Box T\) also entails that the sequence of outcomes does not have any conspicuous patterns. For example, it can’t be 200,000 heads followed by 800,000 tails, or 200,000 repetitions of HHHHT; in either case, it would be easy to specify the exact sequence, so a good systematization of the outcomes would not resort to probabilities at all. Finally, \(\Box T\) plausibly entails that right after a heads outcome, the relative frequency of another heads is not too far from 80%; otherwise a theory that doesn’t treat successive tosses as independent would have greater fit without too much a cost in simplicity.

So there is a tight connection between probabilistic theories and claims about relative frequency and disorder. If a scientist accepts our model T, she will expect an irregular sequence with about 80% heads and 20% tails. If the sequence turns out to be more regular or the frequencies different, the scientist will have to revise her attitudes towards T. It is therefore understandable that many science textbooks endorse some form of the frequency interpretation on which probability claims simply are claims about relative frequency.8

We can also see what is right about the epistemic interpretation. On the supposition that a theory T provides a good systematization of the relevant patterns in the world (i.e., on the supposition that \(\Box T\) is true), a rational agent should generally align her credence with the theory’s probabilities. To illustrate, suppose you know that the best systematization of our coin toss sequence is the theory T that treats the tosses as independent with a fixed probability 0.8 of heads. As we saw, this entails that the sequence is irregular with about 80% heads and 20% tails. Now consider, say, toss number 512. How confident should you be that this particular toss results in heads? In the absence of further relevant information, surely your credence should be about 0.8. Moreover, your credence should be fairly insensitive to information about other outcomes. For example, conditional on the assumption that toss number 511 lands tails, your degree of belief in heads on toss number 512 should still be about 0.8. (See Schwarz (2014) for more details and generalizations of these observations.)

How could you have come to know \(\Box T\), without having surveyed the entire sequence? The short answer is: by induction. Perhaps you have witnessed the first 10,000 tosses, and found an irregular pattern of heads and tails with about 80% heads. All else equal, you would then be justified to assume that the same noisy regularities obtain in the unobserved parts of the sequence.9 Remember also that real scientific theories typically aim for more than a mere summary of actual events. If we know the dynamics of roulette wheels, the method of arbitrary functions explains why it is reasonable to believe that a certain probabilistic model captures the pattern of outcomes even without any direct information about those outcomes.

So the less radical response is that believing or testing a probabilistic theory T is really believing or testing the corresponding proposition \(\Box T\).

The same trick can be used for other occasions where uninterpreted probabilities seem to cause trouble.

For example, physical probability is often thought to be closely related to causation; how does that work if physical probabilities don’t have an interpretation? On the approach I have outlined, physical probability resides in laws or models: if smoking probabilistically causes cancer, then this is because of a suitable probabilistic law linking smoking and cancer. Now observe that where causation is underwritten by laws, it is generally not the content of the law that is doing the work, but the fact that this content obtains as a matter of law. For example, compare a world where by mere coincidence I feel a tickle in my ear whenever you clap your hands with a world where the same connection is a matter of law; only in the second case does your clapping cause my tickle, even though in both cases it is true that whenever you clap I feel the tickle. Likewise for probabilistic causation: the causal connection between the relevant variables or events is established not by the raw probabilities—which don’t have an interpretation—but by the corresponding law claim, which is an ordinary proposition.

For another example, consider explanation. When we explain some phenomenon by citing a probabilistic law or model, again it is arguably not the raw content T of the law that does the explanatory work, but the corresponding proposition \(\Box T\). If it is a mere coincidence that that I feel a tickle when you clap your hands, then this fact, together with your clapping, hardly explains my tickle.

Some may be sceptical that \(\Box T\), an ultimately ‘Humean’ proposition with no built-in modal force, can do the suggested work for causation and explanation: can the fact that \(P(Decay )=0.5\) is part of the best statistical summary of certain patterns in the world really explain the decay behaviour or tritium atoms? Here my proposal shares the burden of other Humean accounts. I will not attempt a full response here, but let me note two things. First, one can explain the diffusion of gas throughout a container by appealing to the probabilities of statistical mechanics even though these do not represent basic non-Humean quantities. Second, if you want to know why the gas diffused, it arguably helps to know that almost any initial configuration of gas molecules would lead to diffusion. On the present account this is (in part) what the probabilities of statistical mechanics capture.

6 Conclusion

I have suggested that none of the currently popular interpretations of probability yield an adequate understanding of probabilistic theories in science. The interpretations all assume that probability claims in science are claims about a particular probabilistic quantity, but it is hard to see what that quantity could be. I have argued that we should stop looking for a candidate. The point of probabilistic theories is not to express facts about some probabilistic quantity, but rather to capture noisy relationships between ordinary, non-probabilistic quantities.

On the resulting picture, probabilistic theories cannot be true or false, except in their non-probabilistic parts, but they can still be evaluated for simplicity, strength, fit and other theoretical virtues. To capture a noisy pattern in the world means to score (comparatively) high in terms of these virtues. If a theory captures a noisy pattern, we could say that it represents or predicts that pattern. In that sense, probabilistic theories do have representational content, even though their probability functions do not have an interpretation. The theory represents a pattern without stating that the pattern exists.

To fully accept a theory is to regard it as a good systematization of the relevant facts. Under normal conditions, this implies expecting a close fit between the theory’s probabilities and actual (as well as counterfactual) frequencies. It also implies adopting the theory’s probabilities as one’s own degrees of belief.

So the best-systems interpretation, the frequency interpretation, and the epistemic interpretation are not entirely off the mark. They all misrepresent the content of probabilistic theories, but they identify important aspects of what a rational agent must believe who accepts a probabilistic theory. Probabilistic laws do not say of themselves that they have various theoretical virtues, but accepting the laws plausibly involves believing that they do.

What about probability statements in ordinary language? Officially my proposal is neutral on this question. I am sympathetic to the view (defended e.g. in Maher 2010) that most ordinary statements about probability are normative epistemic statements. That is, by saying that there is a 90% probability of rain, I would typically recommend a corresponding degree of belief. Since accepting a theory goes hand in hand with taking the corresponding degrees of beliefs to be rationally adequate, we can see how the ordinary sense of ‘probability’ relates to the probabilities in scientific theories, and thus why the latter are called ‘probabilities’.10

Finally, what about the probabilities of statistics, as they figure for example in parameter estimation from noisy data? Again, the proposal I have made does not speak directly to this question. In principle, it is compatible with both Bayesian and frequentist accounts. However, it might offer a new perspective on the interpretation of ‘likelihoods’: the probability of data given some hypothesis. These likelihoods are often (on frequentist accounts, always) derived from general models of the experimental setup. My proposal straightforwardly applies to those models. It suggests that such model-based likelihoods are neither degrees of beliefs nor frequencies. Just as frequentists insist, they track objective features of the world. But they are also closely related to rational degrees of belief, for the reasons I have reviewed. Specifically, on the assumption \(\Box M\) that some statistical model M captures the noisy relationship between hypotheses and experimental data, it is generally rational to set one’s credence in data E given hypothesis H to equal the probability which M assigns to E given H. Bayesian reasoning therefore goes through essentially as before.

Footnotes

  1. 1.

    Albert (2000: ch. 7) suggests that the probabilities of statistical mechanics, and indeed of all higher-level scientific theories, might be reduced to those of GRW quantum physics. A friend of primitive chance might therefore hold that all probabilistic theories in science are in fact—more or less indirectly—concerned with the primitive chances of GRW. But even if we grant Albert’s controversial reductionism about special science (on which the proposal I will put forward is neutral), GRW is a highly speculative theory. What if GRW is not the true theory of fundamental physics? Nobody thinks that this would undermine the principles of statistical mechanics or evolutionary theory. So there must be an interpretation of these theories on which they do not presuppose the truth of GRW.

  2. 2.

    The qualification ‘fundamental’ is important. The future state of my coffee is certainly sensitive to its present temperature, although temperature does not figure in the fundamental laws. This is because temperature is realized by more fundamental physical properties. If the coffee’s temperature had been different, then its micro-state would also have been different, which would have made a difference to the coffee’s future.

  3. 3.
    I have assumed that theories directly specify probabilities to events. Lewis (1980) instead suggests that stochastic dynamical theories specify ‘history-to-chance conditionals’, saying that if H is the history of the world up to time t, then event A at \(t'\) has probability x. It is true that stochastic dynamical theories generally assign probabilities to future states relative to past or present states, but in my view these ‘conditionals’ are better understood as a kind of conditional probability. If we follow Lewis, the hypothesis that a particular tritium atom has an 0.5 chance of decaying within the next 12 years comes out as equivalent to something like (**):
    (**)

    The true history of the world up to now is such that whichever physical theory best combines the virtues of simplicity, strength, fit, etc. specifies that if the history up to now is just like that then there is an 0.5 probability that the tritium atom will decay within the next 12 years.

    If we apply this interpretation to the probability statements in theories like GRW, which on Lewis’s assumption have the form \(H \rightarrow P(Decay )=0.5\), they are equivalent to \(H \rightarrow (**)\). The problems I am about to raise carry over, mutatis mutandis, to this interpretation. In addition, there is at least one further problem: if the best system is anything like GRW, it plausibly implies conditionals of the form \(H \rightarrow P(Decay )=0.5\) for various non-actual histories H such that the best system for H-worlds does not imply \(H \rightarrow P(Decay )=0.5\). The conditional \(H \rightarrow (**)\) [which according to Lewis (1980: 95) is a ‘strong’ conditional like those discussed in Lewis (1973)] is then false, yet the laws of nature are supposed to be true!
  4. 4.

    Some hold that universal truths are made true by—and thus explained by—their instances. In that sense, the fact that opposite charges attract (i.e., that whenever two things have opposite charges they attract one another) is ‘explained’ by a lot of particular facts about things with opposite charge (plus presumably the fact that there are no further things with opposite charge). Others hold that the attraction between opposite charges is explained by a certain relation between universals. Neither type of explanation is the kind people would normally be asking for when they wonder why opposite charges attract.

  5. 5.

    Of course, an agent’s subjective probability function is not ‘uninterpreted’: it represents the agent’s degrees of belief. The point of the analogy is that partial beliefs, like the probabilities in scientific theories, do not simply represent the world as being one way or another.

  6. 6.

    The other virtues may or may not include truth of a theory’s non-probabilistic claims. Certainly good models in special science often involve simplifications or ‘idealizations’ that seem to be false.

  7. 7.

    So there is still an important contrast between scientific realism and anti-realism. It’s just that what is at issue is strictly speaking not the truth (or truth in certain respects) of our best theories T, but the truth (etc.) of the associated propositions \(\Box T\).

  8. 8.

    The connection between scientific probability and relative frequency is sometimes presented as a consequence of the laws of large numbers. But it is not. The connection is a substantive fact that any account of scientific probability should explain. The laws of large numbers are theorems of the probability calculus that hold for every probability function whatsoever, no matter how out of touch it is with the frequencies in the world.

  9. 9.

    As Hume pointed out, there is no logical guarantee that the patterns will continue, and thus no logical refutation of an inductive skeptic who claims that they won’t. In the literature on physical probability, it is sometimes assumed that in order to explain the connection between probability and rational belief, one must also refute the inductive skeptic (see e.g. Salmon 1967; Strevens 1999). Unsurprisingly, the verdict then is that the task is impossible.

  10. 10.

    Evidently, these remarks are not even a sketch of an actual semantics/pragmatics for probability statements in natural language; the details are hard (see e.g. Lassiter 2011; Rothschild 2012).

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Authors and Affiliations

  1. 1.School of Philosophy, Psychology and Language SciencesThe University of EdinburghEdinburghUK

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