Abstract
There are two notions in the philosophy of probability that are often used interchangeably: that of subjective probabilities and that of epistemic probabilities. This paper suggests they should be kept apart. Specifically, it suggests that the distinction between subjective and objective probabilities refers to what probabilities are, while the distinction between epistemic and ontic probabilities refers to what probabilities are about. After arguing that there are bona fide examples of subjective ontic probabilities and of epistemic objective probabilities, I propose a systematic way of drawing these distinctions in order to take this into account. In doing so, I modify Lewis’s notion of chances, and extend his Principal Principle in what I argue is a very natural way (which in fact makes chances fundamentally conditional). I conclude with some remarks on time symmetry, on the quantum state, and with some more general remarks about how this proposal fits into an overall Humean (but not quite neo-Humean) framework.
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Notes
- 1.
Incidentally, reviewed by Itamar when it came out (Bub and Pitowsky 1985). Note that my examples of possible assimilation or confusion between epistemic and subjective probabilities (in the senses just sketched) may well prove to be straw-mannish under further scrutiny, but I just wish to make a prima facie case that these two notions can indeed be confused. The rest of the paper will make the extended case for why we should distinguish them and how.
- 2.
It is unclear to what extent this ‘objective Bayesian’ strategy indeed yields objective probabilities. Consider for instance situations in which information about the physical situation is in principle readily available but is glibly ignored. In this case, even if we use principles such as maximum entropy, one might complain that there is little or no connection between the physical situation and the probabilities we assign. On the other hand, there may be situations in which the physical situation itself appears to systematically lead to situations of ignorance, e.g. because of something like mixing properties of the dynamics. In this case, there may arguably be a systematic link between certain objective physical situations and certain ways of assigning probabilities, which may go some way towards justifying the claim that these probabilities are objective. (Cf. discussions about whether high-level probabilities are objective, e.g. Glynn (2009), Lyon (2011), Emery (2013) and references therein.) If so, note that this will be the case whether or not there is anyone to assign those probabilities. Many thanks to Jossi Berkovitz for discussion of related points.
- 3.
For an example of what Jaynes has in mind, see Heisenberg’s ‘The Copenhagen interpretation of quantum theory’ (Heisenberg 1958, Chap. 3). In Sect. 3.9 below, I shall allude to what I think are the origins of these aspects of Heisenberg’s views, but I agree with Jaynes that (at least in the form they are usually presented) they are hard to make sense of. Note also that the ‘Copenhagen interpretation’ as publicised there appears to have been largely a (very successful) public relations exercise by Heisenberg (cf. Howard 2004). In reality, physicists like Bohr, Heisenberg, Born and Pauli held related but distinct views about how quantum mechanics should be understood.
- 4.
Spontaneous collapse approaches are invariably indeterministic (unless one counts among them also non-linear modifications of the Schrödinger equation, which however have not proved successful), as is Nelson’s stochastic mechanics. Everett’s theory vindicates indeterminism at the emergent level of worlds. And even many pilot-wave theories are stochastic, namely those modelled on Bell’s (1984) theory based on fermion number density. Thus, among the main fundamental approaches to quantum mechanics, only theories modelled on de Broglie and Bohm’s original pilot-wave theory provide fully deterministic underpinnings for it. (For a handy reference to philosophical issues in quantum mechanics, see Myrvold (2018b) and references therein. For more on Nelson see Footnote 40.)
- 5.
16 December is the traditional date for Beethoven’s birthday, but we know from documentary evidence only that he was baptised on 17 December 1770 – Beethoven was probably born on 16 December.
- 6.
With the qualifications mentioned in Footnote 2. Thanks to both Jos Uffink and Ronnie Hermens for making me include more about Jaynes.
- 7.
These (especially the latter) can be skipped by readers who are not particularly familiar with debates on the metaphysics of time or, respectively, on the nature of the quantum state and of collapse in quantum mechanics.
- 8.
While in a Cambridge HPS mood, I cannot refrain from thanking Jeremy Butterfield for (among so many other things!) introducing me to this wonderful paper and all issues around objective probabilities.
- 9.
Again, terminology varies somewhat, but I shall take ‘credences’ to be synonymous with ‘subjective probabilities’.
- 10.
Bayesian conditionalisation is often taken as the unique rational way in which one can change one’s credences. But there may be other considerations competing with it, in particular affecting background theoretical knowledge. This will generally be implicit (but not needed) in much of this paper. Thanks to Jossi Berkovitz for pointing out to me that de Finetti himself was not committed to conditionalisation as the only way for updating probabilities, believing instead that agents may have reasons for otherwise changing their minds (cf. Berkovitz 2012, p. 16).
- 11.
Two proposals of note are the one by Deutsch and by Wallace in the context of Everettian quantum mechanics, who derive the quantum mechanical Born rule from what they argue are rationality constraints on decision making in the context of Everett (see e.g. Wallace 2010, or Brown and Ben Porath (2020)), and the notion of ‘Humean objective chances’ along the lines of the neo-Humean ‘best systems’ approach (Loewer 2004; Frigg and Hoefer 2010). Note that I am explicitly distinguishing between Lewisian chances and Humean objective chances. The latter are an attempt to find something that will satisfy the definition of Lewisian chances, but there is no consensus as to whether it is a successful attempt. (See Sect. 3.10 below for some further comments on neo-Humeanism.) The Deutsch-Wallace approach is often claimed to provide the only known example of Lewisian chances, but again this claim is controversial, as it appears that some of the constraints that are used cannot be thought of as rationality assumptions but merely as naturalness assumptions. On this issue see Brown and Ben Porath (2020) as well as the further discussion in Saunders et al. (2010). The difficulty in finding anything that satisfies the definition of Lewisian chances is (of course) an argument in favour of subjectivism.
- 12.
I shall assume throughout that degrees of belief attach to propositions about matters of fact (in the actual world), which incidentally need not be restricted to Lewis’s ‘particular’ matters of fact (e.g. I shall count as matters of fact also holistic facts about quantum states – if such facts there be). I shall call these material or empirical propositions.
Of course, some of the background assumptions that we make in choosing our degrees of belief may not just refer to matters of fact, but will be theoretical propositions (say, referring to possible worlds). In that case, I would prefer not to talk about degrees of belief attaching to such theoretical propositions, but about degrees of acceptance.
The matter is further complicated by the fact that even full acceptance of a theory need not imply belief in all propositions about matters of fact following from the theory. For instance, an empiricist may believe only a subset of the propositions about actual matters of fact following from a theory they accept (say, only propositions about observable events), thus drawing the empirical-theoretical distinction differently and making a distinction between material propositions and empirical propositions, which for the purposes of this paper I shall ignore. (For my own very real empiricist leanings, see Bacciagaluppi 2019.)
For simplicity I shall also ignore the distinction between material propositions and actual propositions that one makes in a many-worlds theory, where the material world consists of several ‘possible’ branches, even though I shall occasionally refer to Everett for illustration of some point.
One might reflect the distinction between material (or empirical) and theoretical propositions by not conditionalising on the latter, but letting them parameterise our credence functions. Another way out of these complications may be to assign subjective probabilities to all propositions alike, but interpret the probabilities differently depending on what propositions they attach to: in general they will be degrees of acceptance, but in some cases (all material propositions, or a more restricted class of empirical propositions), one may think of them as straightforward degrees of belief. Such an additional layer of interpretation of subjective probabilities will, however, not affect the role they play in guiding our actions (degrees of acceptance will generally be degrees of ‘as-if-belief’). In any case, the stroke notation makes clear that assuming a theory – even hypothetically – is meant to affect our credences as if we were conditionalising on a proposition stating the theory (I believe this is needed to derive some of Lewis’s results).
Note that for Lewis, T will typically include theoretical propositions about chances themselves in the form of history-to-chance conditionals, and because in Lewis’s doctrine of Humean supervenience chances in fact supervene on matters of fact in the actual world, such background assumptions may well be propositions that degrees of belief can attach to (at least depending on how one interprets the conditionals), even if one wishes to restrict the latter (as I have just sketched) to propositions about matters of fact in the actual world.
We shall return to the material-theoretical distinction in Sect. 3.7.
- 13.
Note that X is a proposition about (actual) matters of fact. Thus, chances in the sense of Lewis indeed satisfy the intuition that objective probabilities are properties of objects or more generally determined by actual facts. Thanks to Jossi Berkovitz for discussion.
- 14.
For the classic Newtonian analysis of coin flipping see Diaconis et al. 2007.
- 15.
Or for any other purpose – as a matter of fact, I shall assume throughout a block universe picture. Specifically, in the case of indeterminism one should think of each possible world as a block universe. The actual world may be more real than the merely possible ones, or all possible worlds may be equally real (whether they are or not is inessential to the following, although a number of remarks below may make it clear that I am no modal realist, nor in fact believe in objective modality). But within each world, all events will be equally real. (Note that at a fundamental level the Everett theory is also deterministic and Everettians emphasise that the Everettian multiverse is indeed a block universe.)
- 16.
This is of course the quantum mechanical no-signalling theorem, as seen in the standard collapse picture. (It is instructive to think through what the theorem means in pilot-wave theory or in Everett. Thanks to Harvey Brown for enjoyable discussion of this point.) And of course, mixed quantum states are the classic example of the distinction in quantum mechanics between probabilities that are ignorance-interpretable (‘proper mixtures’) and probabilities that are not (‘improper mixtures’).
- 17.
In this example, it is of course possible to think of mixed epistemic and ontic probabilities as arising not only through ignorance of matters of fact in the ontic context H t, but also through ignorance of the correct theory of chance T C. We might indeed know the exact composition of the radioactive probe, but take ourselves to be ignorant of the correct decay laws for the various isotopes, with different credences about which laws are correct. I shall neglect this possibility in the following, because as noted already in Footnote 12, I prefer not to think of ignorance when talking about theoretical assumptions. In any case, I shall not need this further possible case for the arguments below.
- 18.
I am deeply grateful to Chris Fuchs for many lovely discussions about qBism and subjectivism over a number of years.
- 19.
Whether or not they are will make little difference to my argument (although if they should be purely subjective I would have further concrete examples of subjective ontic probabilities in the sense I need). In Bub’s case, he explicitly states that the no-cloning principle is an assumption and could turn out to be false. That means that the constraints hold only given some specific theoretical assumptions, and are not rationally compelling in the strong sense used by Lewis. In Pitowsky’s case, the constraints seem stronger, in the sense that the ‘quantum bets’ are not per se theoretical assumptions about the world, but are just betting situations whose logical structure is non-classical. Whether or not there are rational constraints on our probability assignments will be a well-posed question only after you and the bookie have agreed on the logical structure of the bet. On the other hand, whenever we want to evaluate credences in a particular situation, we need to make a theoretical judgement as to what the logic of that situation is. That means, I believe, that whether we can have rationally compelling reasons to set our credences in any particular situation will still depend on theoretical assumptions, and fall short of Lewis’s very stringent criterion. See Bacciagaluppi (2016) for further discussion of identifying events in the context of quantum probability, and Sect. 1.2 of Pitowsky (2003) for Itamar’s own view of this issue. Cf. also Brown and Ben Porath (2020).
- 20.
- 21.
The quantum state is thus taken to be a matter of fact, but presumably not a ‘particular matter of fact’ in the sense of Lewis, because of the holistic aspects of the quantum state. (Cf. Footnote 12 above.) Note also that historically the notion of quantum state preceded its ‘statistical interpretation’, famously introduced by Max Born (for discussion see Bacciagaluppi 2008).
- 22.
- 23.
Frigg and Hoefer (2007) have argued that the propensity concept is inadequate to the setting of GRW. Their own preference is to use ‘Humean objective chances’ instead (cf. Frigg and Hoefer 2010). Note that if the alternative subjectivist strategy includes taking the quantum state as a non-probabilistic physical object, there are severe constraints on doing this for the case of relativistic spontaneous collapse theories (see also below, Sects. 3.6 and 3.9).
- 24.
Indeed, we defined these probabilities as ontic, so unless we accept subjective ontic probabilities, we are committed to regarding these probabilities either as objective or (more generally!) as objectively wrong.
- 25.
Many thanks to Ronnie Hermens for noting that this point was not spelled out clearly enough in a previous draft.
- 26.
Note that also epistemic and ontic probabilities as discussed here are naturally seen as conditional (on epistemic and ontic contexts, respectively). That all probabilities should be fundamentally conditional has been championed by a number of authors (see e.g. Hájek 2011 and references therein). The idea in fact fits nicely with Lewis’s own emphasis that already the notion of possibility is indeed to be naturally understood as a notion of compossibility, of what is possible keeping fixed a certain context. (Recall his lovely example in Lewis (1976) of whether he can speak Finnish as opposed to an ape, or cannot as opposed to a native speaker.)
- 27.
And pace Lewis, who explicitly ruled out the possibility of such deterministic chances (Lewis (1986), p. 118). Note that high-level deterministic chances relate to all kinds of interesting questions (e.g. emergence, the trade-off between predictive accuracy and explanatory power, the demarcation between laws and initial conditions, etc.). For recent literature on the debate about deterministic chances, see e.g. Glynn (2009), Lyon (2011), Emery (2013), and references therein.
- 28.
I actually prefer the 2000 archived version to the published paper (Myrvold 2002). More recently, Myrvold (2017a) has shown that any such theories of foliation-dependent collapse must lead to infinite energy production. I believe, however, that Myrvold’s idea that probabilities in special relativity (and in quantum field theory) can be defined conditional on events to the past of an arbitrary hypersurface can still be implemented (see Sect. 3.9 below).
- 29.
I am indebted to both of these proposals and their authors in more ways than I can say. See also my Bacciagaluppi (2010a).
- 30.
Retrocausal theories (in particular in the context of quantum mechanics) have been vigorously championed by Huw Price. All-at-once theories are ones in which probabilities for events are specified given general boundary conditions, rather than just initial (or final) conditions, and have been championed in particular by Ken Wharton. See e.g. Price (1997), Wharton (2014) and the other references in Adlam (2018a). For further discussion see also Rijken (2018) (whom I thank for discussion of Adlam’s work).
- 31.
Of course also in the case of \(E=H_{t'}\), the chance given E is the coarse-graining over all possible intervening histories between t′ and t. But this particular coarse-graining does yield again a Lewisian chance, namely the chance at t′.
- 32.
And, indeed, already in Sect. 3.3 we suggested that ontic probabilities tend to be associated with indeterministic contexts, which we defined in terms of multiple possible futures in our theoretical models compatible with the present (or present and past) state of the world.
- 33.
As mentioned above, one could of course distinguish further between the actual world and the material world (if one is a many-world theorist) or take some material propositions to be theoretical (if one is an empiricist), but for simplicity I neglect these possibilities.
- 34.
Material probabilities by definition support only indicative conditionals: ‘If Beethoven was not born on 16th December, he was born on the 15th’ expresses our next best guess. Instead, when talking about atomic decay we more commonly use subjunctive or counterfactual conditionals, such as ‘Should this atom not decay in the next hour, it would have probability 0.5 of decaying during the following hour’.
- 35.
My thanks to Niels van Miltenburg and to Albert Visser for pressing me on this point.
- 36.
We shall refine this point in our discussion of time-symmetric theoretical models in Sect. 3.8.
- 37.
It may, however, contribute to muddle the waters in the Popper and Jaynes examples, or indeed in modern discussions about ‘typicality’ in statistical mechanics: are we talking about the actual world (of which there is only one), or about possible worlds (of which there are many, at least in our models)? It is also not always clear which aspects of a model are indeed law-like and which are contingent. We shall see a class of examples in Sect. 3.8, but a much-discussed one would be whether or not the ‘Past Hypothesis’ can be thought of as a law – which it is in the neo-Humean ‘best systems’ approach (cf. Loewer 2007); thanks to Sean Gryb for raising this issue. For further discussion of typicality and/or the past hypothesis, see e.g. Goldstein (2001), Frigg and Werndl (2012), Wallace (2011), Pitowsky (2012), and Lazarovici and Reichert (2015).
- 38.
In particular, I agree with Saunders (2002) that standard moves to save ‘relativistic becoming’ in fact undermine such metaphysically thick notions. Rather, I believe that the correct way of understanding the ‘openness’ of the future, the ‘flow’ of time and similar intuitions is via the logic of temporal perspectives as developed by Ismael (2017), which straightforwardly generalises to relativistic contexts, since it exactly captures the perspective of an IGUS (‘Information Gathering and Utilising System’) along a time-like worldline. For introducing me to the pleasures of time symmetry I would like to thank in particular Huw Price.
- 39.
- 40.
Of course, this strategy is analogous to that of modelling time-directed behaviour in the deterministic case using time-symmetric laws and special initial conditions. The same point was made independently and at the same time by (Uffink 2007, Section 7), specifically in the context of probabilistic modelling of classical statistical mechanics. A further explicit example of time-symmetric ontic probabilities is provided by Nelson’s (1966,1985) stochastic mechanics, where the quantum mechanical wavefunction and the Schrödinger equation are derived from an underlying time-symmetric diffusion process. See also Bacciagaluppi (2005) and references therein. The notorious gap in Nelson’s derivation pointed out by Wallstrom (1989) has now been convincingly filled through the modification of the theory recently proposed by Derakhshani (2017).
- 41.
In the case of Nelson’s mechanics, for instance, one has a systematic drift velocity that equals the familiar de Broglie-Bohm velocity (which in fact also changes sign under time reversal). If one subtracts the systematic component, transition probabilities are then exactly time-symmetric.
- 42.
More precisely, it will be uniquely determined within each ‘ergodic class’. (Note that ergodicity results are far easier to prove for stochastic processes than for deterministic dynamical systems. See again the references in Bacciagaluppi (2010b) for details.)
- 43.
Without going into details, these distinctions will be relevant to debates about the ‘neo-Boltzmannian’ notion of typicality (see the references in Footnote 37), and about the notion of ‘sub-quantum disequilibrium’ in theories like pilot-wave theory or Nelsonian mechanics (for the former see e.g. Valentini 1996, for the latter see Bacciagaluppi 2012).
- 44.
This section can be skipped without affecting the rest of the paper. The discussion is substantially influenced by Myrvold (2000), but should be taken as an elaboration rather than exposition of his views. See also Bacciagaluppi (2010a). For Myrvold’s own recent views on the subject, see Myrvold (2017b, 2019).
- 45.
The interaction picture is obtained from the Schrödinger picture by absorbing the free evolution into the operators. In other words, one encodes the free terms of the evolution in the evolution of the operators, and encodes the interaction in the evolution of the state.
- 46.
Another obvious exception is Nelson’s stochastic mechanics (mentioned in Footnote 40), but there it is not the quantum mechanical transition probabilities that are fundamental but the transition probabilities of the underlying diffusion process. While also a number of Bohmians (cf. Goldstein and Zanghì 2013) think of the wavefunction not as a material entity but as nomological, as a codification of the law-like motion of the particles, there is no mathematical derivation in pilot-wave theory of the wavefunction and Schrödinger equation from more fundamental entities comparable to that in stochastic mechanics, and the wavefunction can be thought of as derivative only if one takes a quite radical relationist view, like the one proposed by Esfeld (forthcoming).
- 47.
For a sketch of how I think this relates to the PBR theorem (Pusey et al. 2012), see below.
- 48.
The other main strategy is to consider collapse to be occurring along light cones (either along the future light cone or along the past light cone, as in the proposal by Hellwig and Kraus (1970)), and it arguably allows one to retain the picture of quantum states as physical objects (Bacciagaluppi 2010a). No such theory has been developed in detail, however.
- 49.
Rather related ideas are common in the decoherent histories or consistent histories literature. Griffiths (1984) does not use an initial and a final state, but an initial and a final projection of the same kind as the other projections in a history, so his consistent histories formalism is in fact a no-state quantum theory. Hartle (1998) suggests that the fundamental formula for the decoherence functional is the two-state one, but that we are well-shielded from the future boundary condition so that two-state quantum theory is predictively equivalent to standard one-state quantum theory (i.e. we can assume the maximally mixed state as future boundary condition). If in one-state quantum theory we assume that we are similarly shielded from the initial boundary condition, then by analogy one-state quantum theory is in fact predictively equivalent to no-state quantum theory (i.e. we can assume the maximally mixed state also as past boundary condition). Thus, it could just as well be that no-state quantum theory is fundamental. (Of course we need to include pre-selection on the known preparation history.)
- 50.
The idea behind Myrvold’s proof is extremely simple. Leaving technicalities aside, because of the Lorentz-invariance of the vacuum state, infinite energy production would arise if localised collapse operators did not leave the vacuum invariant on average. Thus, one should require that local collapse operators leave the vacuum invariant. But since a local operator commutes with local operators at spacelike separation, and since (by the Reeh-Schlieder theorem) applications of the latter can approximate any global state, it follows that in order to avoid infinite energy production collapse operators must leave every state invariant. Thus there is no collapse.
One way out that is already present in the literature (and which motivates Myrvold’s theorem) is to tie the collapse to some ‘non-standard’ fields (which commute with themselves at spacelike and timelike separation), as done by both Bedingham (2011) and Pearle (2015) himself. Another possibility, favoured by Myrvold (2018a), is to interpret the result as supporting the idea that collapse is tied to gravity: in the curvature-free case of Minkowski spacetime, the theorem shows there is no collapse. As mentioned, however, I believe that a no-state proposal will also circumvent the problem of infinite energy production, because collapse operators will automatically leave the (Lorentz-invariant) maximally mixed state invariant on average.
- 51.
I should briefly remark on how I think these suggestions relate to the work of Pusey et al. (2012), who famously draw an epistemic-ontic distinction for the quantum state, and who conclude from their PBR theorem that the quantum state is ontic. An ontic state for PBR is an independently existing object that determines the ontic probabilities (in my sense) for results of measurements. And an epistemic reading of the quantum state for PBR is a reading of it as an epistemic probability distribution (in my sense) over the ontic states of a system. Thus, the PBR theorem seems to push for a wavefunction ontology. In a no-state proposal (as here or in Bedingham and Maroney 2017), however, the probabilities for future (or past) collapse events are fully determined by the set of past (or future) collapse events. The contingent quantum state (in fact whether defined along hypersurfaces or along lightcones) can thus be identified with such a set of events, which is indeed ontic, thus removing the apparent contradiction. Whether this is the correct way of thinking about this issue of course deserves further scrutiny.
- 52.
- 53.
For an example of (Born and) Heisenberg using a very unusual choice of ‘collapsed states’ to calculate quantum probabilities, see (Bacciagaluppi and Valentini 2009, Sect. 6.1.2).
- 54.
To this position, I would wish to add a measure of empiricism (Bacciagaluppi 2019). For a similar view, see Brown and Ben Porath (2020). On these matters (and several others dealt with in this paper) I am earnestly indebted to many discussions with Jenann Ismael over the years. Any confusions are entirely my own, however.
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Acknowledgements
I am deeply grateful to Meir Hemmo and Orly Shenker for the invitation to contribute to this wonderful volume, as well as for their patience. I have given thanks along the way, both for matters of detail and for inspiration and stimuli stretching back many years, but I must add my gratitude to students at Aberdeen, Utrecht and the 5th Tübingen Summer School in the History and Philosophy of Science, where I used bits of this material for teaching, and to the audience at two Philosophy of Science seminars at Utrecht, where I presented earlier versions of this paper, in particular to Sean Gryb, Niels van Miltenburg, Wim Mol, Albert Visser and Nick Wiggershaus. Special thanks go to Ronnie Hermens for precious comments on the first written draft, further thanks to Harvey Brown, Alan Hájek and Jenann Ismael for comments and encouraging remarks on the final draft, and very special thanks to Jossi Berkovitz for a close reading of the final draft and many detailed suggestions and comments.
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Bacciagaluppi, G. (2020). Unscrambling Subjective and Epistemic Probabilities. In: Hemmo, M., Shenker, O. (eds) Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34316-3_3
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