New theory about old evidence

1.1 Example: food inspector raising a new hypothesis

Throughout the paper, especially the more technical Sect. 3, it may be helpful to keep in mind a simple example. For this purpose, we offer the following scenario (inspired by an example from Romeijn 2005).

A food safety inspector wants to determine whether or not a restaurant is taking the legally required precautions against food poisoning. She enters the restaurant anonymously and orders a number of dishes. She uses food testing strips to determine for each of the dishes whether or not it is infected by a particularly harmful strain of Salmonella. She assumes that these tests work perfectly, interpreting a positive test result as a Salmonella-infected dish and a negative result as an uninfected one. She also assumes that in kitchens that implement the precautionary practices each dish has a probability of 1% of being infected, whereas this probability rises to 20% in kitchens that do not implement the practices. She orders five dishes from the kitchen and they all turn out to be infected. This prompts her to consider a third hypothesis: the test strips may have been contaminated, rendering all test results positive, irrespective of whether the dish is infected or not.

After considering this third option, the inspector will not order any additional dishes. Instead, she will take the old evidence (that five dishes out of five appeared to be infected) to confirm the new theory (that the test strips were infected) and it seems reasonable enough for her to do so. Our challenge is how to represent this positive confirmation of the old evidence for the new theory within (an extension of) the Bayesian framework.

1.2 Old evidence and new theories

The confirmation-theoretic model of this paper sheds new light on the problem of old evidence and new theories. This problem for Bayesianism was first identified by Clark Glymour (1980). The problem arises from the discrepancy between descriptive, historical examples, in which old evidence does seem to lend positive confirmation to new theories, and the normative, Bayesian position, in which old evidence cannot confirm new theories. In particular, by updating via Bayes’ rule (used here to refer to Bayesian conditionalization), taking into account evidence that has already been conditioned upon cannot change the probabilities. And since all expressions of confirmation hinge on differences in probabilities, it seems that old evidence cannot lead to confirmation of new theories. Many later authors have called Glymour’s problem simply “the problem of old evidence”.¹ A minority of philosophers has stressed the importance of the other side of the problem: “the problem of new theories” (for example Earman 1992). In what follows, we will clarify that both problems can be resolved in open-minded Bayesianism.

New theories pose a bigger problem for Bayesianism than usually recognized.² In fact, without a way of introducing a new theory into the domain of an agent’s degrees of belief, its prior and posterior degrees of belief simply do not show up in the model. In effect, as we will explain in Sect. 3.3.2, those probabilities are set to zero. Either way, for want of a way to express non-zero probability assignments to a new theory, the problem of old evidence does not even occur—or it is worse than the usual presentations suggest. Therefore, we analyze the problem of new theories first and offer a conservative extension of Bayesianism to deal with this problem: a framework for open-minded Bayesianism. In the course of doing so, it will become clear what is missing to deal adequately with old evidence and to determine the confirmation it may give to a new theory. In particular, our model is compatible with Glymour’s observation that in important historical examples old evidence does offer positive confirmation to new theories.

Some proposals for addressing the problem of old evidence (in particular that of Garber 1983) observe that the crucial content that is being learned and that lends positive confirmation to a new theory, is not the old evidence itself, but rather the fact that this new theory implies or explains the old evidence. Recently, Sprenger (2014) has proposed a new solution along these lines. We are sympathetic to this approach.³ However, Sprenger’s results presuppose that the old evidence, the new theory, and the relevant relation between the two are all elements of some algebra (see his Theorems 1 and 2). As such, this approach does not address a more fundamental question: how can a new theory (or a new relation between a theory and a piece of evidence) be incorporated in the algebra? This is the problem of new theories, which is especially pressing in the presence of old evidence, that we tackle here.

1.3 Bayesian confirmation theory

Since the problem of old evidence and new theories is ultimately a problem concerning Bayesian confirmation, we should first be clear on how we intend to measure confirmation of a hypothesis by a body of evidence. This is in itself an interesting problem in formal epistemology, and some reactions to the problem of old evidence are in fact proposals for a new measure of confirmation (e.g., Christensen 1999; Joyce 1999). In qualitative terms, a piece of evidence \(E\) lends positive confirmation to a theory \(T\) if the posterior \(P(T \mid E)\) exceeds the prior \(P(T)\). To turn this into a quantitative notion, different measures of confirmation have been proposed: for instance, the difference or (the log of) the ratio of posterior and prior.

However, our current investigation focuses on how to deal with new theories, which is a problem that besets Bayesianism more broadly, and quite independently of the chosen confirmation measure. Therefore, we will not opt for any such measure, and focus our attention on what they supervene on: the probability assignment over hypotheses themselves. Nothing in our exposition hinges on the precise measure of confirmation that may be grafted onto the probabilistic models.

1.4 The catch-all hypothesis

Our proposal of open-minded Bayesianism relies on the use of a catch-all hypothesis: given a set of explicit hypotheses, we introduce an additional hypothesis that is the negation of the union of the previous hypotheses. Facing the possibility of currently unexplored theoretical alternatives is relevant, not only for the formal framework of Bayesian confirmation theory, but also for the philosophy of science more generally. See for instance the discussion on the pessimistic meta-induction by Sklar (1981), who speaks of “unborn hypotheses”, and by Stanford (2006), who uses the term “unconceived alternatives”. In statistical parlance, the catch-all hypothesis makes good on Lindley’s demand for observing Cromwell’s rule (Lindley 1991 p. 104), which states that prior probabilities of zero or one should only be assigned to logical truths or falsehoods (cf. strict coherence and regularity; see, e.g., Hájek 2012).

We aim to develop a particular way of observing Cromwell’s rule, which can be found already in Shimony (1970). He discussed the idea of a catch-all hypothesis in the context of his “tempered personalist” account of probability: he suggested it as a way to represent open-mindedness, which he regarded as a tempering condition to obtain a weakened form of Bayesianism adequate for scientific inference. (Shimony (1970), p. 96) suggested not to assign numerical weights (priors) to the catch-all (in contrast to the other hypotheses).

Also Earman (1992) discussed the use of a catch-all to make room for later theory change. According to (Earman (1992), p. 196), new theories are “shaven off” from the catch-all hypothesis, which thus “serves as a well for initial probabilities for as yet unborn theories, and the actual introduction of new theories results only in drawing upon this well without disturbing the probabilities of previously formulated theories.” However, he is not satisfied by the proposal of shaving off from a catch-all; according to (Earman (1992), pp. 195–196) it leads to the assignment of successively smaller probabilities to later theories (cf. Romeijn 2004), and shaving off does not give an adequate description of scientific revolutions (in the Kuhnian sense) that involve radically new theories. These reservations do not apply to the way in which we formalize the notion of a catch-all, as we will explain in Sect. 4.

1.5 Assigning open-minded probabilities

When Bayesian ideas are applied within the sciences, the domain of the probability function tends to have a small scope: it is used to compare parametric models that apply to a single, well-delineated target system. In philosophy, however, we often speak as if the domain of the probability function captures every thinkable thought. In particular, in philosophy of science and Bayesian confirmation theory, the probability function assigns values to scientific theories.

If all later changes to the probability assignment are to be due to conditioning, as standard forms of Bayesianism prescribe, we have to be able to specify the domain in such a way as to include all possible scientific theories, including those that are yet to be developed. Nevertheless, it may happen that genuinely new scientific theories do emerge. It is unclear how those can be incorporated in a domain that has to be defined upfront.

In Sect. 2, we will make explicit what the domain of the probability function is on the standard account. Since probability functions assign values to scientific theories as well as to particular pieces of evidence, we have to define a domain that can represent all these objects, even though they are of very different kinds. Specifying this domain provides us with a good opportunity to formalize the notion of the catch-all hypothesis, and how it is used to change the domain of the probability function.

In Sect. 3, we will introduce two forms of open-minded Bayesianism, called vocal and silent, which both employ a catch-all hypothesis. Both are based on the idea that we can remain open-minded about our probabilities by employing sets of probability functions rather than single functions. But both approach the (incomplete) assignment of probabilities in slightly different ways. Also the rule for updating on new theories takes a different form in both contexts.

In Sect. 4, we evaluate the proposals and offer a hybrid approach that alternates between silent and vocal episodes.

2.1 Domain of the probability function

Dynamics: time stamps In Bayesian confirmation theory, we model the rational degrees of belief of an agent by a probability function. To capture the dynamics of the agent’s degrees of belief, we consider a succession of probability functions, indexed by a time stamp: \(P_t\) is the probability function that represents the rational degrees of belief of the agent at time \(t\). In standard Bayesianism, these belief states are linked by Bayes’ rule, as detailed below.

2.2 Evidence and updating

Before we can start applying probability theory, we have to fix a particular sample space (or set of atomic events), \(\varOmega \), which is chosen such that any result of a measurement can be represented as a subset of \(\varOmega \). The sample space can be a Cartesian product of sets, which allows us to represent very different types of empirical data.⁶ We represent (actual and hypothetical) pieces of evidence as elements of an algebra on the sample space, \(\mathcal {A}(\varOmega )\). This set is usually called the event space, but in the current context it is better to call it the ‘evidence space’.

Dynamics: Bayes’ rule If an agent receives evidence \(E\) at \(t=n\), then Bayes’ rule prescribes that the agent has to adopt a new probability function \(P_{t=n}\) that is equal to the posterior of the agent’s immediately preceding probability function: \(P_{t=n}(\cdot )=P_{t=n-1}(\cdot \mid E)\), which can be computed via Bayes’ theorem.

2.3 Explicit hypotheses and the catch-all

In the Bayesian framework, probability functions range over evidence and hypotheses. Hence, in addition to specifying \(\varOmega \) and \(\mathcal {A}(\varOmega )\), we need to define a set of hypotheses, \(\mathcal {H}\), and an algebra over this set, \(\mathcal {A}(\mathcal {H})\). The hypotheses are only specified up to their empirical content. The scientific theories that motivate them are not brought into view. The way to characterize an empirical hypothesis, \(H\), is by specifying a likelihood function \(P( \cdot \mid H)\) ranging over the evidence space, \(\mathcal {A}(\varOmega )\). Because the empirical content of hypotheses is spelled out in terms of probability functions over the data, the hypotheses are called statistical.⁷

Under a hypothesis we may also subsume an entire family (i.e., a set) of likelihood functions, which have the same form except for a different value of a parameter (or vector of parameters).⁸ Henceforth, we will treat all hypotheses as sets of probability functions on the domain \(\mathcal {A}(\varOmega )\). Hypotheses that correspond with singleton sets will be called elementary hypotheses, others will be called composite. Observe that the hypotheses in \(\mathcal {H}\) need not be elementary in this sense.

Like the elementary events in \(\varOmega \), the hypotheses in \(\mathcal {H}\) need to be mutually exclusive and jointly exhaustive. However, merely exhausting the union of the hypotheses in \(\mathcal {H}\), which is the set of hypotheses that are being considered at a given point in time, may not suffice. In particular, it does not suffice once a new hypothesis emerges, because in that case we want to involve a hypothesis outside \(\bigcup _{H \in \mathcal {H}}H\). As indicated before, if we do not offer a domain in which possibilities outside \(\mathcal {H}\) can be denoted, we cannot begin to formulate the problem of old evidence and new theories.

Let us consider a collection of \(N+1\) hypotheses (with \(N\) a positive integer) that are mutually exclusive and jointly exhaustive: this partition of \(\varTheta \) contains \(N\) explicitly formulated hypotheses, \(H_0,\ldots ,H_{N-1}\), and one catch-all, \(\overline{\varTheta _N}\). By an ‘explicitly formulated’ hypothesis, \(H_i\), we mean an empirical hypothesis that is produced by a scientific theory. We do not discuss in detail the scientific theories themselves, or even how they lead to statistical hypotheses.¹⁰

Dynamics: shaving off In the previous subsection, we have seen that the incorporation of evidence leads to an update of the probability function governed by Bayes’ rule. Standard Bayesianism lacks an analogous procedure for revising the probability function in light of a new hypothesis. We will now discuss how the presence of the catch-all allows us to represent the dynamics of the set of hypotheses. This prepares us for the proposal of open-minded Bayesianism in the next section.

After a new scientific theory has been developed, the statistical hypothesis it produces may be added to the partition of \(\varTheta \) by “shaving off” from the catch-all (by the terminology of Earman 1992, p. 196) . At this point in time, the former catch-all may be decomposed into an additional explicitly formulated hypothesis \(H_N\) (disjoint from the earlier hypotheses) and a new (smaller) catch-all, \(\overline{\varTheta _{N+1}}\). So, the algebra on \(\varTheta \times \varOmega \) changes.¹¹

2.4 Summary of key ideas

We briefly recapitulate our approach so far and our use of the following terms: scientific theory, statistical hypothesis, sample space, evidence, and catch-all.

A scientific theory together with background assumptions produces an empirical, or statistical, hypothesis. (How this happens requires engaging with the details of a scientific theory, which falls outside the scope of our current framework.) Such an empirical or statistical hypothesis is a set, possibly a singleton, of probability functions. In order to compare hypotheses produced by different theories in the light of a common body of empirical data (and thus to compare their measures of confirmation or evidential support), their probability functions need to have a common domain. This domain is called the evidence space: it is an algebra on a sample space (which may be a Cartesian product set to allow for the representation of mutually independent measurable quantities).

The union of all statistical hypotheses produced by the currently available scientific theories \((\varTheta _N)\) does not exhaust the set of all probability functions on the evidence space \((\varTheta )\).¹² The complement of the former set relative to the latter set is called the catch-all hypothesis \((\overline{\varTheta _N})\): unlike the other hypotheses, it is not produced by a scientific theory, but rather it results from a meta-theory. The catch-all hypothesis is included to express that many other hypotheses are conceivable, each associated with a probability assignment or a set of such assignments over the evidence.

With the idea of a catch-all hypothesis in place, we can now turn to a full specification of open-minded Bayesianism. The inclusion of a catch-all hypothesis makes room for modeling the introduction of new hypotheses, namely by shaving them off from the catch-all. But this in itself is not sufficient: we still need to specify how shaving off influences probability assignments over the hypotheses. This is the task undertaken in the next section.

3.1 Vocal and silent open-mindedness

In open-minded Bayesianism, hypotheses are represented as sets of probability functions. If prior probabilities are assigned to the functions within a set, then a single marginal probability function can be associated with the set. But without such a prior probability assignment within the set, the set specifies so-called imprecise probabilities (see, for instance, Walley 2000).

We first clarify probability assignments over explicitly formulated hypotheses. In standard Bayesianism, prior probabilities are assigned to the hypotheses, which are all explicitly formulated. We can furthermore assign priors over the individual probability functions contained within composite hypotheses, if there are any. We call such priors within a composite hypothesis sub-priors. The use of sub-priors leads to a marginal likelihood function for the composite hypothesis.¹⁴ Upon the receipt of evidence we can update all these priors, i.e., those over elementary and composite hypotheses as well as those within composite hypotheses.

Now recall that in open-minded Bayesianism, the space of hypotheses also contains a catch-all, which is a composite hypothesis encompassing all statistical hypotheses that are not explicitly specified. In standard Bayesianism, this catch-all hypothesis is usually not mentioned, and all probability mass is concentrated on the hypotheses that are formulated explicitly. Within the framework of open-minded Bayesianism, we will represent this standard form of Bayesianism by setting the prior of the catch-all hypothesis to zero.¹⁵

3.2 A conservative extension of standard Bayesianism

As detailed in the foregoing, we aim to represent probability assignments of an agent that change over time. An agent’s probability function therefore receives a time stamp \(t\). Informally, this is often presented as if the probability function changes over time, but it is more accurate to say that the entire probability function gets replaced by a different probability function at certain points in time. Accordingly, subsequent functions need not even have the same domain.

Standard Bayesianism has one way to replace an agent’s probability function once the agent learns a new piece of evidence: Bayes’ rule. It amounts to restricting the algebra to those sets that intersect with the evidence just obtained. Equivalently, it amounts to setting all the probability assignments outside this domain to zero. If at time \(t\) an agent learns evidence \(E\) with certainty, Bayes’ rule amounts to setting \(P_{t=n}\) equal to \(P_{t=n-1}(\cdot \mid E)\). If \(E\) is the first piece of evidence that the agent learns, this amounts to restricting the domain from an algebra on \(\varTheta \times \varOmega \) to an algebra on \(\varTheta \times E\) and redistributing the probability over the remaining parts of the algebra according to Bayes’ theorem.

In addition to this, open-minded Bayesianism requires a rule for replacing an agent’s probability function once the agent learns information of a different kind: the introduction of a new hypothesis. This amounts to expanding the algebra to which explicit probability values are assigned (from an algebra on \(\varTheta _N \times E\) to an algebra on \(\varTheta _{N+1} \times E\)). Or in other words, it amounts to refining the algebra on \(\varTheta \times E\). On both views, the new algebra is larger (i.e., it contains more sets). What is still missing from our framework is a principle for determining the probability over the larger algebra. In analogy with Bayes’ rule, one natural conservativity constraint is that the new probability distribution must respect the old distribution on the preexisting parts of the algebra.

Viewed in this way, our proposal does not introduce any radical departure from standard Bayesianism. Open-minded Bayesianism respects Bayes’ rule, but this rule already concerns changes in the algebra, namely reductions. The only new part is that we require a separate rule for enlarging the algebra (extending \(\varTheta _N\) or refining the partition of \(\varTheta \)) rather than for reducing it (restricting \(\varOmega \)). The principle that governs this change of the algebra again satisfies conservativity constraints akin to Bayes’ rule. As detailed below, silent and vocal open-minded Bayesianism will give a slightly different rendering of this rule.

3.3 Updating due to a new hypothesis¹⁷

In this section, we consider how the probability function ought to change upon the introduction of a new hypothesis after some evidence has been gathered. We first consider an abstract formulation of a reduction and extension of the domain, as well as an example of such an episode in the life of an epistemic agent. After that, we consider both versions of open-minded Bayesianism as developments of the standard Bayesian account.

4.1 Evaluation of open-minded Bayesianism

It may be argued that open-minded Bayesianism fails to provide us with the required normative guidance. In the silent version, it only concerns suppositional reasoning and hence cannot inform our unconditional beliefs. In metaphorical terms, the worry is that the agent cannot keep hiding behind the conditionalization stroke. In the vocal form, the same worry arises in relation to the use of factors with indefinite numerical values, which are introduced to represent the prior and likelihood of the catch-all hypothesis, but which soon ‘infect’ all probability assignments and measures of confirmation. Either way, it may seem that the agent must come clean on her absolute commitments at some point.

We respond to this worry by biting the bullet. If we want to allow new theories to enter the conceptual scene, then we will have to provide room for this in our formal framework. There are attempts to accommodate (other forms of) theory change in a Bayesian framework that employ fully specified probability assignments (e.g., Romeijn 2004, 2005). In this paper, by contrast, we have offered a framework that creates room for new theories by leaving part of the probability assignment unspecified. We accept that this leads to a model that only concerns conditional belief.

We should emphasize that we are not alone in preaching an open-minded form of Bayesianism. We already mentioned the proposal for tempered Bayesianism by Shimony (1970), who suggested the use of a catch-all hypothesis in this context. This suggestion was also discussed by Earman (1992), who introduced the evocative terminology of shaving off new hypotheses from the catch-all. Furthermore, our proposal of humble Bayesianism is related to earlier work by Salmon (1990) and Lindley (1991). Morey et al. (2013) recently introduced what they call humble Bayesianism in a debate over the nature of statistical model comparisons.

The latter paper lends further support to open-minded Bayesianism. The point of Morey et al. (2013) is that an agent may want to use Bayesian methods to evaluate statistical models, without buying into the conviction, implicit in the standard Bayesian framework, that one of the theories under consideration is true. After all, a standard Bayesian will have the probabilities of the hypotheses under consideration add up to one, and so judges herself to be perfectly calibrated (cf. Dawid 1982). The standard Bayesian is overly confident, hence a more open-minded form of Bayesianism seems called for.

The price to pay is that the epistemic attitudes for which the framework of the open-minded Bayesian provides the norms are different: they have a conditional nature. Whether we spell out the details using a vocal or a silent open-mindedness, the normative framework tells the agent what to believe only if she temporarily supposes, without committing to it, that the true theory is among those currently under consideration.

4.2 The old evidence problem

Now that we have bitten the bullet, we better make sure that we do so for good reasons. In this section, we argue that open-minded Bayesianism provides a new handle on the problem of old evidence, by explaining how old evidence can be re-used.

In his encyclopedia entry on Bayesianism, Talbott (2008) summarizes the Bayesian problem of new theories as follows: “Suppose that there is one theory \(H_1\) that is generally regarded as highly confirmed by the available evidence \(E\). It is possible that simply the introduction of an alternative theory \(H_2\) can lead to an erosion of \(H_1\)’s support. [...] This sort of change cannot be explained by conditionalization.” It is precisely this “erosion” of support that can be captured by the update rule for open-minded Bayesianism, since both approaches make the agent reconsider the posteriors of the old hypotheses. The strong point of open-minded Bayesianism is that this reconsideration of the posteriors does not render the agent probabilistically incoherent.

When writing about the operation of shaving off new hypotheses, (Earman (1992), p. 195) worried that a point may be reached “where the new theory has such a low initial probability as to stand not much of a fighting chance.” This worry, however, does not apply to our framework. Notice that we do not assign an explicit value to the prior of the current theoretical context. We may think of the prior associated with the catch-all hypothesis as a number extremely close to unity—and the humbler we are, the closer to unity we can imagine it to be. At any rate, no matter how large the discrepancy between the posteriors of the old hypotheses and the new one, the impact that the decomposition of the catch-all has on the catch-all’s posterior will remain unknown, or indefinite. Of course, once we pin down a value for the probability \(\tau _N\), the worry of Earman becomes a live one. But lacking such a definite value,²⁴ the problem that the catch-all gets crowded out by explicit hypotheses does not arise.

There are, however, differences in how the vocal and silent approaches to open-minded Bayesianism deal with reassessing the posteriors, and in what role they give to old evidence. The vocal approach requires us to assign a prior to the new hypothesis \(H_N\) after the fact, and to compute its current posterior on the basis of this assignment. The other posteriors are obtained via a renormalization.²⁵ This approach requires us to evaluate probabilities retroactively: priors have to be set post hoc, for hypotheses that were not known at the time.²⁶ To our mind this need not be a cause of concern though. One cannot unlearn the evidence that has been gathered, but it is still possible to use base rates or other sources of objective information to determine the priors retroactively.²⁷

The silent approach, by contrast, requires us to assign a posterior to the new hypothesis \(H_N\) without offering an explicit recourse to the prior probability assignments over the old hypotheses. The point here is rather subtle. It is in virtue of a prior probability assignment of \(\tau _N\) to the old catch-all \(\varTheta _N\) that we can meaningfully claim, as part of the vocal approach, that the prior of the old catch-all is decomposed into the prior of a new hypothesis \(H_N\) and the prior of a new catch-all \(\varTheta _{N+1}\). Since the silent approach remains silent precisely on this prior, it is hard to see how we can retroactively decompose it. So in this approach, it is not clear whether old evidence ever confirms new theories. Unless we have set the value of \(P_{t=2}(H_N \mid \varTheta _{N+1})\) by means of a reconstruction that ultimately depends on \(P_{t=0}(H_N \mid \varTheta _{N+1})\), its value is not obtained via conditionalization on \(E\). In silent Bayesianism, the old evidence is therefore not given a new role.

Now that we have discussed the role of evidence in two forms of open-minded Bayesianism, it is time to take stock. Both approaches suffer some drawbacks. The vocal proposal comes with the complication of a heavy notational load that hampers the evaluation of the degree of confirmation. The silent proposal allows too much freedom in the assignment of a posterior to the new hypothesis—so much freedom, that it is not clear that the old evidence has any impact. For these reasons, we propose a hybrid approach to open-minded Bayesianism, that combines the best elements of both.

On our hybrid proposal, the open-minded Bayesian remains in the silent phase,²⁸ except for the times at which her theoretical context changes. Unlike a standard Bayesian, the open-minded Bayesian is allowed to change the algebra to which probabilities are assigned and thus to assign non-zero probabilities to the new hypothesis, which is impossible without a catch-all. Then she enters the vocal phase: she engages in assigning a prior to the new hypothesis (retroactively) and computing its posterior given the evidence (also retroactively) and renormalizing the other priors.

Open-minded Bayesianism thus offers a particular perspective on the use of old evidence for confirming a new theory. On the conceptual level, it shows how our perception of evidence and confirmation changes if we move from one theoretical context to another. Relative to one set of hypotheses, the data were telling towards one particular candidate hypothesis, and so counted as evidence that confirms this candidate. But with the inclusion of a new hypothesis, the data may tell against the formerly best candidate, and so count as evidence that disconfirms it. We take it to be a virtue of our model that it brings out this context-sensitivity of evidence and confirmation.

4.3 Illustration of the hybrid approach

To make our proposal for a hybrid approach more vivid, we apply it to the food inspection example. Initially, when the food inspector implicitly assumes her equipment to be working properly, she can be described by the silent approach to open-minded Bayesianism. Within the initial context, she only needs to take into account two explicit hypotheses: the kitchen is clean or it is not. She assigns prior probabilities to these hypotheses and she computes posteriors, but these assignments are conditional on her implicit assumption that the testing strips are uncontaminated (as well as the many other background assumptions collected in the theoretical context). So far, she acts much like any Bayesian would; her open-mindedness will surface only when provoked.

The result, that five dishes out of five appear to be infected, was initially unlikely on both of her explicit hypotheses. (Recall that the initial likelihood was \(10^{-10}\) in the case of a clean kitchen and \(3.2 \times 10^{-4}\) in the case of an unclean kitchen.) Computing the posterior probabilities, which implicitly requires us to assume that the correct hypothesis is among the two hypotheses being considered, leads to a value close to zero \((3.1 \times 10^{-7})\) for a clean kitchen and a value near to unity \((1 - 3.1 \times 10^{-7})\) for an unclean kitchen. If the priors were equal (or at least of the same order of magnitude), then on any measure of confirmation, the evidence provides very strong confirmation for the hypothesis that the kitchen was unclean.

The observation that it is highly unlikely even for an unclean kitchen to produce five infected dishes may suggest that there is an even better hypothesis ‘out there’ that has not yet been taken into account. Indeed, seeing the result prompts the inspector to reconsider one of her implicit assumptions and she turns its negation into a new theory (and associated statistical hypothesis): the testing strips may not have been clean after all (\(\hbox {bias} = 1\)).²⁹ (Of course, this is still but one out of many other alternative hypotheses.) Our framework for open-minded Bayesianism is able to represent this formally.

In the vocal phase, the agent shaves off her third hypothesis from the catch-all and revises her probability assignments: she retroactively assigns a prior to the new hypothesis, adjusts the priors of the two old hypotheses by a suitable factor, and computes the likelihood of the old evidence on the new hypothesis (as described in Sect. 3.3.3). All this leads her to reassess the posteriors of the old hypotheses and to assign a posterior to the new hypothesis. Assuming equal priors, the final result is this: within the new theoretical context, the posterior of the new hypothesis given the old evidence is more than three thousand times higher than that of the hypothesis that was best confirmed within the old theoretical context. Irrespective of the details of the confirmation measure and assuming priors of at least equal orders of magnitude, this implies that the old evidence strongly confirms the new hypothesis and disconfirms the others. This illustrates that it is the shift in theoretical context itself that may cause old evidence to confirm a new hypothesis.

Once the agent is satisfied that, for the evidence currently at hand, the new theoretical context includes all the relevant hypotheses, she may start to conditionalize all her findings on this context and thereby enter a new silent phase. The remaining catch-all hypothesis need not be mentioned again until new doubts arise.

In Kuhnian terminology, the silent version of open-minded Bayesianism is sufficient for describing episodes of normal science (and if the conditionalization on the theoretical context remains implicit, it is indistinguishable from the usual Bayesian picture), but the vocal version of open-minded Bayesianism is required to model revolutionary changes in the theoretical context.

4.4 Further research

With the foregoing, we believe we have only scratched the surface of the matter at hand. Many avenues for further research lay open for exploration. In what follows, we briefly mention a number of these avenues. With this we showcase our ongoing research on this, we invite the reader to join in, but mostly we indicate where we ourselves feel that our account is lacking.

One important consideration that has received relatively little attention in the foregoing concerns degrees of confirmation. Our goal with this paper was to show that we can accommodate the introduction of a new theory and hence a new empirical hypothesis in the Bayesian framework, and that old evidence can play a role in the determination of the posterior probability of this new hypothesis without violating probabilistic coherence. We have been mostly silent on how the posteriors may be used to compute a degree of confirmation, so that the impact of old evidence can be expressed more precisely: any such story will supervene on the probability assignments. However, a complete account of open-minded Bayesianism might involve more detail on degrees of confirmation.

Another aspect of the process of theory change targeted in this article certainly deserves a more detailed normative treatment: the decision to introduce a new theory. In the foregoing, we have treated this decision as completely external to the model. However, we also indicated that the search for new theories may be motivated by a so-called statistical model selection criterion, e.g., by a measure of the predictive performance of the agent, or by some other score that attaches to the data and the hypotheses currently under consideration. We think that our account, which may provide rationality constraints on the transition from one theoretical context to another, can be combined fruitfully with an account of how theoretical contexts are evaluated and selected.

Furthermore, we should stress that we have only considered one type of theory change—a change that may be captured by shaving off new hypotheses from a catch-all hypothesis.³⁰ In general, theory change may lead to other types of change to the domain of the probability function, \(\mathcal {A}(\varTheta \times \varOmega )\), in various ways. For one, we have not explicitly considered changes in the space \(\varOmega \) of empirical possibilities. Notice that such changes are generally more radical than changes in the theoretical realm: theories obtain their empirical content in terms of hypotheses that are formulated by means of \(\varOmega \). One captivating question concerns the exact reach of our account of new theory and old evidence. Specifically, can we assume at the outset that \(\varTheta \) and \(\varOmega \) are rich enough to accommodate all conceivable theory changes? An answer to this question requires us to survey a rich landscape of theory changes as moves in an encompassing space of possible theories.³¹

We would like to mention one other aspect to theory change that is related to two issues discussed above, namely the decision to introduce a new theory and the type of theory change effected by that. It concerns the notion of awareness. Hill (2010) and Dietrich and List (2013) have argued that a decision problem obtains new dimensions when the agent is made aware of considerations that were previously not live to her. We think that roughly the same can be said about the epistemic problems an agent faces, and that the foregoing offers a natural model for an agent that becomes aware of a theory while performing a predictive, or more generally an epistemic task. It seems natural to combine the frameworks for modeling awareness.

Finally, we briefly mention two possibilities that open-minded Bayesianism offers, when it is combined with ideas on relative infinitesimals (in the sense of Wenmackers 2013). On one side of the spectrum, the framework allows us to model radically skeptical yet empiricist epistemic attitudes: all the priors and posteriors of explicit hypotheses, old and new ones, may be very small, indeed infinitesimally small, compared to the probabilities associated with the catch-all. That is, we may choose \(\tau _N\) to be some number very close to one. Despite that, a particular theory may have a large prior or posterior relative to the other theories in the theoretical context. The framework thus allows us to model a radical sceptic who is nevertheless sensitive to differences in empirical support. On the other side of the spectrum, the framework of open-minded Bayesianism allows us to model practical certainty without spilling over into dogmatism. We may be aware of the existence of certain hypotheses, but we might choose not to include them in our considerations: they may seem irrelevant to the kinds of evidence under study (assuming statistical independence), they are deemed highly unlikely,³² including them requires too high a number of computations, or other pragmatic reasons. However, upon receiving falsifying or strongly disconfirming evidence, we might want to reconsider some of these omissions.³³ The catch-all hypothesis with an infinitesimal prior may then serve as a reservoir for the hypotheses that seemed dispensable at one point in time, but that later on turn out to be relevant. Falsifying or strongly disconfirming evidence may lead to a situation in which the probability of the catch-all is no longer regarded as a relative infinitesimal: the marginal likelihood becomes so small that it becomes comparable to the probability of the catch-all.³⁴

The above list of research topics indicates that our resolution to the problem of old evidence and new theories leaves much to be done. However, the list also suggests that the framework of open-minded Bayesianism provides access to several interesting aspects of belief dynamics that fall outside the scope of standard Bayesianism. We call to mind what Sue says in (Earman (1992), p. 235): “By all means keep an open mind, but not so open that your brain falls out.” It seems to us that open-minded Bayesianism does precisely that.