In the previous section, we have introduced the formal framework of open-minded Bayesianism. It is a form of Bayesianism that requires the set of hypotheses to include a catch-all hypothesis. In the current section, we develop the probability kinematics for open-minded Bayesianism. Two versions will be considered: vocal and silent. The two approaches suggest slightly different rules for revising probability functions upon theory change.Footnote 13
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.Footnote 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.Footnote 15
Let us turn to open-minded Bayesianism itself. To express that we are prepared to revise our theoretical background, we assign a strictly positive prior to the catch-all. However, we agree with Shimony (1970) that it is not sensible to assign any definite value to the prior of the catch-all. Since the catch-all is not based on a scientific theory, the usual “arational” considerations (to employ the terminology of Earman 1992, p. 197) for assigning it a prior, namely by comparing it to hypotheses produced by other theories, do not come into play here. Moreover, it seems clear that the catch-all should give rise to imprecise marginal likelihoods as well, which suggests that we should refrain from assigning sub-priors to its constituents, too. (Recall that the algebra on \(\varTheta \times \varOmega \) cannot pick out any strict subset of the catch-all.) These considerations lead us to consider two closely connected forms of open-minded Bayesianism, which both avoid assigning a definite prior to the catch-all:
-
Vocal open-minded Bayesianism assigns an indefinite prior and likelihood to the catch-all hypothesis, \(\overline{\varTheta _N}\). We represent its prior by \(\tau _N \in ]0,1[\) and its likelihood by \(x_N(\cdot \mid E)\). To ensure normalization over all hypotheses (including the catch-all), the priors assigned to the explicitly formulated hypotheses are set equal to the value they would have in a model without a catch-all now multiplied by \((1-\tau _N)\).
-
Silent open-minded Bayesianism assigns no prior or likelihood to the catch-all hypothesis, not even symbolically. To achieve this, all probabilistic statements are conditionalized on the algebra on \(\varTheta _N\) (shorthand for \(\varTheta _N \times \varOmega \)). \(\varTheta _N\) represents the union of the hypotheses in the current theoretical context. From the viewpoint of the algebra on \(\varTheta \times \varOmega \), the probability assignments are incomplete.
In both cases, we deviate from the standard Bayesian account in that we give a strictly positive prior to the catch-all, and then allow opinions to be partially unspecified: vocal open-minded Bayesianism retains the entire algebra but uses symbols without numerical evaluation as placeholders, whereas silent open-minded Bayesianism restricts the algebra to which probabilities are assigned (leaving out the catch-all).Footnote 16 Formally, the partial specification of a probability function comes down to specifying the epistemic state of the agent by means of a set of probability assignments (cf. Halpern 2003; Haenni et al. 2003).
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.
Updating due to a new hypothesis
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.
Reducing and enlarging: setting the stage
The epistemic episode that we aim to model has three stages:
\((t=0) N\)
explicit hypotheses At time \(t=0\), the theoretical context of the agent consists of \(N\) explicit hypotheses: \(T_N = \{ H_0,\ldots ,H_{N-1} \}\). The union of the hypotheses in \(T_N\) is \(\varTheta _N\). The catch-all is the complement of the latter (within \(\varTheta \)): \(\overline{\varTheta _N}\).
\((t=1)\)
Evidence
\(E\) At time \(t=1\), the agent receives evidence \(E\). The initial likelihood of obtaining this evidence given any one of the hypotheses \(H_i\) (\(i \in \{0,\ldots ,N-1\}\)) is a particular value \(P_{t=0}(E \mid H_i)\).
\((t=2)\)
New hypothesis
\(H_N\) At time \(t=2\), a new scientific theory is introduced, which produces a statistical hypothesis that is a subset of \(\overline{\varTheta _N}\); call this additional hypothesis \(H_N\). The new set of explicit hypotheses is thus \(T_{N+1} = \{ H_0,\ldots ,H_{N-1},H_N \}\). The union of the hypotheses in \(T_{N+1}\) is \(\varTheta _{N+1} \supset \varTheta _N\). The new catch-all is the complement of \(\varTheta _{N+1}\): \(\overline{\varTheta _{N+1}} \subset \overline{\varTheta _N}\). In other words: in the algebra on \(\varTheta \), the old catch-all \(\overline{\varTheta _N}\) is replaced by two disjoint parts, \(H_N\) and \(\overline{\varTheta _{N+1}}\). The new explicit hypothesis \(H_N\) is shaven-off from the old catch-all, \(\overline{\varTheta _N}\), leaving us with a smaller new catch-all, \(\overline{\varTheta _{N+1}}\).
Our first question is how the agent ought to revise her probability assignments at \(t=2\). The second question is whether the old evidence (\(E\) obtained at \(t=1\)) can lend positive confirmation to the new hypothesis (\(H_N\) formulated at \(t=2\)). We will consider these questions in the context of standard Bayesianism and both forms of open-minded Bayesianism. As will be seen, the probability assignments that result from open-minded Bayesianism will show the relevant similarities with those of standard Bayesianism: within \(\varTheta _N\), both have the same proportions among the probabilities for the hypotheses \(H_i\).
Food inspection example While reading our general treatment of the three stages, it may be helpful to keep in mind the example of Sect. 1.1. In this example, the number of explicit hypotheses is \(N=2\). The hypotheses \(H_0\) (meaning, informally, “the kitchen is clean”) and \(H_1\) (“this kitchen is not clean”) can be made formal in the following way: the distribution of infections follows a binomial distribution with bias parameter \(p_0=0.01\) (\(H_0\)) or with bias parameter \(p_1=0.2\) (\(H_1\)). The sample space is the same for both hypotheses: \(\varOmega = \{0,1\}^\mathbb {N}\), where 0 means that a dish tested negatively and 1 means that a dish tested positively. In this case, the evidence takes the form of initial segments of the sequences in the sample space (cylindrical sets of \(\{0,1\}^\mathbb {N}\)).Footnote 17 At \(t=1\), the inspector tests five dishes and receives as evidence an initial segment of five times ‘1’. The initial likelihood of obtaining this evidence \(E\) given hypothesis \(H_0\) is
$$\begin{aligned} P_{t=0}(E \mid H_0)=p_0^5=10^{-10}, \end{aligned}$$
and given hypothesis \(H_1\) the initial likelihood of the evidence is
$$\begin{aligned} P_{t=0}(E \mid H_1)=p_1^5=3.2 \times 10^{-4}. \end{aligned}$$
At \(t=2\), the inspector considers a new hypothesis, \(H_2\), which can be modeled as a binomial distribution with \(p_2=1\).
No update rule for standard Bayesianism
Standard Bayesianism works on a fixed algebra on a fixed set \(\varTheta _N \times \varOmega \). On this view, none of the probabilities can change due to hypotheses that are external to \(\varTheta _N\).
(
\(t=0\)
) N explicit hypotheses Each explicit hypothesis receives a prior probability, \(P_{t=0}(H_i)\). If we assume that, initially, the agent is completely undecided with regard to the \(N\) hypotheses, she will assign equal priors to them: \(P_{t=0}(H_i)=1/N\) (for all \(i \in \{0,\ldots ,N-1\}\)).Footnote 18
(
\(t=1\)
) Evidence E The marginal likelihood of the evidence can be obtained via the law of total probability:
$$\begin{aligned} P_{t=0}(E) = \sum _{j=0}^{N-1} P_{t=0}(H_j) \ P_{t=0}(E \mid H_j), \end{aligned}$$
which is about \(1.6 \times 10^{-4}\) for the example. The posterior probability of each hypothesis given the evidence can be obtained by Bayes’ theorem:
$$\begin{aligned} P_{t=0}(H_i \mid E)=\frac{P_{t=0}(H_i) \ P_{t=0}(E \mid H_i)}{P_{t=0}(E)}\mathrm \ (for\ all\ i \in \{0,\ldots ,N-1\}). \end{aligned}$$
In the example, this is about \(3.1 \times 10^{-7}\) for \(H_0\) and \(1 - 3.1 \times 10^{-7}\) for \(H_1\). According to Bayes’ rule, upon receiving the evidence \(E\), the agent should replace her probability function by \(P_{t=1}=P_{t=0}(\cdot \mid E)\). The inspector should now assign a probability to \(H_1\) that is more than three million times higher than the probability she assigns to \(H_0\). So, in the example, the confirmation is positive for \(H_1\) and negative for \(H_0\).
(
\(t=2\)
) New hypothesis
\(H_N\) Suppose a new hypothesis is formulated: some \(H_N \in \overline{\varTheta _N}\). In terms of the example: the inspector was in a situation in which she could have received evidence with a much higher initial probability than that of the evidence she actually received, and we might imagine that this makes her decide to take the hypothesis \(H_2\) concerning infected test strips into consideration. Now since, in general, the new hypothesis \(H_N\) is not a part of the theoretical context, \(T_N\), the intersection of
\(H_N\)
with
\(\varTheta _N\)
is empty. Hence, the probability assigned to \(H_N\) is zero, simply because \(P(\overline{T_N})=0\). And since the prior of this hypothesis is zero, the confirmation of this hypothesis is zero as well. In other words, standard Bayesianism simply does not allow us to represent new hypotheses (other than by the empty set). In this sense, the ensuing problem of old evidence does not even occur: new theories cannot be taken into account in the first place.
Update rule for vocal open-minded Bayesianism
Vocal open-minded Bayesianism employs a refinable algebra on a fixed set \(\varTheta \times \varOmega \). In this view, none of the previous probability assignments change upon theory change, but additional probabilities can be expressed and earlier expressions can be rewritten accordingly.
(
\(t=0\)
) N explicit hypotheses Each explicit hypothesis receives a prior, \(P_{t=0}(H_i)\) (and, where appropriate, sub-priors). The proposal of vocal open-mindedness is to assign an undefined prior, \(\tau _N \in (0,1)\), to the catch-all hypothesis, \(\overline{\varTheta _N}\):
$$\begin{aligned} P_{t=0}(\overline{\varTheta _N})=\tau _N. \end{aligned}$$
No subsets of the catch-all receive (sub-)priors at \(t=0\), but certain subsets of the catch-all will receive a prior later on. To ensure normalization over all hypotheses (including the catch-all), the priors assigned to the explicitly formulated hypotheses are set equal to the value they had in the model without a catch-all now multiplied by \((1-\tau _N)\); for each \(i \in \{ 0,\ldots ,N-1 \}\):
$$\begin{aligned} P_{t=0}(H_i) = (1-\tau _N) \ P_{t=0}(H_i \mid \varTheta _N). \end{aligned}$$
Although the value of \(\tau _N\) is unknown, the \(N+1\) priors sum to unity. In the example, we have as prior of the catch-all \(P_{t=0}(\overline{\varTheta _2})= \tau _2\) and as prior for the two explicit hypotheses \(P_{t=0}(H_0)=1/2 \times (1-\tau _2)=P_{t=0}(H_1)\).
The likelihood functions of the explicit hypotheses \(H_i\) are the same as in the usual model. Regarding the likelihood of the catch-all, the proposal is to represent it by an undefined weighted average of functions in \(\varTheta \setminus \varTheta _N\): \(P_{t=0}( \cdot \mid \overline{\varTheta _N}) = x_N(\cdot )\).
(
\(t=1\)
) Evidence E The marginal likelihood of the evidence has an additional term as compared to the standard model:
$$\begin{aligned} P_{t=0}(E) = \sum _{j=0}^{N-1} P_{t=0}(H_j) \ P_{t=0}(E \mid H_j) \ + \ \tau _N \ x_N(E). \end{aligned}$$
Due to the presence of undetermined factors associated with the catch-all, \(P_{t=0}(E)\) cannot be evaluated numerically. As a result, also the updated probability function, \(P_{t=1}(\cdot )=P_{t=0}(\cdot \mid E)\), contains unknown factors. These are the posteriors for \(H_i\) (for all \(i \in \{0,\ldots ,N-1\}\)):
$$\begin{aligned} \begin{array}{lll} P_{t=0}(H_i \mid E) &{} = &{} \frac{P_{t=0}(H_i) \ P_{t=0}(E \mid H_i)}{P_{t=0}(E)} \\ &{} = &{} \frac{(1-\tau _N) \ P_{t=0}(H_i \mid \varTheta _N) \ P_{t=0}(E \mid H_i)}{\sum _{j=0}^{N-1} (1-\tau _N) \ P_{t=0}(H_j \mid \varTheta _N) \ P_{t=0}(E \mid H_j) \ + \ \tau _N \ x_N(E)}. \end{array} \end{aligned}$$
Although this expression cannot be evaluated numerically, some comparative probability evaluations can be computed since the unknown factors cancel. In particular, the ratio of two posterior probabilities assigned to explicit hypotheses can still be obtained; for \(i,j \in \{0,\ldots ,N-1\}\):
$$\begin{aligned} \frac{P_{t=1}(H_i)}{P_{t=1}(H_j)} = \frac{P_{t=0}(H_i \mid \varTheta _N) \ P_{t=0}(E \mid H_i)}{P_{t=0}(H_j \mid \varTheta _N) \ P_{t=0}(E \mid H_j)}. \end{aligned}$$
In the example, it can still be established that after receiving evidence \(E\) the inspector should assign a probability to \(H_1\) that is more than three million times higher than the probability she assigns to \(H_0\). Similarly, we can still establish that both hypotheses have a very small likelihood for the evidence that is obtained. And this may be enough to motivate the introduction of a new hypothesis.
In the context of vocal open-mindedness, any expression of the belief change will contain unknown factors, and the implications are worse than for the posteriors: if the change is measured as the difference between posterior and prior, both terms have different unknown factors (\(\frac{1-\tau _N}{P_{t=0}(E)}\) and \(1-\tau _N\), respectively).
(
\(t=2\)
) New hypothesis
\(H_N\) Recall that the old catch-all \(\overline{\varTheta _N}\) is replaced by two disjoint parts: the hypothesis that is shaven off, \(H_N\), and the remaining part of the catch-all, \(\overline{\varTheta _{N+1}}\). Finite additivity suggests to decompose the prior that was assigned to \(\overline{\varTheta _N}\) into two corresponding terms:
$$\begin{aligned} \tau _N = P_{t=0}(H_N) \ + \ \tau _{N+1}, \end{aligned}$$
where \(P_{t=0}(H_N)\) is the prior of the new hypothesis \(H_N\) and \(\tau _{N+1} \in ]0,\tau _N[\) is the (indefinite) prior of the remaining catch-all \(\overline{\varTheta _{N+1}}\), both of which are assigned retroactively. Although the value of \(\tau _{N+1}\) is unknown, the \(N+2\) priors sum to unity.
The priors for the hypotheses in \(T_N\) can thence be written in three ways:
$$\begin{aligned} P_{t=0}(H_i)&= (1-\tau _N) \ P_{t=0}(H_i \mid \varTheta _N) \\&= (1-\tau _{N+1}) P_{t=0}(H_i \mid \varTheta _{N+1}) \\&= (1-\tau _{N+1}) \ (1-P_{t=0}(H_N \mid \varTheta _{N+1})) \ P_{t=0}(H_i \mid \varTheta _N), \end{aligned}$$
where \(P_{t=0}(H_N \mid \varTheta _{N+1})\) is some definite number \(\in ]0,\tau _{N}[\). For instance, if we had a uniform prior over \(T_N\) and we want to keep a uniform prior over \(T_{N+1}\), we have to set \(P_{t=0}(H_N \mid \varTheta _{N+1})=\frac{1}{N+1}\).
Now that \(H_N\) is an explicit hypothesis, its likelihood is a definite function \(P_{t=0}(\cdot \mid H_N)\) (also specified retroactively). In the example, the likelihood for obtaining the evidence \(P_{t=0}(E \mid H_2)\) is 1 on the new hypothesis. We assign an undefined likelihood to the new catch-all: \(P_{t=0}( \cdot \mid \overline{\varTheta _{N+1}}) = x_{N+1}(\cdot )\). This allows us to rewrite the previous expression obtained for the marginal likelihood:
$$\begin{aligned} P(E)&= \sum \nolimits _{j=0}^{N-1} (1-\tau _{N+1}) \ ( 1-P_{t=0}(H_N \mid \varTheta _{N+1}) ) \ P_{t=0}(H_j \mid \varTheta _N) \ P_{t=0}(E \mid H_j)\\&+ \ P_{t=0}(H_N) \ P_{t=0}(E \mid H_N) \ + \ \tau _{N+1} \ x_{N+1}(E), \end{aligned}$$
where the last two terms equal \(\tau _N \ x_N(E)\).
At this point, we can also rewrite the expressions for the posteriors (for all \(i \in \{0,\ldots ,N-1\}\)):
$$\begin{aligned} P_{t=2}(H_i) = \frac{(1-\tau _{N+1}) \ (1-P_{t=0}(H_N \mid \varTheta _{N+1})) \ P_{t=0}(H_i \mid \varTheta _N) \ P_{t=0}(E \mid H_i)}{P(E)}. \end{aligned}$$
Moreover, we can now assign a posterior to \(H_N\):
$$\begin{aligned} P_{t=2}(H_N) = \frac{(1-\tau _{N+1}) \ P_{t=0}(H_N \mid \varTheta _{N+1}) \ P_{t=0}(E \mid H_N)}{P(E)}. \end{aligned}$$
Although it is still not possible to evaluate these posteriors numerically, we can compute new comparative probability evaluations for ratios involving \(H_N\). For all \(i \in \{0,\ldots ,N-1\}\):
$$\begin{aligned} \frac{P_{t=2}(H_N)}{P_{t=2}(H_i)} = \frac{P_{t=0}(H_N \mid \varTheta _{N+1}) \ P_{t=0}(E \mid H_N)}{( 1-P_{t=0}(H_N \mid \varTheta _{N+1}) ) \ P_{t=0}(H_i \mid \varTheta _N) \ P_{t=0}(E \mid H_i)}. \end{aligned}$$
In the case of uniform priors, additional factors cancel:Footnote 19
$$\begin{aligned} ( 1-P_{t=0}(H_N \mid \varTheta _{N+1}) ) \ P_{t=0}(H_i \mid \varTheta _N)&= \left( 1-\frac{1}{N+1}\right) \ \frac{1}{N} \\&= \frac{1}{N+1} \\&= P_{t=0}(H_N \mid \varTheta _{N+1}). \end{aligned}$$
And so, in the case of uniform priors, we obtain:
$$\begin{aligned} \frac{P_{t=2}(H_N)}{P_{t=2}(H_i)} = \frac{P_{t=0}(E \mid H_N)}{P_{t=0}(E \mid H_i)}. \end{aligned}$$
For the example, we can compute \(\frac{P_{t=2}(H_2)}{P_{t=2}(H_1)} = \frac{1}{p_0^5}=\frac{1}{3.2 \times 10^{-4}}=3,125\). So, in the new theoretical context (\(T_3\)) the posterior of the new hypothesis (\(H_2\)) given the old evidence \(E\), namely the sequence of five positive tests, is more than three thousand times higher than that of the hypothesis that was best confirmed (\(H_1\)) within the old theoretical context (\(T_2\)).Footnote 20
At \(t=1\), no degree of belief can be expressed for \(H_N\), but at \(t=2\) the degrees regarding \(H_N\) at \(t=1\) can be expressed and the expressions for the old hypotheses \(H_i\) can be rewritten. We are still left with two terms that have different unknown factors, which do not simply cancel out.Footnote 21 At any rate, degrees of confirmation can be evaluated if we first condition the probability assignments on the current theoretical context, \(\varTheta _{N}\). We return to this point below.
Update rule for silent open-minded Bayesianism
Silent open-minded Bayesianism employs an algebra on a set \(\varTheta _N \times \varOmega \), which may be extended to \(\varTheta _{N+1} \times \varOmega \) (and beyond). On this view, when the theoretical context changes, new conditional probabilities become relevant to the agent.
Let us briefly motivate the silent version as an alternative to vocal open-mindedness. We have seen that the vocal version comes with a heavy notational load. Given that, in the end, we can only compute comparative probabilities, it seems desirable to dispense with the symbolic assignment of a prior and a likelihood to the catch-all hypothesis. Silent open-mindedness achieves this by conditioning all evaluations on \(\varTheta _N\), the union of the hypotheses in the theoretical context. This allows us to express the agent’s opinions concerning the relative probability of \(H_{i}\) and \(H_{j}\) (for any \(i, j \in \{0,\ldots ,N-1\}\)) without saying anything, not even in terms of free parameters, about the absolute probability that they have. Opinions about the theories in the current theoretical context \(T_N\) are thus comparative only.
(
\(t=0\)
) N explicit hypotheses Instead of assigning absolute priors to \(P_{t=0}(H_i)=P_{t=0}(H_i \mid \varTheta )\), silent Bayesianism suggests to only assign priors that are conditionalized on the theoretical context, \(P_{t=0}(H_i \mid \varTheta _N)\).
(
\(t=1\)
) Evidence E Since \(H_{i} \subseteq \varTheta _{N}\), the likelihoods of explicit hypotheses are statistically independent of the theoretical context:
$$\begin{aligned} P_{t=0}(E | H_{i} \cap \varTheta _{N}) = P_{t=0}(E | H_{i}). \end{aligned}$$
Silent open-mindedness suggests not to assign a likelihood to the catch-all. This “probability gap” is not problematic (by the terminology of Hájek 2003), since all the other probability assignments are conditionalized on \(\varTheta _N\). The agent can update her comparative opinion in the usual Bayesian way, as long as she conditionalizes everything on this context:Footnote 22
$$\begin{aligned} P_{t=1}(H_{i} \mid \varTheta _{N}) = P_{t=0}(H_{i} | E \cap \varTheta _{N}) = P_{t=0}(H_{i} | \varTheta _{N}) \ \frac{P_{t=0}(E | H_{i})}{P_{0}(E | \varTheta _{N})}. \end{aligned}$$
(
\(t=2\)
) New hypothesis
\(H_N\) After a new hypothesis has been introduced, the silently open-minded Bayesian has to start conditionalizing on the expanded (union of the) theoretical context \(\varTheta _{N+1}\) rather than on \(\varTheta _N\). Once \(H_N\) gets formulated, its likelihood will be known too. We require that the probability evaluations conditional on the old context \(\varTheta _N\) do not change. In this way, we cohere with standard Bayesianism and with the vocal open-minded variant.
We can treat \(P_{t=2}(H_N \mid \varTheta _{N+1})\) much like a ‘postponed prior’, and give it a value based on arational considerations that are not captured by constraints within the (extended) Bayesian framework. In particular, we can engage in the kind of reconstructive work as is done in vocal open-mindedness, but this is not mandatory here. We might also determine the posterior probability of \(H_N\) and so reverse-engineer what the prior must have been to make this posterior come out after the occurrence of \(E\). In any case, when moving to a new context, the other posteriors need to be changed accordingly (such that the \(N+1\) posteriors sum to unity): \(P_{t=2}(H_i \mid \varTheta _{N+1}) = (1-P_{t=2}(H_N \mid \varTheta _{N+1})) P_{t=1}(H_i \mid \varTheta _N)\). So, the move from \(T_N\) to \(T_{N+1}\) essentially amounts to a kind of recalibration of the posteriors.
Importantly, we can compute all known confirmation measures using the priors and posteriors that are conditional on a particular theoretical context. Once the context changes, this clearly impacts on the confirmation allotted to the respective hypotheses. The price for this transparency is of course that we can only establish the confirmation of a hypothesis relative to a theoretical context \(\varTheta _N\). The natural use of a degree of confirmation thus becomes comparative: it tells us which hypothesis among the currently available ones is best supported by the evidence, but there is no attempt to offer an absolute indication of this support.