1. Vassend on the Interpretive Problem

Vassend (Reference Vassend2019) formulates the interpretive problem by noting that Bayesian methods often assign nonzero probabilities to hypotheses that are known to be false. This is inconsistent with the standard interpretation that Bayesian probabilities express degrees of belief in the truth of a hypothesis given evidence. Vassend argues that this problem can be solved only by giving a new semantics for Bayesian inference.

Let Θ be a set of rival and mutually exclusive hypotheses indexed by a parameter θ, and let X be the set of possible outcomes of an observation or experiment. Then a Bayesian statistical model specifies the prior probability p(θ) of θ in Θ and the likelihood $p(x/\theta )$ of θ on x in X, and calculates the posterior probability $p(\theta /x)$ of θ given x by means of Bayes’s theorem as proportional to the product $p\left(\theta \right)p(x/\theta )$. Assume now that background information K implies that each hypothesis θ in Θ is false. Then, given K, p(θ) as the degree of belief in the truth of θ is zero, which implies by Bayes’s theorem that $p(\theta /x)$ is zero as well. In this situation Bayesian inference collapses.

Vassend’s verisimilitude interpretation suggests that the prior p(θ) is understood as

(1)$\begin{array}{}\text{the probability that}\theta \text{is closest to the truth out of the hypotheses in}\Theta \text{.}\end{array}$

If a verisimilitude measure v is available, then according to (1) p(θ) is the probability that θ maximizes v. Even though typically v is maximized by a single hypothesis in Θ, so that disjunctive hypotheses do not satisfy the condition (1), the probability of members of Θ can be formally extended to their exclusive disjunctions by the addition law.

The counterfactual interpretation, which Vassend attributes to Sprenger’s (Reference Sprenger2017) unpublished paper, states that the prior probability of θ should be understood as the conditional probability $p(\theta /b)$, where b is the counterfactual assumption that one of the hypotheses in Θ is true. In statistical regression analysis, b might be the false idealizing assumption that the functional dependency of two quantities can be expressed by a linear function.

While Sprenger seems to think that these two alternatives are separate interpretations of probabilities, Vassend’s contribution is to argue that his verisimilitude interpretation is in fact “intertranslatable” with the counterfactual interpretation. The reason why these approaches are “two sides of the same coin” comes primarily from the fact that the semantics of counterfactuals involve a similarity ranking s of possible worlds, and the functions v and s can be defined in terms of each other.

Vassend’s project has an aim similar to Shimony’s (Reference Shimony and Colodny1970) “tempered personalism,” which wishes to assign nonzero “degrees of rational commitment” to “seriously proposed hypotheses,” even if their degree of belief in the ordinary sense is zero or extremely small. Shimony too typically has in mind situations in which the true theory is not included among the relevant hypotheses, but his treatment is clearly different from Vassend’s.

2. The Problem with Sharp Hypotheses

Let us start the assessment of Vassend’s article by noting a special issue with sharp hypotheses. These are typically statements that pick out a point in a continuum, so that their probability measure is zero. Vassend thinks that his set of hypotheses Θ can be assumed to be finite “without loss of generality” (Reference Vassend2019, 699), but this is potentially misleading, since, for example, in the estimation of real-valued parameters Θ is a continuous subset of the infinite class R of real numbers. Then the prior probability density p(θ) may have a nonzero value for each point θ in Θ, but the probability of a sharp hypothesis θ is nevertheless zero. This situation is different from Vassend’s interpretive problem, since all sharp hypotheses in Θ receive zero probabilities—independently of the question whether Θ includes a true element. But the interpretive problem is in a sense independent of the problem with a sharp hypothesis, since it reappears as the task of making sense of nonzero probability densities for point hypotheses and nonzero probabilities for regional hypotheses, when all the hypotheses are known to be false.

When Vassend illustrates the interpretive problem in practice in his section 3, he chooses the class Θ so that it includes all functions of the form $Y=\alpha {(\mathrm{1,010}-X)}^n+\epsilon$ between quantities X and Y. In section 6 he simplifies this equation to straight lines. In both cases Θ is a continuous subset of the real space (three parameters in R³ or two parameters in R²), so that the counterfactual probability of its individual members is zero.

Even though sharp hypotheses have zero probability, their degree of probable approximate truth may be nonzero (see Niiniluoto Reference Niiniluoto1987, 280). This degree can be defined for a real-valued hypothesis θ as the probability of a small interval u(θ) around θ, that is, the sum of all values p(θ) for θ in u(θ). The same idea can be generalized to hypotheses that are real-valued curves. This concept, which has important applications in Bayesian probably approximately correct learning as a method in machine learning (see Niiniluoto Reference Niiniluoto, Hajek, Valdés-Villanueva and Westerståhl2005), is clearly different from Vassend’s verisimilitude interpretation of probability.

3. Verisimilitude Measures

Standard expositions of the similarity approach to verisimilitude define this concept relative to a cognitive problem B, which consists of a partition of mutually exclusive and jointly exhaustive hypotheses h_i in some interpreted language L (see Oddie Reference Oddie1986; Niiniluoto Reference Niiniluoto1987). Then there is one and only one element h* in B that is true (in the actual world). This h* is the unknown element of the problem B, and the task of the investigator is to identify it. If a metric or distance function d is defined between the elements of B, and d is extended to measure the distances of disjunctions of hypotheses h_i to an element in B, then a verisimilitude measure v is obtained for any theory H expressible in L: the degree of truthlikeness v(H, h*) of H depends on its distance to the target h*.

The choice of the distance function d depends on the type of hypotheses in the cognitive problem B. Vassend’s section 6 illustrates this in the case in which the compared hypotheses are real-valued curves in R², but his attempt is not successful. Niiniluoto (Reference Niiniluoto1987, 385–86) proposes to measure the distance between two real-valued quantitative laws f and g by the Minkowski metrics for function spaces:

(2)$\begin{array}{}{L}^p\left(f,g\right)={\left[\int {\left|f\left(z\right)-g\left(z\right)\right|}^p\mathrm{dz}\right]}^{1/p},\end{array}$

where integration is defined over the domain of the functions f and g. Here L^p for small values of p reflects the area between the curves f and g, while for infinite p it gives the maximum distance between the values of f and g. Apart from this maximum distance, Vassend mentions the minimum distance between f and g (Reference Vassend2019, 708), but for a good reason this is not included in the Minkowski family (2). Indeed, the minimum distance as a measure of verisimilitude is here worthless: if any curve that intersects the true curve g has maximal verisimilitude, then we have no useful discrimination or ordering among false hypotheses by this criterion. As a curve is a conjunction of an infinite number of claims, also the maximum distance is a crude measure, and the Manhattan metric ($p=1$) and the Euclidean metric ($p=2$) are more plausible.

4. Verisimilitude with a False Presupposition

According to Vassend’s verisimilitude interpretation (1), p(θ) is the probability that θ is closest to the truth among hypotheses in Θ, when background knowledge implies the falsity of all elements of Θ. But this means that Θ is not a cognitive problem in the standard sense, since there is no element in Θ that serves as the true target of the verisimilitude measure v. Thus, the condition (1) that p(θ) is the probability that θ maximizes the verisimilitude measure v is not well defined.

A solution to this problem has been proposed by Niiniluoto (Reference Niiniluoto1987, 259–62). Let $B_b=\left\{h_i\right|i\epsilon I\}$ be a cognitive problem in language L relative to a false presupposition b. Thus, b implies that the elements h_i in B_b are mutually exclusive and jointly exhaustive, but they all are false. If s is a metric on the space of L-structures (possible worlds), then choose the target h*[b] as the element of B_b that is true in the b-world that is minimally distant from the actual world. If there is more than one such minimally distant L-structure where b is true, choose the target as the disjunction of the corresponding statements in B_b.Footnote ¹ Using the semantics of counterfactuals by Lewis (Reference Lewis1973), this choice can be expressed by saying that the target h*[b] of a cognitive problem B_b with a false presupposition b is the most informative statement in the language of B_b that would be true if b were true. By this choice, the elements of B_b and their disjunctions can be compared for verisimilitude v_b by their distance to the target in B_b. And the rule (1) now states that the prior p(h_i) of h_i in B_b expresses the probability that h_i is identical with this target h*[b].

Another solution is implicit in Vassend’s paper.Footnote ² Assume that B_b is a subclass of a larger cognitive problem B with the true target h*, where B is defined without the counterfactual assumption b. Then the members of B_b can be compared for their verisimilitude by the measure v for B (instead of v_b for B_b), even though the target h* is not included in the set B_b. There may be issues about the choice of B, and the comparison of verisimilitude orderings v and v_b, but for the purposes of Vassend’s project they are irrelevant, since both v and v_b are maximized in B_b by the hypothesis h*[b]. According to both solutions, the probability that an element h_i of B_b is closest to the truth is the probability that h_i is h*[b]. Note also that if b is true, then h*[b] is identical with h*.Footnote ³

Vassend correctly notes that similarity rankings of possible worlds and verisimilitude rankings of propositions induce each other (Reference Vassend2019, 712). It is also correct that this sort of link was used by Hilpinen (Reference Hilpinen, Przelecki, Szaniawski and Wojcicki1976) in his pioneering account of the similarity approach to truthlikeness (n. 15). But Vassend’s exposition here is flawed. Hilpinen employed Lewis’s qualitative notion of similarity spheres as a primitive, while Niiniluoto (Reference Niiniluoto1987) explicitly defines a quantitative measure by structural similarity considerations.Footnote ⁴ Moreover, if theory H is a set (disjunction) of possible worlds, then the verisimilitude of H should not be defined by the minimum distance of H from the actual world, as Vassend (Reference Vassend2019, 712) assumes.Footnote ⁵ Hilpinen defined the notion of approximate truth by the minimum distance, but as a definition of verisimilitude this is known to be hopeless. Indeed, Hilpinen argued convincingly that the comparative notion of truthlikeness should take into account at least the maximum distance as well, but this gives only a partial ranking of hypotheses. A quantitative version of Hilpinen’s proposal was given by Niiniluoto’s (Reference Niiniluoto, Butts and Hintikka1977) min-max measure, and more elaborate alternatives like Oddie’s (Reference Oddie1986) average measure and Niiniluoto’s (Reference Niiniluoto1987) min-sum measure are still debated (e.g., Oddie Reference Oddie and Zalta2014; Niiniluoto Reference Niiniluoto2020).

5. Counterfactual Probabilities and Expected Verisimilitude

The preceding section gives support to Vassend’s contention that probabilities (1) under the verisimilitude interpretation are equivalent to counterfactual probabilities, when the set of hypotheses Θ is based on a false presupposition. It is also correct that thereby Bayesian probabilities depend on the pragmatic factor related to the choice of the verisimilitude measure, since different distance measures s and s′ could lead to different targets h*[b] in B_b.

But it should be remembered that after all the Bayesian agent does not know which hypothesis is the true one (in the standard case) or would be true (in the counterfactual case), so that she may choose the priors freely as long as probability axioms are satisfied. In the standard special case in which Θ includes a true target h*, a hypothesis θ has the minimum distance from the truth if and only if θ is identical with h*. In the counterfactual case, the best hypothesis is identical with the target h*[b]. Condition (1) is not an operational rule for finding the maximally truth-like hypothesis, but it suggests that prior probabilities should express the agent’s best guesses about the location of the unknown target h* or h*[b] within the relevant cognitive problem.

Formally, let H be a theory in a cognitive problem B, and let $\epsilon >0$ be a small real number. Let U_ε(H) be the set of hypotheses h_i in B such that $v\left(H,h_i\right)\ge 1-\epsilon$. Then one may define probable verisimilitude in the following way: the probability ${\mathrm{pv}}_{1-\epsilon }\left(H/e\right)$ that the degree of truthlikeness of H is at least $1-\epsilon$ equals the probability of U_ε(H) (see Niiniluoto Reference Niiniluoto1987, 279). Then ${\mathrm{pv}}_{1-\epsilon }\left(H/e\right)$ has the maximum value 1 if and only if the posterior probability is wholly concentrated on U_ε(H). It is thus possible that more than one hypothesis h_i in B receives the maximum value, and it may happen that $p\left(h_i/e\right)=0$ but ${\mathrm{pv}}_{1-\epsilon }\left(h_i/e\right)=1$. But when $\epsilon \to 0$, we have in the limit ${\mathrm{pv}}_{1-\epsilon }\left(h_i/e\right)\to p\left(h_i/e\right)$ for all i.

There is a further concept that behaves formally in a way different from probability, that is, expected verisimilitude $\mathrm{ver}\left(H/e\right)$, introduced by Niiniluoto (Reference Niiniluoto, Butts and Hintikka1977):

(3)$\begin{array}{}\mathrm{ver}\left(H/e\right)=\sum _{i\epsilon I}p\left(h_i/e\right)v\left(H,h_i\right)\text{,}\end{array}$

where e is the available evidence. Here v(H, h_i) is the degree of truthlikeness H would have if h_i were true, which is weighted by the posterior probability $p\left(h_i/e\right)$ of h_i given e, and the sum goes over all hypotheses h_i in B. If Θ is a continuous space, then in (3) p is a probability density, and the finite sum operator Σ is replaced by the integral

(4)$\begin{array}{}\mathrm{ver}\left(H/x\right)={\int }_{\Theta }p(\theta /x)v(H,\theta )d\theta \mathrm{.}\end{array}$

(See Niiniluoto Reference Niiniluoto1987, 269). Definitions (3) and (4) are tools for estimating unknown degrees of truthlikeness, when we have an epistemic probability distribution over the relevant alternatives h_i in B. Unlike posterior probability $P\left(H/e\right)$, the value of $\mathrm{ver}\left(H/e\right)$ may be nonzero and even high, when e contradicts H (274). If posterior probability is almost completely allocated to a particular hypothesis h_j in B, so that h_j is believed to be true, then the expected verisimilitude of other hypotheses h_i are approximately equal with their closeness to this presumed truth h_j (273).

If a cognitive problem B_b is defined relative to a counterfactual presupposition b, so that the realist condition ¬b implies the falsity of all of its elements h_i, then $p(h_i/e\text{\&}\neg b)=0$ and $\mathrm{ver}(h_i/e\text{\&}\neg b)=0$ for any true evidence e. For example, h_i might describe the ballistic behavior of projectiles near the surface of the air, where b is the idealizing assumption that the resistance of air has no influence, and e describes observations under real conditions on air. Niiniluoto (Reference Niiniluoto1987, 286) argues that still we may have $p\left(h_i/e^{\prime }\text{\&}b\right)>0$, where e′ tells what the data e would have been in the idealized situation b. Thus, e′ is obtained—perhaps using theoretical background assumptions—by subtracting the influence of air from the observational data e (e.g., Suppes Reference Suppes, Nagel, Suppes and Tarski1962) or by trying to realize the idealized circumstances in controlled experiments or computer simulations (e.g., study of free fall in a vacuum). For empty evidence, this positive prior probability $p\left(h_i/b\right)$ is an instance of Sprenger’s (Reference Sprenger2017) counterfactual interpretation of conditional probability. Applying (3) with counterfactual probabilities, we may have $\mathrm{ver}\left(h_i/e^{\prime }\text{\&}b\right)>0$ and likewise $\mathrm{ver}\left(H/e^{\prime }\text{\&}b\right)>0$ for a disjunctive theory H in B_b. An alternative approach is to transform the idealized hypotheses into a realist one $h_i^{\prime }$ by concretization, that is, by adding the influence of the air to the description h_i (see Niiniluoto Reference Niiniluoto1987, 116–17). Then we may have $p(h_i^{\prime }/e\text{\&}\neg b)>0$ and $\mathrm{ver}(h_i^{\prime }/e\text{\&}\neg b)>0$. In other words, idealized hypotheses may be confirmed by idealized evidence, and factual hypotheses by factual evidence.

These considerations illustrate the important feature of Bayesian modeling that counterfactual assumptions may have a significant role in the determination of both prior and posterior probabilities—as well as evidence-based estimates of verisimilitude.