4.1. A Metanormative Principle for Epistemic Norms

The above general insight is underspecified for our purposes, however. To apply it here, we will articulate a metanormative principle that captures its spirit in the context of evaluating the rationality of credences as the agent obtains new evidence. I am interested in whether our relative assessment about a credence function is appropriately consistent, either over time or under equivalent representations of the agent’s doxastic state. In particular, consider what follows. If we render a verdict about the rationality of A’s credences, relative to B’s credences, at time 1, and between time 1 and time 2 the only relevant aspect of the situation that changes is that a new piece of evidence is obtained, and A and B both update their beliefs diligently to account for the new evidence, then this much seems clear: at time 2, it does not seem appropriate for our assessment of the relative rationality of A and B’s credences to shift. In other words, since A and B have both diligently updated their beliefs, the verdict rendered at time 2 should be consistent with the verdict rendered at time 1. For lack of a better term, I will call this principle the requirement of comparative consistency.

In short, if A’s priors are less incoherent than B’s, then, for any observation, A’s posteriors should not be more incoherent than B’s.Footnote ⁴ In some respects, comparative consistency is like a stability condition on norms of epistemic rationality. It says that if you now believe A’s credences to be more rational than B’s credences, you should not change that assessment unless you have an appropriate reason to do so—such as a new piece of evidence suggesting that one of them acted irrationally (or less rationally) in some way. In other words, the verdict about comparative rationality should not shift arbitrarily. While this is only a bare-bones sketch of the notion, the goal of this project is not to fully articulate and argue for a metanormative theory. Rather, I hope the principle is sufficiently plausible for now.

4.2. Updating Incoherent Credences

I will show through a simple example that approximate coherence is not comparatively consistent. The updating situation is ordinary. There are no issues of uncertain evidence that would trigger Jeffreys’s rule or some alternative to it or of misleading evidence, and so on. Further, our agents will indeed diligently update. The notion of comparative consistency prohibits a verdict that deems one of them more rational before the update and less so following the update.

Suppose we have two agents, A and B, who have opinions about the unknown bias of a certain coin, which may take any value $\theta \in \left[\mathrm{0,\; 1}\right]$. These opinions are expressed by their credence functions for θ, denoted by p_A(θ) and p_B(θ), respectively. A and B may not be coherent, which means that p_A(θ) and p_B(θ) can fail to be probability distributions. However, in the case we will consider, these functions remain additive. This is similar to the kind of incoherence that De Bona and Staffel use to describe graded incoherence (violations of normalization). Suppose for illustration that A and B will perform a simple experiment. In particular, they are going to toss the coin once. The assumption that our agents will toss the coin once is adopted for ease of exposition. It will be clear that the problem can arise for any finite number of tosses. We will denote the result of the coin toss with the random variable X, which can take the value 0 (for heads) or 1 (for tails). After observing the result they will update by Bayes’s rule to get the posterior credence functions $p_A\left(\theta \right|x)$ and $p_B\left(\theta \right|x)$. For ordinary coherent priors, updating would be straightforwardly accomplished through the familiar expression of Bayes’s rule, as follows:

$\begin{array}{}\text{(1)}& p\left(\theta \right|x)=\frac{f\left(x\right|\theta \left)p\right(\theta )}{m\left(x\right)},\end{array}$

where

$\begin{array}{}\text{(2)}& m\left(x\right)={\int }_0^{1}f\left(x\right|\theta \left)p\right(\theta )d\theta \end{array}$

is the marginal credence function for $X=x$ over all possible values of the unknown quantity θ and $f\left(x\right|\theta )$ is the data-generating distribution.

For incoherent agents, however, m(x) may or may not be a valid probability function. In ordinary Bayesian inference, this is usually not an issue because the function m is not a function of the unknown quantity of interest, θ, but is instead a function of the data, x. Accordingly, we would proceed as usual, treating this function as a pseudomarginal distribution, focusing on the components that are functions of θ, and taking the posterior to be proportional to the prior times likelihood:

$\begin{array}{}\text{(3)}& p\left(\theta \right|x)\propto f(x\left|\theta \right)p\left(\theta \right)\text{.}\end{array}$

This is indeed what I will do.

But whereas for ordinary credences we might, if desired, normalize the posterior so as to rescale the resulting distribution into [0, 1], for our purposes here doing so is not innocuous. Our main subject of interest is the incoherence in the agent’s credence function. Accordingly, I do not want to automatically rescale because I am taking it for granted, as De Bona and Staffel do, that there is something noteworthy about the agent’s imperfect doxastic state, and I want to preserve that aspect of the situation under updating. Accordingly, I want to take the prior distribution to a posterior distribution as faithfully as possible without arbitrarily wiping out the initial incoherence and thereby changing the agent’s subjective representation of uncertainty. Since the agent’s beliefs remain additive, the Bayesian posterior is a faithful posterior that maintains prior proportions of probabilities while reflecting the new evidence. Further, inferences on θ and predictions about X remain the same whether we normalize or not. After updating, I want to compare relative incoherence under different possible observations $X=x$ to see whether the ranking is stable. That will be the strategy in the next section.

Ultimately, however, we will see what it takes to normalize each credence function. Rather than hiding the ball, so to speak, we will do things one step at a time. Recall that for a fixed value of $X=x$, the quantity m(x) is a constant. The core of the problem will be that for credence functions that can be normalized, it is a finite normalizing constant, whereas for credence functions that cannot be normalized, the problem we identify persists for any real number assigned to this quantity.

Comparative consistency requires that our normative assessment of the quality of A and B’s new credence functions should not depend on whether the coin lands on heads or tails. Since their mental states are the same in all relevant respects in both the heads world and the tails world, and their updating behavior is the same, and neither has done anything to suggest a change in our verdict regarding their relative rationality, our postexperiment normative assessment about the rationality of their credences should not depend on the observed outcome. It should be consistent with our preexperiment verdict.

4.3. Reflection and Expectation

The updating strategy in equation (3) is far from arbitrary. Rather, it is a more general expression of the ordinary statement of Bayes’s rule as given in equation (1). Huttegger (Reference Huttegger2014, Reference Huttegger2017) argues that in order for an update rule to count as a reasonable response to a learning experience it must satisfy the principle of reflection or, more specifically, the martingale principle. And drawing on Zabell (Reference Zabell1982), he further shows that, given some fairly modest assumptions (called exchangeability, regularity, and sufficientness), the general expression in equation (3) is the only one that does so for the kind of case we are considering (single parameter inference for an unknown proportion; see Huttegger Reference Huttegger2017, chaps. 5 and 6). This remains true whether the agents are coherent or not, provided their credences remain additive.

In our context, the martingale principle requires that the conditional expected value of the probability of heads/tails on the $n+1$th toss, given the result on the nth toss, is equal to the probability of heads/tails on the nth toss. We can show that the update rule in equation (3) indeed satisfies the martingale principle, as follows. To simplify, I will express the point using the first and second tosses.

First, we need to define the predictive distribution of an as yet unobserved outcome of X, which we will denote as $\tilde{X}$, given the outcomes of X that have so far been observed, which we will denote with the lowercase x. This prediction takes the following form:

$\begin{array}{}\text{(4)}& \mathrm{Pr}(\tilde{X}=1|x)={\int }_0^1\mathrm{Pr}(\tilde{X}=1\left|\mathrm{x,}\theta \right)p\left(\theta \right|x)d\theta ={\int }_0^1\theta p(\theta \left|x\right)d\theta =E\left[\theta \right|x]\mathrm{.}\end{array}$

That is, the probability that $\tilde{X}=1$ (the coin lands on tails) is given by the posterior mean of θ after conditioning on the tosses that have been observed. Before any tosses have been observed, we would simply use the prior mean for θ, given by E[θ]. This would be the predicted probability that $X_1=1$. Under the above assumptions, both the prior and the posterior mean can be computed from the kernel of the prior/posterior distribution—that is, we do not need to determine the normalizing constant. Now we want to show that the expected value of the probability that the second toss lands on tails, given the result of the first toss, is equal to the probability that the first toss lands on tails. That is,

$\begin{array}{}\text{(5)}& E\left[\mathrm{Pr}\right(X_2=1\left)\right|X_1=1]=\mathrm{Pr}(X_1=1)\mathrm{.}\end{array}$

We know from equation (4) that the quantity $\mathrm{Pr}(X_2=1)|X_1=1]$ is given by the posterior mean; that is, $E\left[\theta \right|X_1=1]$. And by the law of iterated expectation, we know that $E\left[E\right[\theta |X_1=1]]=E[\theta ]$. The same point holds for any sequence of outcomes X ₁, … X_n.

Thus, both expressions of Bayes’s rule (eqq. [1] and [3]) remain faithful to the martingale principle and more generally reflection. This has to be true because they are proportional to each other, and as a result inferences on θ and associated predictions about X remain unchanged. The constant, $c=m\left(x\right)$, rescales the distribution of θ, but the scale is irrelevant to inference. What matters is the relative density assigned to different regions of the parameter space. But, if we move to some other non-Bayesian update rule, it is not clear how we could satisfy the martingale principle. I discuss the Bayesian updating assumption further in section 5.2 and the normalization point in section 5.3.

4.4. A Counterexample

Suppose that A and B’s credence functions before the experiment are given by

$\begin{array}{}\text{(6)}& p_A\left(\theta \right)=\frac{1}{1-\theta },p_B\left(\theta \right)=\frac{1}{\sqrt{\theta (1-\theta )}}\mathrm{.}\end{array}$

These credences are not coherent. It is not entirely clear how we should interpret incoherent credence functions. If we assume that they indeed encode an agent’s partial beliefs, as De Bona and Staffel suggest, then the interpretation of agent A would be that A is very confident the coin is tails biased, whereas the interpretation of agent B would be that B is just as confident that the coin is biased but indifferent as to which direction it is biased toward. However, for any plausible measure of divergence, B’s credence function is more rational (better) because A’s credence function is a nonfinite measure over the parameter space [0, 1] whereas B’s credence function is an unnormalized probability. Specifically,

$\begin{array}{}\text{(7)}& {\int }_0^1\frac{1}{1-\theta }d\theta =+\infty ,{\int }_0^1\frac{1}{\sqrt{\theta (1-\theta )}}d\theta =\pi \mathrm{.}\end{array}$

Suppose we assess the degree of A and B’s incoherence by applying the well-known Kullback-Leibler divergence. Between p_i(θ) and p_j(θ), it is defined as

$\begin{array}{}\text{(8)}& \mathrm{KL}\left(p_i\right|\left|p_j\right)={\int }_{\Omega }{p}_i\left(\theta \right)\log \frac{p_i\left(\theta \right)}{p_j\left(\theta \right)}d\theta \mathrm{.}\end{array}$

Nothing in this argument depends on KL divergence being the ‘right’ measure of divergence. I use this measure for illustration, as it is well known and corresponds to the Shannon measure of information entropy (Shannon Reference Shannon1948a, Reference Shannon1948b) and to the strictly proper logarithmic scoring rule (Gneiting and Raftery Reference Gneiting and Raftery2007). Applying this expression to the bias of the coin and rearranging, we have

$\begin{array}{}\text{(9)}& \mathrm{KL}\left(p_i\right|\left|p_j\right)={\int }_0^1{p}_i\log p_{i}d\theta -{\int }_0^1{p}_i\log p_{j}d\theta \mathrm{.}\end{array}$

For any coherent candidate credence function p_C, $\mathrm{KL}\left(p_A\right|\left|p_C\right)$ is either undefined or ∞, since the left-hand side of the difference is ∞ and the right-hand side is either ∞ or 0 (if C’s credences are uniform on θ, $p_C=1$). Meanwhile, $\mathrm{KL}\left(p_B\right|\left|p_C\right)$ is a finite number. For example, if C’s credences are uniform, then $\mathrm{KL}\left(p_B\right|\left|p_C\right)=4.35$. Therefore, our initial assessment must be that B’s credences are better. A’s credences are not coherent, and they cannot be made coherent—there is no finite normalizing constant that would turn A’s credence function into a valid probability distribution. If instead we use simple absolute value distance (or squared distance, for that matter), then the divergence between A and the closest coherent distribution (indeed, any coherent distribution) is likewise ∞ since $x/(1-x)$ is not bounded above on [0, 1]. To summarize,

Consider, next, what happens after our agents perform a simple experiment. The distribution of the coin is given by ${\theta }^x{(1-\theta )}^{1-x}$. Accordingly, when the coin lands on tails, the posterior density is proportional to the prior density multiplied by θ, and when the coin lands on heads, the posterior density is proportional to the prior density multiplied by $1-\theta$.

Case 1: The coin is tossed and it lands on tails.—If the coin lands on tails, we obtain the following posterior credences:

$\begin{array}{}\text{(10)}& p_A\left(\theta \right|X=1)=\frac{\theta }{1-\theta },p_B(\theta |X=1)=\sqrt{\frac{\theta }{1-\theta }\mathrm{.}}\end{array}$

In this case, A’s credences are given by the odds for heads, whereas B’s credences correspond to their square root. As a result,

$\begin{array}{}\text{(11)}& {\int }_0^1\frac{\theta }{1-\theta }d\theta =+\infty ,{\int }_0^1\sqrt{\frac{\theta }{1-\theta }}d\theta =\frac{\pi }{2}\mathrm{.}\end{array}$

Approximate coherence gives the verdict that A remains awful and B remains better. To use the shorthand from above: $B>A$. This is consistent with the verdict rendered before the coin was tossed. So far so good—approximate coherence is comparatively consistent. Now consider what happens if the coin lands on heads.

Case 2: The coin is tossed and it lands on heads.—If the coin lands on heads, we obtain the following posterior credences:

$\begin{array}{}\text{(12)}& p_A\left(\theta \right|X=0)=1,p_B(\theta |X=0)=\sqrt{\frac{1-\theta }{\theta }\mathrm{.}}\end{array}$

In this case, A’s credences are uniform. The density is a straight line at $y=1$ for every value of θ between 0 and 1. Meanwhile, B’s credences correspond to the odds for tails—that is, they are reciprocal to B’s credences when the coin lands on heads. As a result,

$\begin{array}{}\text{(13)}& {\int }_0^{1}1d\theta =1,{\int }_0^1\sqrt{\frac{1-\theta }{\theta }}d\theta =\frac{\pi }{2}\mathrm{.}\end{array}$

A has become perfectly coherent, whereas B remains as incoherent as B would be if the coin landed on tails. The verdict according to approximate coherence is now the reverse: A’s credences are better and B’s credences are worse. Approximate coherence is no longer comparatively consistent. Insofar as approximate coherence is a standard for making comparative rationality judgments between agents—or between their credence functions—then B is in effect punished for being unlucky. The possible results after updating on a single coin toss are summarized as follows:

The verdict according to approximate coherence, as between whether A has more rational credences than B, depends on whether the coin lands on heads or tails, a feature of the problem that is external to both agents and over which they have no control.

In short, we have what follows: if credences that sum to 1 are less incoherent than those that do not, and posteriors that can be normalized by a (finite) constant are less incoherent than those that cannot be, then approximate coherence violates comparative consistency. This problem is guaranteed to occur for any finite number n of tosses if we observe a sequence of n heads or tails. In particular, if we observe a sequence of n tails, then approximate coherence will give the verdict that B is better and A is awful, whereas if we observe a sequence of n heads, approximate coherence will give the verdict that A is better (although not necessarily perfect) and B is worse. So while the problem is increasingly improbable as n increases, it is a problem nonetheless and can occur for any finite n.

5.1. Misleading Evidence

First, it is worth emphasizing that the issue is not about misleading evidence—that is, a situation in which the agent receives some evidence about the outcome of the coin toss that may or may not correspond to what actually happened. The outcomes of the toss in either case are by hypothesis accurately observed and recorded. Rather, the concern is that if a credence function about the bias of a coin is said to be relatively more rational/irrational conditional on heads, it should be equally rational/irrational conditional on tails. There is no relevant distinction to be made between the heads state and the tails state that would motivate a shift in the evaluative verdict regarding the agents’ comparative rationality.

5.2. Bayesian Updating

Second, and more importantly, one might question whether incoherent agents should update along Bayesian lines. The updating strategy adopted here is a standard application of Bayes’s rule for single parameter inference in the case of an unknown proportion with an improper prior (e.g., Lindley and Phillips Reference Lindley and Phillips1976). I presuppose that the agents’ update should be a reasonable response to a learning experience, and thus the update rule should satisfy the martingale principle (Huttegger Reference Huttegger2017). It is possible, of course, to insist that for subideally rational agents, everything goes out the window—they cannot update along Bayesian lines, and more generally they violate the martingale principle and reflection. One might defend this on the ground that when you deviate from rationality in one respect, it is overall better to deviate from rationality in other respects as well, in the spirit of the theory of the second best (Lipsey and Lancaster Reference Lipsey and Lancaster1956). But the very idea of approximate coherence suggests that when one is not ideal, it is better to be closer to the ideal than to be further away from it.

Further, the prior I have used is one with a distinguished history in Bayesian statistics—it is the Jeffreys’s prior (for person B) and its transformation (for person A; Jeffreys Reference Jeffreys1946). Both are closely related to the uninformative (and incoherent) Haldane prior (Haldane Reference Haldane1932). It is fair to say, then, that this project starts from the observation that incoherent priors are an important part of the Bayesian paradigm. Indeed, in many types of problems, both invariant (in the sense of Jeffreys) and uninformative priors are often incoherent. For example, a conventional uninformative prior for the mean (μ) and variance (σ²) of a normal likelihood is a prior that is uniform on (μ, log σ) ( Gelman et al. Reference Gelman, Carlin, Stern, Dunson, Vehtari and Rubin2013, eq. [3.2]). This prior is incoherent.

However, one might insist that an incoherent agent ought not update before making the prior coherent. While this might seem like a reasonable request, it would arbitrarily restructure the agent’s representation of uncertainty in the absence of new evidence and significantly limit the scope of Bayesian inference.

In some cases, the request might be feasible. In particular, when the prior integrates to a finite constant, the position would be that we must normalize before updating. Even here, the demand is not innocuous. It would create substantial impediments to inference. Often, we normalize after updating in order to avoid computing the marginal distribution of the data (in our case x), which is not a function of the parameter of interest (in our case θ). This request would make inference less tractable in difficult problems. But suppose we grant that this is what rationality requires of imperfect agents, so to speak. There is a further problem still.

Sometimes, the prior is incoherent in such a way that its integral does not converge (i.e., is improper). For example, the prior I used for person A. If we demand that this prior must be made coherent, the request is impossible to meet. As a result, if we took this position, it would commit us to prohibiting the use of improper priors, such as the Jeffreys’s and Haldane priors for inference involving proportions. Suppose we grant this too. There is yet a further problem, which is that proper priors on one space are often improper on simple transformations of that space. So it is not clear how we should understand such a request—that is, the request that improper priors must not be used in Bayesian inference.

For example, instead of reasoning about our problem using a prior for the unknown parameter θ, it is often useful in Bayesian inference to reason about such a problem using a prior for the natural parameter of the exponential family representation of the data-generating process. In our case, that would be the log odds for θ, $\phi =\log (\theta /(1-\theta ))$, because ${\theta }^x{(1-\theta )}^{1-x}=\exp \left[\log \right(\theta /(1-\theta ))x-\log (1-\theta \left)\right]$, which is of the form $\exp [\phi x-A(\phi \left)\right]$ for $A=\log (1+e^{\phi })$. In the absence of information about φ, a natural starting point would be a prior that is uniform for all its possible values. However, using the change of variable formula, a uniform prior for φ would lead to a prior for θ that is proportional to $1/\left(\theta \right(1-\theta \left)\right)$, which is not integrable (i.e., is improper). Yet the two distributions should be equivalent in terms of corresponding to the same representation of uncertainty. In other words, if we insist that no updating can be done until the prior is made proper, then making the prior proper on the transformed space will often imply an improper prior on the original space. Thus, whether one is permitted to apply Bayes’s rule will then be sensitive to the parameterization of the problem. Yet it is the same underlying inference problem. Consistent with these remarks, Robert justifies the use of improper priors as follows: “the inclusion of improper distributions in the Bayesian paradigm allows for a closure of the inferential scope (figuratively as well as topologically)” (Reference Robert2007, 28).

Perhaps it will be helpful to explain by analogy the strategy adopted in this article. Suppose we have an imperfectly rational agent, not in the sense we are currently considering (imperfect coherence) but in the sense of imperfect precision. When agents are imprecise, we can identify their upper and lower probabilities for an event and thus compute their interval estimate for the event. Seidenfeld and Wasserman (Reference Seidenfeld and Wasserman1993) describe a phenomenon where an agent starts with an interval estimate for an event, obtains some evidence, updates the initial set of priors by Bayesian conditioning, and ends up with a posterior interval that strictly contains the prior interval. This is the well-known phenomenon of probabilistic dilation. By identifying this phenomenon, Seidenfeld and Wasserman (Reference Seidenfeld and Wasserman1993), among others, suggested there is a problem either with imprecise probability models or with the assumption that one should update them by Bayes’s rule because, intuitively, one’s interval should become narrower—or, at least, not wider—after undergoing a learning experience. By framing the problem in terms of this potential incompatibility—made explicit by the occurrence of dilation—they encourage us to think carefully about how the imprecise framework fits with the standard Bayesian updating picture. Since then, there have been many proposed solutions of the apparent conflict. My goal in this article is to highlight something similar for approximate coherence—that is, to identify a potential conflict between the approximate coherence framework and standard Bayesian updating, but whereas Seidenfeld and Wasserman’s (Reference Seidenfeld and Wasserman1993) underlying normative principle for interval estimates is that they should not dilate under updating, my underlying normative principle is that comparative judgments about rationality should be consistent under updating.

5.3. To Normalize or Not?

Third, one might question why I do not automatically normalize the agent’s credences as part of the update. Instead, I break up the analysis as follows. First, I derive the posterior, taken to be prior times likelihood, then I show what it would take to normalize it. For example, in equation (10), p_A is A’s nonnormalized posterior and P_B is B’s nonnormalized posterior. In the next step, equation (11), I show just how incoherent they are by highlighting what it would take to normalize them—p_B sums to $\pi /2$, and p_A is nonconvergent. Thus, p_B can be normalized by dividing by the marginal probability, which is equivalent here to the integral of the unnormalized posterior treated as a function of x. That is, $1/(\pi /2)\times (\pi /2)=1$. But for p_A, there is no finite normalizing constant, which is what makes it epistemically “worse.”

I do this in order to be faithful to De Bona and Staffel’s project and create room for evaluating deviations from coherence across updates. In other words, I do not normalize because our starting point is that there is something interesting about the doxastic state of an agent whose credences violate normalization. Another way to put the point is as follows: if we take incoherent priors and normalize them automatically as part of the update rule, then we erase the agent’s incoherence by the same stroke. No matter how incoherent one is, the update rule wipes it all out. In that case, approximate coherentism under updating would only apply to violations of additivity, and one could only violate the normalization axiom in one’s ur-prior (the stylized prior that we take to exist at the very beginning of agents’ doxastic life and before they collect any evidence). Moreover, normalizing as part of the update rule would make my own conclusion somewhat circular. I would suggest that there is not much room for approximate coherence judgments under updating because my update rule eliminates any incoherence.

Now, we could normalize right away and draw a similar overall conclusion. The conclusion would be that credences that cannot be normalized by a finite constant are beyond repair, whereas those that can be normalized are all essentially on the same footing from the perspective of epistemic rationality. The fact that one normalizable credence is closer to coherence than another normalizable credence is epistemically insignificant. Thus, the spirit of the conclusion would be the same: what really matters under updating is the binary question of whether the posterior can be normalized, not the graded question of how far from coherence it happens to be. This is what I have tried to show.

One might still object: Could not the same type of problem arise for ordinary coherence if we do not normalize as part of the update? It could, but this would be the result of forcing the approximate coherence model into the ordinary coherence framework. In other words, it is not a problem for the ordinary coherentist. The ordinary coherentist is not interested in ranking credence functions in terms of their degrees of incoherence. As a result, it is not problematic for the ordinary coherentist to normalize as part of the update—normalizing does not jeopardize the relevance of coherence as a normative principle for evaluating the epistemic rationality of a person’s credences. The same is not true for approximate coherence because, as mentioned, normalizing wipes out precisely that feature of the person’s doxastic state that approximate coherence is designed to help us model and evaluate.

For the ordinary coherentist, being incoherent is epistemically bad because one’s credence function is then accuracy dominated. But the ordinary coherentist is not in the business of telling us how to move to a coherent credence function. Thus, if we ask the ordinary coherentist should the incoherent agent update and normalize, the answer ought to be yes. And if we ask, should the coherent agent update and normalize, the answer should likewise be yes. And neither answer undermines the role of ordinary coherence as an evaluative norm for the rationality of an agent’s credences.

But suppose that in the ordinary coherence framework we have one coherent agent, who updates and normalizes, and one incoherent agent, who updates but does not normalize. Is this a problem for the ordinary coherentist? It is not because in this case they are doing different things. Still, we can take note that the coherent agent will remain coherent. And the incoherent agent will be sometimes coherent and sometimes not. Thus, the nonideal agent will never become epistemically better, and the ideal one will remain ideal. It is the graded nature of the approximate coherence framework that suggests itself to making ongoing comparisons under updating, but as I hope to have shown, these ongoing comparisons can be misleading.

5.4. Synchronic or Diachronic Norms?

One might further object to the concerns articulated here on the ground that De Bona and Staffel only assess the rationality of a credence function at a time. To what extent, then, should they be persuaded by this notion of comparative consistency over time? Note, as I flagged in section 4, that while I articulate comparative consistency in terms of what happens before, and after, a learning experience, the temporal framing is a heuristic for illustrating the problem. The notion of comparative consistency is at a minimum a relation among a person’s doxastic attitudes. In the preceding paragraphs, I suggested that it would be difficult to banish improper priors from the Bayesian landscape because a proper prior under one representation of a problem commits an agent to an improper prior under a slightly different representation of the same problem. Here, a similar suggestion is appropriate: comparative consistency applies between a prior (unconditional) credence function and a posterior (conditional) credence function. While the prior/posterior language suggests a temporal perspective, we can also understand the notion in terms of the agent’s conditional commitments. My current credence function commits me to a certain credence function in a heads world and to a certain credence function in a tails world. That is, our commitments can be understood from a time-slice-centric perspective of rationality, to borrow Hedden’s (Reference Hedden2015) and Moss’s (Reference Moss2015) expression. From an evaluative perspective, the heads and tails worlds, or the estimates conditional on heads and the estimates conditional on tails, are indistinguishable. Yet approximate coherentism’s assessment differs among them.

In the large literature pertaining to the proper scope of the requirements of rationality, Broome (Reference Broome2007) argues that they should be understood as state requirements. In the epistemic context, state requirements require the agent’s doxastic attitudes to be a certain way at a given time (by comparison, Kolodny [Reference Kolodny2005] argues for a process interpretation). The notion of comparative consistency suggested here can be understood within a state-based picture of the requirements of rationality. It is a notion of consistency between unconditional doxastic attitudes and the conditional doxastic attitudes one is thereby committed to. The concern on this framing would be that comparative judgments about relative rationality at a time are sensitive to normatively innocuous features of a state (i.e., one’s conditional credence given the event heads vs. one’s conditional credence given the event tails). Note that this is a minimalist conception of comparative consistency. It is of course possible to understand the concept diachronically, in terms of the temporal framing. I only wish to highlight that such a conception is not necessary in order to find the idea compelling.