Thesis

As agents, we learn from evidence and use what we have learned to make predictions. We learn about processes in the world well enough to become skillful predictors about what will happen next. Whether we are predicting how a person will behave or how the weather will change, our abilities as predictors are crucial to our flourishing as agents in the world. And yet, we often encounter new circumstances – situations in which we have not yet collected evidence. Every new time and every new place, in fact, are new circumstances in the broadest sense of the term. Despite changing circumstances, however, we can still make successful predictions. Apparently, we learn enough about the world to successfully transfer what we have learned from past evidence over to new circumstances. On the other hand, since the changing circumstances are not always irrelevant to our predictive success, it is not easy to say how our expectations should behave across changing circumstances.

It will serve us, then, to speak more clearly about predictions and expectations in order to get a precise grasp on what goes on when we enter new predictive territory.

In order to more precisely formulate how an agent learns from evidence, philosophers have utilized the notion of a credence. Roughly speaking, a credence is a degree of belief in some proposition about the world. Formally, an agent’s credence in a proposition is the subjective probability that the agent assigns to the truth of the proposition. It is represented as a number between 0 and 1 (inclusive). When my credence in a proposition is 1, I am certain that the proposition obtains. When my credence in a proposition is 0, I am certain that it does not obtain. When I toss what I take to be a fair coin, I assign a credence of ½ to the proposition that it lands on heads.

Credences are crucial for modeling changes in our doxastic states because they can precisely capture changes in my degree of belief. More specifically, credences enable specific theories about how observations bear on doxastic states. For instance, if we learn something that we take as evidence for a certain proposition about the world, we become more confident in that proposition. In such a case we say that our credence in that proposition increases.

Without a more detailed theory of doxastic change, the precise impact of observations on credences is underspecified. Yet, Bayesian Conditionalization—the predominant theory of doxastic change—does specify the precise impact of observations on our credences. Bayesian Conditionalization provides a formula to compute how our credences in competing hypotheses change in light of new evidence. The computation is known as Bayesian updating, since the computation models how credences change when we update them on new evidence.

Bayesian updating requires that some proposition (or hypothesis) makes predictions about the observed event, which is then taken to be evidence for or against the proposition or hypothesis. (From now on I will use the term “hypothesis” instead of “proposition” without any loss of generality. That being said, we will have to be careful about what constitutes certain hypotheses when we interpret the thought experiments in this paper.) According to the theory, when a hypothesis makes an event E more likely than the competing hypotheses, we become more confident in that hypothesis over its

competitors upon the observation of E (in proportion to how much better its prediction for E was). When a hypothesis says nothing about some observed event, the observation of that event does not affect our confidence in the hypothesis. Formally:

P'

(

)

(

|

)

(

|

)

(

)

P

(

)

Bayesian updating has been used most centrally in Confirmation Theory in the Philosophy of Science, but the framework is being increasingly applied to the study of norms of practical reasoning in Rational Decision Theory.1 _{In general, updating is of value wherever we seek out the most predictively} successful hypothesis. Bayesian updating can play a role wherever past predictive success of some hypothesis has some epistemic bearing on the future predictive success of that hypothesis.

Inference from past observations is fundamental to any inductive theory; it is not particular to the Bayesian framework. However, the Bayesian theory does implicitly apply a particular way in which past observations bear on beliefs, as follows: an agent’s credence in a hypothesis after updating on some evidence (known as a posterior credence) becomes the initial (or prior) credence for future predictions made by that hypothesis. In other words, if my confidence in a hypothesis is increased after some observation, I will be precisely that much more confident in that hypothesis for the next predicted event. I’m going to call this aspect of the Bayesian theory Credal Transfer. Formally:

Credal Transfer: For a hypothesis H and sequentially observed events E and then F, where P(H) serves as the prior for updating based on E and P’(H) represents the posterior after updating based on E, the value of P’(H) serves as the prior for H when updating on the next predicted event F.

Currently, the mode of Credal Transfer as implied by the Bayesian theory is fairly simple: credences carry over without any adjustment or qualification from one observation to the next. The posterior credence from one observation always becomes the prior credence for the next prediction. So, if after many observations of gravitational interactions I have a high credence in a theorem of General Relativity (say, a credence of .95), then my prior confidence in this theorem for the next event will be just this high credence. If this observation is correct, and supposing that after updating on this correct prediction my new credence is .96, then I will have .96 as the prior credence for the next predicted event. It is a simple stepwise transfer.

Credal Transfer plays important roles in strengths of the Bayesian theory. A particular advantage of Bayesian updating is that, given certain plausible assumptions such as a large number of observations, as an agent updates serially on evidence according to the Bayesian methods, the agent’s credence in the true hypothesis will approach 1.2 _{This result implies that, by following Bayesian updating, the agent will} be most confident in the best hypothesis in the long-run. The guarantee depends in part on the implicit assumption of Credal Transfer, for if credences from one observation aren’t required to transfer over to the next prediction, then the credences couldn’t be constrained over serial observations. Without any restriction on how credences should transfer, the credences might be arbitrarily adjusted between observations. Then, there would be no guarantee of convergence to the best hypothesis. For instance, after many successful predictions, an agent might have a high credence in a theorem of General

Relativity. Such a credence in this best hypothesis would be in the process of converging to 1. Yet, if our theory permits the agent to arbitrarily change her credence in this theorem of General Relativity, then

1 _{Weisberg, Varieties of Bayesianism, beginning on page 527.}

there is no guarantee that the credence will continue to converge. Rather, the credence might be changed to a lesser value, or the credences across the theories of gravity might be reset to an indifferent distribution. Currently, Credal Transfer plays the role of constraining credences across events, so it seems like we need to assume Credal Transfer (or something very much like it) in order to support arguments for convergence guarantees.

The epistemic and practical significance of Credal Transfer is also revealed when considering one’s future expectations. Before we have observed any evidence, we may be indifferent across the competing hypotheses under consideration. Then, we are indifferent between the different predictions made by the competing hypotheses.3 _{However, after having observed some evidence, we will believe} more strongly in the hypotheses that were more predictively successful. So, we will no longer be indifferent across these competing hypotheses. When a hypothesis makes better predictions than its competitors, we increase our credence in that hypothesis, lending it epistemic preference.

Thus, we will prefer the predictions made by the hypotheses that were more predictively successful in proportion to their past predictive success. This can be stated more precisely. Our theory of credences implies that an agent’s confidence in a hypothesis translates directly into the agent’s

confidence in the predictions made by the hypothesis. Hypotheses assign likelihoods to the events that they predict, and our overall confidence in the occurrence of an event is the expected likelihood of the event. Each hypothesis contributes to the expected likelihood of an event according to how likely the event is given the hypothesis and how confident we are in the hypothesis. In this way, our confidence in a hypothesis translates into our confidence in its predictions.4

As rational agents, we take these predictions seriously. Our own expectations are defined by the predictions made by the hypotheses, and our decisions will depend on these expectations. It is therefore important to our epistemic and practical success that we revise our credences in the appropriate ways.

Notably, if we do not utilize Credal Transfer (or something very much like it), then our

expectations will not be based on all of the available information because the information on which the un-transferred posterior credences were based no longer has any impact on our current credence in the hypothesis. In such a case, we might be failing to account for evidence which should be taken to indicate that one hypothesis is better than others. For instance, if we were to ignore cases of gravitational interaction under extreme circumstances, we might be ignoring evidence that supported General Relativity over a Newtonian theory. In such a case, we would be disadvantaged compared to an agent who included such evidence. Such a failure is taken as epistemically and rationally impermissible because any agent who forewent this evidence would not be maximizing utility as an epistemic agent.5

This discussion is closely related to literature on the Principle of Total Evidence,6 _{but I will not discuss} this principle in greater detail.

So it would appear that Credal Transfer is a crucial assumption for successful belief dynamics. However, the story is not finished. While Credal Transfer does play an important role in learning from the past, it can lead to problems when applied ubiquitously. Let’s consider a specific instance of its use.

Suppose you’re a leading meteorologist on a team developing weather models. You’ve been working on a new model with data from the East Coast of the United States. You feed in certain inputs from around the region: humidity, air pressure gradients, and temperature readings. The model takes the inputs and gives you certain predictions. You compare these predictions to the world, and they’re spot-on: every time the model says it will rain in Virginia, it rains in Virginia. Every time the model says it will snow in New York, it snows in New York. And so on for various weather predictions around the East Coast. Appropriately, you become very confident in the predictive success of this model. What’s more, you should! Your high credence in this model is justified by its past successes.

Then, your friend calls and asks for a prediction. She lives in North Carolina and she’d like to go hiking. She knows you’re a great meteorologist, and she’d like to know if you have better information than the local weather channel. Well, since you are highly confident in your state-of-the-art model, you are confident that you do have better information. You run the numbers and the output is that clear skies in North Carolina are very likely this coming Saturday. I take it, then, that you would be highly confident in telling your friend that she can depend on Saturday for her excursion. Formally speaking, you apply your posterior credences to the new prediction. Such an instance I take to be an appropriate application of Credal Transfer. For, without Credal Transfer, you wouldn’t be able to make a confident endorsement of this prediction. You’d have to tell your friend that you didn’t have any useful

information about what was going to happen. But that’s silly, of course, because you do have useful information. You have a model of proven accuracy.

Now your cousin calls. He lives in Washington, and he wants to know about the weather on the San Juan Islands this weekend. You’ve never tested the model on weather conditions on the West Coast, but you retrieve the appropriate data (humidity, air pressure gradients, and temperature readings from the West Coast) and feed it into your model. Your model tells you that clear skies are very likely at the San Juan Islands. So, what do you tell your cousin? Do you endorse this prediction just as confidently as you endorsed the prediction made by the model for North Carolina?

I think, rather, that you should be somewhat tentative. After all, while predictive successes about rain on the East Coast might indicate future predictive successes for those same kinds of

prediction, we might not take such evidence to indicate predictive power of this model for weather on the West Coast. The difference can be described in terms of justification. In the former case, the past successful predictions are indicative that the model makes reliable predictions about rain on the East Coast. These successful predictions are then taken to justify a belief that the model will continue to make reliable predictions about rain on the East Coast. However, one might reason (rightly, I think) that these successful predictions do not provide evidence that the model will be reliable for other kinds of predictions, such as those about weather on the West Coast. So, to have the same high confidence that these predictions will be correct would be unjustified. However, on the face of it, the Bayesian

formalism doesn’t distinguish between different types of predictions and so it obscures the specificity of justification. If this prima facie criticism bites, we’ll need to refine the Bayesian treatment of credences in order to accurately describe how our beliefs work in this case.

These problematic cases are not rare. We could generate similar intuitions using theories of fundamental physics. Consider our confidence in a hypothesis about how massive bodies interact, which has been supported by numerous observations in fairly common circumstances. We might not hesitate to transfer our updated credence over to predictions about interactions in common circumstances, but it’s arguable that we ought to be more tentative in our expectations for interactions under extreme circumstances, such as near a black-hole or during moments just after the Big Bang. It’s epistemically possible that the hypothesis fails to account for conditions that undermine predictions in these extreme circumstances, and so arguably we should be less confident in the predictions for events in these circumstances. However, simple Credal Transfer can’t accommodate such adjustments.

Similarly, we could generate an instance using Newstein. I borrow and modify the case of Newstein, which the Bayesian can and arguably should be able to account for.7 _{Newstein is a brilliant} scientist who asserts two claims within her field of research, P and Q. Suppose we have no independent reasons to believe either P or Q. Even so, upon the observation that P, we will become more confident that Q is also the case. This seems intuitively correct: Newstein has made one correct prediction, and she has made another related assertion, so we become more confident that this other related assertion will also be correct. The Bayesian can account for this inductive reasoning, and transferring credences accords with intuition.

However, suppose that Q were not a statement within Newstein’s field of research. That is, suppose Q were about some very different field of inquiry from P. Should our confidences change in just the same way as in the case where P and Q are in the same scientific field? The Bayesian treatment of credences doesn’t distinguish between these two cases, so it suggests that our confidences should change in just the same way. In other words, our credences transfer uniformly to all new predictions, despite our intuitions for the contrary. (I will consider these two new cases as isomorphic to the original weather models case, at least with respect to the broad stokes of this analysis.)

What’s going wrong when we transition to new types of predictions? Why shouldn’t our confidences carry over in just the same way that they carry over to new predictions of the same type? For an analysis of what’s going wrong, let’s examine the weather model more closely.

Since the model makes correct predictions about rain on the East Coast, we infer that its

predictive apparatus captures some kind of structure exhibited by the weather systems that produce the successfully predicted events. In this case, we can think of the model as capturing a functional

relationship between atmospheric conditions which are taken as input, and the predicted events which are the computed output of the model. More specifically, we see that the model takes certain inputs (regional humidity, temperature and pressure gradients, etc.) to predict what will happen in the future. After a series of successful predictions on the East Coast, we become confident that the function

described by the model is accurate because the weather system on the East Coast exhibits this particular functional relation by precipitating the predicted events.

However, it is plausible that this particular functional relation is not exhibited in other locations. The same atmospheric events (outputs) may result from different explicit conditions (inputs) and so the function described by this model may not capture the structure of weather systems in other locations. For instance, on the West Coast, the Jet Stream behaves differently, humidity and pressure gradients behave differently, etc. So, the function may utterly fail to make correct predictions, or it may make worse predictions than some other model. That the model fails to make correct predictions on the West Coast is a result of the model failing to accurately capture the structure of the weather systems in these locations.

That we infer that the model captures the structure of the weather systems on the East Coast is inductively significant. We increase our credence in this model because we infer that the model captures the structure of the weather systems (in this region). We then take this structure-capturing as sufficient grounding for an increased confidence in the future predictive success of the hypothesis.

Crucially, the structure of the weather system that is explicitly captured by the model is not isolated. Rather, the relation of this structure to the precipitated events may depend on background factors that are not explicitly captured by the model. I will use the terms “factors” or “background factors” to denote those aspects of the world which affect the predictive success of the hypothesis, and “circumstances” to denote all of those conditions surrounding a predicted event which may indicate changes in predictively-affective factors surrounding the event. The success of the weather model depends on circumstances insofar as its success requires that certain background atmospheric factors are held constant; and yet, these background factors do not hold in all circumstances. The failure of the model in new circumstances would result from a change in the predictively-affective background factors. That predictively relevant changes in the factors across different circumstances are epistemically

possible means that such failures across circumstances are also epistemically possible.

Since it is plausible that the background factors on which the weather systems depend could be different in different regions, Credal Transfer needs to be sensitive to differences in such circumstances. Presently, credences can seemingly cut across circumstances – in such instances, credences are

inappropriately transferred to new predictions, the expected successes for which are not justified by the past predictive successes.

Importantly, upon seeing this analysis, a keen Bayesian will deny that her theory is committed to any inappropriate transfer of credences by denying that credences cut across circumstances where the underlying factors have changed. The Bayesian will point out that we must be careful about how to interpret the weather model as a hypothesis. The claim is that, when we’ve sorted out the real

consequent of the conditional, and so we are not committed to the predictions made by the model when the factors have changed. By avoiding the inappropriate transfer of credences in this way, we can preserve the validity of Credal Transfer in many instances. So, we can preserve the strengths of the Bayesian apparatus, such as in convergence guarantees or by preserving the use of relevant evidence, when background factors are not in flux.

Moreover (this reply goes on to say), not only is the silence of the Bayesian theory in circumstances where the background factors have changed not a problem – it is in fact a theoretical virtue. For, when the factors have changed, we will be so tentative about our expectations that we should just shrug, as it were, and make no claims about what will happen. The silence of the Bayesian theory in these instances represents an appropriate epistemic humility – this shrug.

Of course, for the Bayesian to maintain that such a reply will prevent all theoretical commitment to inappropriately transferred credences, she will have to establish that a hypothesis always builds in the appropriate background-sensitivity. I won’t argue that this is theoretically impossible, but there are two serious practical problems associated with this qualification of the application of Credal Transfer. First, while the hypothesis proper may not make inappropriate predictions, the agent may not be aware of the proper hypothetical content associated with the model. In other words, an agent may not know how to specify the factors in the antecedent of the aforementioned conditional content. Because the specific factors on which the model’s success depends are then not known, the agent will not be able to properly discern appropriate from inappropriate applications of Credal Transfer. Second, even if the agent can specify the factors in the antecedent of the conditional, the agent will additionally have to verify whether these factors underlie anticipated events in order to justifiably apply Credal Transfer to events in new circumstances. These practical problems pose serious limitations on the applicability of Credal Transfer, thereby reducing the range of use of the theory.

I think it is also worth noting that by so avoiding the problem of unjustified transfer, the Bayesian has committed herself to a theory of proper credal dynamics which is not strictly internalist. That is to say, the theory states that it is possible for an agent to be unaware of (or, to not have access to) whether or not the application of Credal Transfer is appropriate. In particular, the theory is

committed to the possibility that, in situations where an agent cannot spell out the factors on which the predictive success of the hypothesis depends, the agent won’t be able to use the theory to constrain credences and hence expectations. So, for anyone looking for a theory about how an agent should change credences when under ignorance of underlying factors, this account will be will dissatisfying.

Practical concerns aside, the theoretical point of the Bayesian reply is important step in the right direction. In light of this reply, I argue that the problem of when and how to transfer credences is not a theoretical problem for the Bayesian, but a serious practical problem for the agent who wishes to apply the theory. So, the inappropriate application of Credal Transfer is not a theoretical failure of the

Bayesian theory. Rather, its appropriate application becomes the responsibility of the rational agent who seeks to successfully make use of the Bayesian theory. This responsibility will include the discovery of those factors on which hypotheses depend, and of how those factors vary across circumstances.

events in new circumstances, she will have to revisit her hypothesis space with fresh eyes in order to make any predictions. In other words, even if the model makes predictions about events in new

circumstances, she will not be able to systematically use her posterior credences (those credences which have been updated on evidence from other circumstances) to constrain her expectations, since the posterior credences are only assigned to the hypothesis proper, and not to the model alone. The lack of constraint results from the silence of the Bayesian theory in these new circumstances due to the factor-sensitive conditional content of the hypothesis proper. So, if the agent wants any predictions about events in new circumstances, she will have to assign priors to a new hypothesis, one combining the same model with a new conditional for the relevant background factors. Since the Bayesian theory (famously) says little or nothing about how an agent ought to assign priors, she must simply use her best judgment. I will return to a discussion of this theoretical limitation, but first let’s consider again how changing circumstances should affect an agent’s expectations from her perspective.

The epistemic import of the circumstantial-sensitivity of hypotheses arises when the agent has doubt that the circumstances of past predictive success are the same as for the anticipated events. Looking ahead to predictions about the West Coast, we may doubt that the circumstances are the same. If the circumstances are different, the events’ relation to the inputs may be different than in previous circumstances. So, for all the agent knows, the prediction for rain on the West Coast may depend on different factors for its fruition than do predictions for rain on the East Coast. The background conditions that bring about rain on the West Coast are probably different, and so the predictive apparatus needed for correct predictions in these circumstances may not be the same as the predictive apparatus which has been supported by correct predictions for rain on the East Coast.

The locution “may” is epistemically significant: here, it indicates that for all the agent knows, it is possible that the predictive strength of the hypothesis in previous circumstances could come apart from the predictive strength in the new circumstances. Hence, for all the agent knows, there may be reasons for credences not to carry over simpliciter (as implied by simple Credal Transfer). That there may be reasons for credences not to carry over is enough for credences to not carry over. The doubt means that the credences should change.

Because the predictive success of a hypothesis may be sensitive to circumstances, credences in hypotheses should somehow take into account the circumstances of the evidence. Since a model may be successful in some circumstances but not others, correct predictions may count as evidence for predictive success only within some margin of circumstantial parameters. An immediate problem is how the agent is supposed to learn circumstantial parameters and therefore how to pick out those events for which a model has predictive strength. I’ll call this the problem of circumstantial awareness. Another problem associated with the circumstantial-sensitivity of hypotheses is that it is difficult to analyze exactly how credences should change across circumstantial boundaries even when the boundaries are clear and distinct. So, predictively-affective factors may have clearly changed, but this doesn’t

necessitate any particular effect on a model. It may become less successful, but it might be more successful when these factors have changed! So, what to expect is up in the air. I’ll call this the problem of transfer. The two problems disambiguate the underlying problems of applying Credal Transfer.

As discussed, the Bayesian can build in additional content in the form of a conditional proposition that is sensitive to factors. When the agent is aware of the specific content of the

addition of this content will delimit the applications of Credal Transfer to those instances in which the evidence ought to carry over unqualifiedly – those instances in which the background conditions are similar in the predictively relevant ways. For those events where the background factors are different, some subset of the hypothesis (such as the model) may be taken to make a prediction inappropriately, but strictly speaking the hypothesis makes no such prediction. So, the theory doesn’t permit the transfer of credences over to expectations for circumstances in which the underlying factors have changed because strictly speaking there is no prediction in these new circumstances.

But, when the agent isn’t aware of the specific factors at play, the agent must rely on a vaguer characterization of changing circumstances. The question is then no longer about which predictions are in fact dependent on the same apparatus as successful predictions from previous circumstances. We now ask about which new predictions should be taken to be supported by evidence from previous circumstances. A solution to this question will be useful for agents under ignorance about how exactly the relevant factors have changed and what their impact would be if they have changed.

In order to properly account for circumstantial variation, such an applicable solution must maintain that a credence in a hypothesis is only fully specified when it is indexed to a particular

circumstantial background. Similar to the merely theoretical solution, such indexing can be achieved by adding the conditional content that delimits the hypothesis to the circumstances of evidential support (but not necessarily by specifying the underlying factors). Formally, let’s denote such an indexed credence as PC(HM) = P(HM in C) for hypothesis HM in circumstances C. In the weather model case, HM denotes the weather model, and C denotes the circumstances in which the model is evidentially supported. The weather model consists in the propositions, equations, etc., that make up the full predictive apparatus under consideration. These propositions, equations, etc. define the theoretical structure that captures the structure of the world. By including the circumstantial content as an

essential part of the hypothesis proper, we are formally committed only to the predictions of HM within circumstances C. In comparison to the solution of including an additional factor-sensitive conditional, this solution is different only in that it does not require the factors to be specified. Rather, this solution is framed in terms of circumstances so that an agent can more easily apply the theory.

Crucially, circumstances are accessible to an agent in a way that background factors may or may not be, since circumstances are merely indicators for changes in the factors surrounding an event. For instance, the West Coast is different in certain salient ways: the ocean is on the west side of the land rather than the east side, the ocean currents are different, the mountain ranges are larger, etc. Even for an agent who is not a meteorologist, the differences may point toward changes in relevant underlying factors. These differences are what I take to be circumstantial differences. In the case of Newstein, the circumstances are the fields of inquiry: because we know that biology is so different from physics, we may not trust Newstein’s predictions about biology. Yet, we don’t have to know exactly how the underlying factors have changed, and we don’t have to be able to specify the parameters of events for which factors are different. Rather, we merely need a vague access to or awareness of circumstantial differences. On this analysis, the absence of any indication that the circumstances are different permits the agent to transfer credences unqualifiedly. An interesting question concerns what an agent should be required to know before being rationally permitted to transfer credences in this way.

good hypothesis—one that has been supported by past observations under fairly typical conditions (such as under Earth-like gravitational fields). What should our expectations be when looking forward to interactions under extreme conditions, such as near the edge of a Black Hole?

The analysis reveals that past observations support the inference that the predictive apparatus of our current theory captures the structure of the interactions of massive bodies in Earth-like

circumstances. Thence, it is appropriate to use credences based on these observations to formulate expectations about other events in similar circumstances. However, we would not have similar expectations for events in wildly different circumstances (such as for events around a Black Hole). Whereas we may have a high confidence in one particular outcome for normal events (i.e. events in Earth-like conditions), we will not have such a high confidence for atypical events. Since we have limited Credal Transfer to those cases where the relevant predictive apparatus is very likely to be the same (since the factors are not likely to be relevantly different within circumstantial boundaries), we have prevented inappropriate transfer. Our theory of belief change now matches our intuitions to a greater extent: it does not produce overconfidence in new circumstances.

Yet, we now encounter the problem of transfer: what expectations should we have in these new circumstances? We have prevented inappropriate Credal Transfer by isolating a credence based on one circumstantial background from observations in other circumstances (but we have preserved Credal Transfer within circumstantial parameters). So, the evidence from one circumstantial background will have no bearing on credences for other circumstantial backgrounds. Do we accept total circumstantial separation, and simply revert to indifference across hypotheses when entering different circumstances?

Contra this concession, in some instances we may think that past success in one set of circumstances is somewhat indicative of success in different circumstances. That a theory of how massive bodies interact has succeeded in Earth-like conditions might increase our confidence that the theory will work elsewhere without transferring purely. That a weather model has succeeded on the East Coast may provide some indication that it will also succeed on the West Coast. We might be more tentative in each of these cases, but that doesn’t mean the circumstantial success has no epistemic bearing whatsoever on other circumstances. So, it appears that to totally isolate credences according to the circumstances of evidential support is to go too far. Can we develop a more nuanced approach?

I will explore this consideration by returning to the case of Newstein. Newstein has made two predictions, P and Q. Supposing P to have come out true, and Q to have been a related statement, our inductive theory and our intuitions agree that we should increase our confidence that Q will be the case. However, if Q is not closely related to P – perhaps Q is a claim about evolutionary biology – then our intuitions come apart from credences that gloss over circumstantial differences. A

circumstance-insensitive theory of credences holds that we should be just as confident in a Q that is not related to P as if it were closely related to P. I take it that our intuitions suggest otherwise. Does the indexing of

credences to circumstances solve the problem?

Newstein, as a brilliant physicist, has demonstrated her expertise by correctly predicting P. We infer from this brilliant prediction that Newstein has a deep understanding of the nature of the world at the level of phenomenon such as P, for such understanding is required for successfully predicting P. This understanding can be modeled according to my analysis: Newstein’s understanding captures the structure of the world relevant to correctly predicting P. We are then justified in transferring our

related to P, this is the same underlying understanding and so our transferred credences are justified. In the case where Q is not closely related to P, the relevant understanding is different, so we do not transfer our credences.

So, what about in-between cases? What if Q were not completely unrelated to P, but also wasn’t in exactly the same domain. Perhaps Q is about chemical or astronomical facts, and is therefore more closely related to a background understanding required for making a correct prediction about a fact of physics than for making a correct prediction about a fact of evolutionary biology. Perhaps we are unsure about the breadth of Newstein’s expertise, and while we’re pretty sure she knows very little about evolutionary biology, perhaps she does know enough about chemistry or astronomy to make such a correct prediction. How does such an in-between case bear on circumstantial isolation?

The issue at hand is that the breadth of predictive success of a hypothesis (i.e. the parameters of the circumstances within which the hypothesis will be successful) is difficult to know from past

successes. So, it is difficult to know what credences to have when transferring into new circumstances. While successful past predictions may indicate that some future predictions will be successful, it isn’t easy to tell which future predictions are evidentially supported. We may find it easy to say that when anticipating weather events on the West Coast using the model that has been supported by successful predictions on the East Coast, we should be more tentative than we would be for similar events on the East Coast. And yet, what about for predictions in the Midwest? Or for those just across the Appalachian Mountains? Do we gradually become more conservative in our estimations, or must we simply draw a line somewhere on the map?

Ideally, we want a way to transfer credences in a systematically adjustable manner. When anticipating a predicted event, we may think that the past successes should make us just as confident in this particular prediction as we would be by applying simple Credal Transfer. Or, we may think that the past successes should only make us somewhat confident—a kind of partial transfer. Or, we may think that the past successes should have no bearing on the new prediction, eliminating transfer altogether. Currently, the Bayesian theory is silent when transferring credences across hypotheses (even when the model is in common), but we can perhaps uncover further constraints as addenda to our theories of credal dynamics in order to develop a more informative theory of how to transfer credences.

Let’s briefly consider a few possibilities for theoretical constraints on the assignment of credences in new circumstances. The following possibilities should be understood as addenda to the Bayesian formalism, extending it into territory in which it is currently hands-off.

A second possibility, already shown to be counterintuitive, would be to transfer credences purely, irrespective of circumstantial boundaries. Pure transfer results in unjustified expectations.

A third possibility is a systematic combination of the two previous extremes: akin to Jeffrey conditionalization, we could use a weighted average of our previous credences with the indifferent credal distribution, thereby forming more moderate expectations.8 _{So, if P(H}

M in C) is the posterior from the former circumstances and P(HM indiff.) = 1/N (for a hypothesis space of N competing hypotheses) represents the credence in HM according to an indifferent distribution, then P(HM in Cnew) = (x)*P(HM in C) + (1-x)*(1/N).

But what parameter could serve as the weighted value x? Assuming that this method is a weighted average, then x must be between 0 and 1. Thus, the first two unsatisfactory methods serve as the extremes of this method. If we set x to 0, then we will have set the credences to an indifferent distribution. If we set x to 1, we will have transferred credences unqualifiedly.

Already, this extension of the Bayesian treatment is a substantive addition, and one that is arguably intuitive. For, it prevents us from being more confident in a hypothesis in new circumstances than we were in previous circumstances that were evidentially supportive of the hypothesis. Similarly, it prevents us from being less confident in a hypothesis in new circumstances than we were for previous circumstances that reduced our overall confidence in the hypothesis (given equal priors before updating in those circumstances). At first glance, these are reasonable constraints: our new credences ought to fall somewhere in between the distribution that results from an absence of evidence and the

distribution that incorporates all of our previous evidence. Such new credences are thus the result of a kind of partial transfer of evidential support.

Given that an agent knows nothing about the conditions underlying new circumstances or about the success of this hypothesis outside of the previous circumstances, I find these constraints plausible. These are fairly broad constraints, but it is difficult to set any more specific constraint on credences.

If we try to find a more specific constraint for the weighting, two good candidates would be “the degree to which the agent thinks that the new circumstances are similar” and “the probability that the underlying factors are different in these new circumstances.” And yet, both of these are problematic.

The first constraint is problematic because the degree to which the new circumstances are the same as the old circumstances may be very high (e.g. only one important factor is different), but even a small difference may make the predictive apparatus wildly wrong (even, consistently wrong). If one changes a very small factor in a weather system, the system can produce very different results. So, the “degree of similarity” maps very poorly onto the degree of expected success.

The second possible constraint is problematic because the chances that the underlying factors are different doesn’t incorporate the degree of success if the circumstances are in fact different. So, suppose an agent is be certain that the circumstances are relevantly different. On this attempt to constrain the weighting, the credences would revert to an indifferent distribution (no transfer

whatsoever). And yet, even if the underlying factors are in fact different, the predictive success may not be totally thrown off. It may be only slightly less successful. Thus, “probability of difference” also maps poorly onto expected success.

In general, I see no way presently to accommodate a more specific constraint on the systematic variation of credences across circumstantial boundaries into our theories of credal dynamics in an unproblematic way. The issue is that the epistemic bearing of changing circumstances on hypotheses will depend greatly on the particular structures of the world. How a weather model will perform in new circumstances will be sensitive to the particularities of atmospheric processes, but how a model of fundamental physics performs in new circumstances will depend on very different kinds of factors. The extent to which we put weight in the claims of Newstein may depend on particular observed details from which to infer the breadth of her understanding. So, to formulate a general solution to trans-circumstantial learning is perhaps beyond the realm of a philosophical theory of belief dynamics. For now, these theories of belief change serve us as general frameworks for learning, delegating the details to more specific levels of understanding (such as the special sciences).

While it is difficult to give general prescriptions for transferring credences across circumstances, our theories of credal dynamics can be highly informative within circumstantial parameters. I’ve argued that while the incautious application of Credal Transfer can be problematic, we can and should hold on to Credal Transfer as an important aspect of our theory of credal dynamics. For, if we make no

commitments to any kind of Credal Transfer, then our theory says too little about what credences we should have. Rather, we want an informative theory of credal dynamics: a theory that provides useful constraints on appropriate credences; a theory that does not delegate too much to the subjective assignment of priors. The conclusion is that Credal Transfer should be taken as a principle with a particular domain of applicability. When the full hypothetical content of a hypothesis is explicated (i.e. when there are no unspecified background factors), then Credal Transfer is ubiquitously valid. So, the Bayesian theory can remain committed to qualified and careful Credal Transfer without running into any problems, even when taken as a theory of credal dynamics. It is only for an agent who is unaware of the background factors that the application of Credal Transfer becomes practically difficult, but this is merely a problem for the theory insofar as it should be useful. Yet, even an agent under ignorance is arguably justified in transferring credences whenever the circumstances seem not to have changed.

Buchak, Lara (2010). “Instrumental Rationality, Epistemic Rationality, and Evidence-Gathering,” Philosophical Perspectives 24: 85–120.

Good, I.J. (1967). “On the Principle of Total Evidence,” The British Journal for the Philosophy of Science 17(4): 319-321.

Greaves, Hilary and Wallace, David (2006). “Justifying conditionalization: Conditionalization maximizes expected epistemic utility,” Mind, 115:607–632, 2006.

Hawthorne, James. “A Better Bayesian Convergence Theorem,” last accessed March 2015. http://fitelson.org/few/few_04/hawthorne.pdf

Jeffrey, Richard: 1965, The Logic of Decision, McGraw-Hill, New York; (1983), 2nd ed., University of Chicago Press, Chicago.

Strevens, Michael (2001). “The Bayesian treatment of auxiliary hypotheses,” British Journal for the Philosophy of Science 52(3), 515–537.