On what should we conditionalise?

The orthodox Bayesian account of conditionalisation suggests that conditionalisation is a response to a certain kind of epistemic accident. We begin with a credence function C. We get sensorily irradiated, or something, and newly come to have credence 1 in a number of propositions. How should we rationally respond to this occurrence? Answer: conditionalise! Take the logically strongest proposition which newly has credence 1, p, and adopt as your new credence C+, one’s old credence function conditionalised on p: C+ = C(·|p).

(Jeffrey worries about this in many places, but his worry isn’t a worry about the idea of experience causing propositions newly to have probability 1, but rather that sometimes there is no sentence which expresses the proposition that newly has probability 1. But he notes ‘Perhaps it is the fact that there surely is definite evidence that prompts and justifies the probability shift in [a case of updating on an uncertain sentence], that makes some people think tgere must be an evidence sentence (observation sentence) that will yield the new belief function by conditionalization’ (Jeffrey, ‘Bayesianism with a human face‘, §1). But this worry is motivated by a weird view—the view that credence is a sentential attitude. But if credence has anything to do with belief and knowledge, it had better be a propositional attitude, and there is certainly a proposition—a set of worlds—associated with the content of each of the observations Jeffrey adduces as good cases for Jeffrey conditionalisation. Moreover, though it is of purely academic interest given this observation, many of these propositions can be expressed in English. In Jeffrey’s celebrated case of dining by candlelight, surely we can adduce as evidence the observation sentence ‘the tablecloth is that colour‘, indicating the colour of the tablecloth. Jeffrey gives us no reason to think this demonstrative observation sentence is somehow defective.)

This account—conditionalisation as the rational response to an involuntary act of evidential accomodation—fits well with certain empiricist tendencies that aren’t far below the surface of most Bayesians. So Jeffrey (in ‘Probable Knowledge‘) for example characterises ordinary (non-Jeffrey) conditionalisation by first characterising ‘observational’ propositions as those ‘of which the direct effect will be to change [an agent’s] degree of belief in the proposition to 0 or 1’, and then claiming that

If there is a proposition E such that the direct effect of the observation is to change his degree of belief in E to 1, then for any proposition A in Henry’s preference ranking, his degree of belief in A after the observation will be the conditional probability prob(A|E), where prob is his belief function before the observation.

He then goes on to discuss some cases in which ‘the “direct” effect of the observation can be described as informing the observer of the truth of E. Where this is possible, the relevant passage of sense experience causes the observer to believe E.… where epistemologists have spoken of observation as a source of knowledge, I want to speak of observation as a source of compulsory belief to one or another degree’.

But why should we think of conditionalisation as involving updating in response to new compulsorily acquired beliefs? Why not, instead, think of conditionalisation as updating in response to evidence?

That thought sounds just as plausible as the standard Bayesian view. Maybe the standard Bayesian view is even an attempt to capture it, though involuntary doxastic effects of observation don’t look much like viable candidates for evidence, even empirical sensory evidence. But many of the claims made on behalf of Bayesianism look better with this thought in play. One gets some new evidence E, and conditionalises on it. One doesn’t need the evidence to cause any alteration in one’s credential state; one simply recognises that one has E as evidence, and rationally comes to update one’s other beliefs in light of that evidence. How that evidence bears on one’s other beliefs is a matter of how one judges things to stand with those other beliefs in light of E; that judgement is encoded by one’s conditional probabilities on E. So new evidence E prompts you to rationally adopt those conditional probabilities as unconditional, because of the evidence that one newly has. Conditionalisation looks like a rational response to evidence, rather than simply the best way to accommodate an unavoidable byproduct of experience.

As Williamson argues (Knowledge and its Limits, ch. 9), this explanation only works if one’s evidence is true. It would not be reasonable to adopt as one’s unconditional credences, one’s old credences conditional on a false proposition. Any conception of evidence as potentially justifying our beliefs—any conception on which it is a norm of belief that we should proportion our belief to the evidence—surely is reasonable only if what is evidence is true. For we have no reason to proportion our belief to accommodate falsehoods, and every reason to proportion them to truths. Some falsehoods do seem like evidence admittedly, enough like them that we may well mistakenly conditionalise on them. But no one would think that was the rational course of action, even if it is excusable due to (perhaps unavoidable) ignorance. One’s posterior credences are reasonable only if they result from making best accommodation to the truth. The central idea is that one’s evidence provides one with reasons for other beliefs; falsehoods may seem to provide such reasons, but really don’t.

Presumably evidence needs other features than truth. Maybe a proposition needs to have high enough credence to be evidence (or if, like me, you think the notion of belief is needed alongside the notion of credence in accounting for our doxastic life, maybe it simply needs to be believed to be evidence). Williamson himself thinks evidence has to be known. But whatever further features are needed, there will be generally for us a bunch of propositions that constitute our evidence at each time. Conditionalising on all our evidence will be equivalent to conditionalising on our new evidence, if we’ve previously conditionalised on our old evidence. And that looks like a reasonable thing to do.

On this view, there is no need to become certain of one’s evidence before conditionalising on it—it is the act of conditionalising which makes one certain (since of course if one’s new evidence is F, then C+(F)C(F|F) = 1). We don’t need, as is commonly assumed, two stages—the first non-inferential stage of becoming newly certain of something (perhaps in some purely mechanical causal way, as Jeffrey suggests), the second ‘the probabilistic inference of conditionalizing’ (Talbott, ‘Bayesian Epistemology‘, §2). (I myself discussed the two stage proposal: Philosophy of Probability, p. 117.) There are rather two epistemically significant stages: the acquisition of new evidence; and the recalibration of belief in light of new evidence. Epistemology done as if only one of these is susceptible to epistemic norms will be incomplete.

Also we might try to find a way out of the problem of old evidence. Maybe all conditionalisation is updating the original priors in line with our conditional judgements about how evidence bears on hypotheses. In that case, all of our evidence, whether newly acquired or not, is relevant for the evaluation of hypotheses. Some piece of evidence E confirms H relative to a total body of evidence Ω iff C (H| Ω) > C (H|Ω\{E}). Of course, one needs to opt for something like this only if one finds the standard Bayesian account of confirmation lacking on this front. I myself have some sympathy with less conciliatory approach:

if the evidence really is old, then all the scientists already believe it, and have already updated their credences in the hypothesis in question. If there seems to be additional confirmation in these cases, it is because those scientists have imperfect access to their own credences; or are incoherent, since (despite already knowing e) they are more confident in h after noticing e, even though no new evidence has arrived. These [less conciliatory] Bayesians say: if confirmation doesn’t have a close connection with learning, that only undermines its importance—for the main aim of scientific inference isn’t to see what confirms what for its own sake, but to discover what we should believe, and what we believe is constrained by the theory of subjective probability. (Eagle, Philosophy of Probability, p. 215)

But the main upshot of any such view of what we should conditionalise on is the potential of the view to explain the normative force of conditionalisation. Diachronic Dutch book arguments are all very well, if it is presupposed that we are updating on evidence; but what if we involuntarily come to have credence 1 in a falsehood? It simply does not look reasonable to respond to this occurrence by conditionalising. So it can’t always be rational to conditionalise on propositions one is newly certain of. The full story of Bayesian conditionalisation must hold that the first part of the process is significant for ascriptions of rationality too, which it can be if we update on evidence.

The view also allows the Bayesian to give a fuller account of epistemic rationality. In some setups—in Lewis’ presentation of the diachronic Dutch book, for example—the propositions on which we conditionalise are explicitly experiential: ‘mutually exclusive and jointly exhaustive propositions that specify, in full detail, all the alternative courses of experience you might undergo between time 0 and time 1’ (Lewis, ‘Why conditionalize?‘). But why should experience be so significant? Ordinary theories of confirmation and disconfirmation don’t give any special role to the results of observation as opposed to evidence gathered in other ways. Maybe most of our evidence is in fact observational, but that doesn’t mean that its observational character is more important than its evidential character. Mathematical propositions are evidence for us, but not observational evidence. More generally: updating on the content of the genuine oracle’s pronouncements would be reasonable, and certainly more conducive to epistemic success than updating on the observation that certain pronouncements were made. Of course, if observational evidence were rich enough—if, for example, my observation involved the proposition that the genuine oracle said that p—I could infer p (‘the genuine oracle said that p‘ is a factive operator). So it might not make much difference, if we have the correct rich conception of the content of observational evidence (resisting the temptation to ‘psychologise’ evidence as Williamson puts it in The Philosophy of Philosophy). But the Bayesian emphasis on the causal consequences of sensory irradiation certainly tempts unwary philosophers into standard kinds of internalist empiricist errors about the basis of our beliefs; using a properly understood notion of evidence in conditionalisation does not lead to such temptation.

(This post had its origin in some conversations with Katherine Munn.)


One thought on “On what should we conditionalise?

  1. Patrick Maher proposes this sort of view in his “Subjective and Objective Confirmation” — modulo his specific view there on what one’s evidence is (viz., the propositions one knows “directly by experience”, which you rightly object to at the end of your remarks). If you mod-out by that bit, then his view in that paper sounds similar to what you’re suggesting (and he makes many of the same observations I think). See:


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