Abstract
Conditionalization is a norm that governs the rational reallocation of credence. I distinguish between factive and non-factive formulations of Conditionalization. Factive formulations assume that the conditioning proposition is true. Non-factive formulations allow that the conditioning proposition may be false. I argue that non-factive formulations provide a better foundation for philosophical and scientific applications of Bayesian decision theory. I furthermore argue that previous formulations of Conditionalization, factive and non-factive alike, have almost universally ignored, downplayed, or mishandled a crucial causal aspect of Conditionalization. To formulate Conditionalization adequately, one must explicitly address the causal structure of the transition from old credences to new credences. I offer a formulation of Conditionalization that takes these considerations into account, and I compare my preferred formulation with some prominent formulations found in the literature.
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Notes
I discuss cases where P(E) = 0 in (Rescorla 2018).
The possibility of misplaced certainty bears upon some well-known arguments that seek to justify credal norms. The familiar Dutch Book arguments for Conditionalization and Reflection—as presented, say, by Skyrms (1987)—assume that the agent becomes newly certain of a true proposition. Furthermore, Schoenfield (2017) shows that expected accuracy arguments for Conditionalization go through only if we assume that the agent is certain she will become newly certain of a true proposition. My discussion suggests that these arguments are problematic. I will discuss the implications of misplaced certainty for Dutch book arguments and expected accuracy arguments in future work.
(5) incorporates a “total evidence” requirement: E is the strongest proposition with Pnew(E) = 1. This requirement is the usual way that authors specify a unique privileged conditioning proposition (up to logical equivalence) among all those propositions that receive new credence 1. The total evidence requirement favors E rather than some weaker logical consequence of E as the conditioning proposition. (6) incorporates no total evidence requirement. Instead, (6) specifies a unique privileged conditioning proposition by invoking the causal structure of credal evolution. If the new credal assignment to E mediates the transition from Pold to Pnew, then no distinct proposition plays that same mediating role. (6) therefore pinpoints E rather than any weaker logical consequence as the appropriate conditioning proposition.
To capture the intuitive thought more fully, one must strengthen I.4 so as to prohibit any sequence of arrows from C to Y that bypasses X. This is one reason why I.1–I.4 are necessary but not sufficient for C = c to be an intervention on X with respect to Y.
In (Rescorla 2014), I argue that interventions shed light upon mental causation.
One can imagine an agent who automatically assigns credence 1 to any logical truth, without bothering to recompute that credence in light of new evidence. For such an agent, Pnew({E, ¬E}) is not causally relevant to Pnew(H) when H contains only logical truths. This alternative causal structure is also consistent with (10).
In the modified Jeffrey counterexample, D = 1 controls Pnew({E1, E2}), so Fig. 11 contains no arrow into Pnew({E1, E2}) from any variable other than D. One can also construct counterexamples in which D = 1 does not control Pnew({E1, E2}) but is still an actual cause of Pnew({E1, E2}) having the value it has. For reasons of space, I will not explore such counterexamples here.
Theorists who reject the selective approach may consider revising my definition of “mediation” in an effort to achieve similar results. One idea is to supplement M.2 with a further clause M.2.g along the following lines: If the causal link between D and Pnew(H) had been broken, so that D was no longer causally relevant to Pnew(H), and if Pold and Pnew(E) had had the same values as they in fact do, then Pnew(H) would not have had value α. The effect of M.2.g is to allow actual causes that make no real difference to Pnew(H). To illustrate, let D be a binary variable whose two values reflect whether you take a pill that sets Pnew(H) = β ≠ α (1 signifies that you take the pill; 0 signifies that you do not). Suppose that D = 0. Then D violates M.2.g: you could break D’s causal influence on Pnew(H) by ingesting a second pill that counteracts the credence-altering properties of the first pill, and doing so would not change Pnew(H) because you did not in fact ingest the first pill. Thus, even if we count D = 0 as an actual cause of Pnew(H) = α, the revised definition of “mediation” still allows us to say that the new credal assignment over E mediates the credal transition. I leave further exploration of such revisionary stratagems to those who reject the selective approach.
Williamson’s proof requires the assumption that, for all propositions H, Pnew(H) = 1 if PE (H) = 1. The assumption is satisfied here, since E is the strongest proposition F such that Pnew (F) = 1 and the strongest proposition F such that PE (F) = 1. Diaconis and Zabell (1982) prove a result along the same lines for probability measures over subsets of a countable outcome space. In this setting, the requisite partition {Ei} may be countably infinite.
Elsewhere, Joyce (2019) uses similar causal language to formulate Conditionalization. He says that you should conditionalize on E when you have “a learning experience whose sole immediate effect” is to render you certain of E. This formulation is relatively close to my own: it uses causal language, and it seems to be non-factive. However, it does not handle modified Jeffrey-style counterexamples along the lines of Fig. 11.
When explaining Jeffrey Conditionalization, Jeffrey (1983, p. 173) assumes that the change from Pold to Pnew “originates” in the partition propositions {Ei}. He then defines “origination” in terms of the invariance condition: Pold( . \(|\) Ei) = Pnew( . \(|\) Ei), for all i. Jeffrey’s discussion epitomizes how authors often recognize that causal structure is important yet balk at incorporating explicitly causal vocabulary into the official formulation of Conditionalization or Jeffrey Conditionalization.
Titelbaum calls this requirement “Limited Conditionalization.” He reserves the label “Conditionalization” simpliciter for a similar requirement that omits the subset clause Ck ⊆ Cj (p. 52). He also articulates a generalized requirement, “Generalized Conditionalization,” designed to cover the cases of memory loss and apparent memory loss discussed in Sect. 7 below. None of Titelbaum’s three formulations attempts to handle Jeffrey-style shifts. Thus, my criticism in the main text applies equally well to all three formulations.
Here we see a major advantage of my formulation over Meacham’s evidence-based formulation. Whatever conception of “evidence” one adopts, memory loss cases look like counterexamples to Meacham’s formulation. Take Talbott’s breakfast example, and let E be the cumulative evidence acquired between July 15, 2015 and July 15, 2016. Meacham’s formulation requires that your new credences on July 15, 2016 be given by conditionalizing on E. Conditionalizing on E would yield credence 1 in the proposition I ate granola for breakfast on July 15, 2015, which seems inappropriate given that you cannot remember what you ate for breakfast that day. Thus, Meacham’s approach yields implausible results in memory loss cases.
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Acknowledgements
I presented versions of this material at Carnegie Mellon University and Stanford University. I thank all the audience members present on those occasions, especially Dmitri Gallow, Clark Glymour, Thomas Icard, and Teddy Seidenfeld. I also thank Christopher Meacham and two anonymous referees for this journal for comments that improved the paper significantly.
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Rescorla, M. On the proper formulation of conditionalization. Synthese 198, 1935–1965 (2021). https://doi.org/10.1007/s11229-019-02179-9
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DOI: https://doi.org/10.1007/s11229-019-02179-9