Abstract
I begin by discussing a conundrum that arises when Bayesian models attempt to assess the relevance of one claim to another. I then explain how my formal modeling framework (the “Certainty Loss Framework”) manages this conundrum. Finally, I apply my modeling methodology to respond to Namjoong Kim’s objection to my framework.
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Notes
Let me emphasize that this problem is not unique to CLF—it is faced by every Bayesian modeling approach. In practice, Bayesians often begin writing down equations without explicitly specifying the modeling language they’re using, or how fine-grained they’ve decided to make their sets of possible worlds. But somewhere in there a decision must be made. CLF has the advantage of forcing a modeler to make her modeling language explicit before any modeling begins. This draws attention to the conundrum about judgments of relevance I’ve just described, but the problem was there all along.
Bostrom was working with a version of Sleeping Beauty like the one Kim calls Sleeping BeautyB.
Notice, by the way, that the initial Sleeping Beauty debate between Elga and Lewis was explicitly cast by Lewis as an argument over whether certain types of claims were relevant to Beauty’s confidence in Heads. This is exactly the kind of debate that CLF allows us to adjudicate on purely formal grounds, instead of weighing up relevance intuitions.
(Titelbaum 2013) also provides tools to assist in applying this methodology. In Chapter 8 I prove a series of results that limit the conditions under which adding sentences to a model’s language can undermine that model’s verdicts. For instance, if a claim never goes from less-than-certain to certain or vice versa for the agent over the course of a story, adding a claim representing that sentence will not alter extant verdicts (Theorem E.2 in Chapter 8). Results like this allow us to anticipate how most potential additions to a model’s language will alter (or fail to alter) that model’s verdicts without doing extensive further calculations.
Some technical details about how this would go: Construct a model \({M^{C2+}}\) that is an expansion of model \({M^{C2}}\), with MON the only atomic sentence added to the modeling language. At \(t_m\) Beauty assigns a nonextreme credence to MON, and there is no claim represented in the modeling language of \({M^{C2}}\) that she is certain at that time has the same truth-value as the claim represented by MON. So \({M^{C2}}\) is not a proper reduction of \({M^{C2+}}\), and not all the verdicts of \({M^{C2}}\) (or \({M^{C2-}}\)) will be verdicts of \({M^{C2+}}\). In particular, Kim’s verdict that
\(P_w^{C2-}(HEADS)= P_m^{C2-}(HEADS)\)
will have no anologue verdict in \({M^{C2+}}\).
Strictly speaking Kim says that I have to explain why Beauty has to take MON into consideration, not just argue that she needs to do so. While it would be nice to have explanations of the rational requirements in each story we analyze, if we’re simply looking to settle what rationality requires then Kim’s demand goes too far. On CLF’s methodology, the fact that adding MON changes the verdicts of \({M^{C2}}\) demonstrates that those verdicts should not be trusted. To respond to Kim’s charge that CLF produces contradictory results for Sleeping BeautyB and Sleeping BeautyC, I simply need to demonstrate that if the framework and its methodology are applied as stated in my (2013), the results obtained for the two stories are consistent.
Kim actually offers this point to demonstrate that a principle he calls “Universal Inclusion” is too strong. I don’t endorse Universal Inclusion; I will assess whether the same point could be used to show that the Multiple Models Principle is too strong.
Kim actually focuses on a sentence representing the claim “It is now 8 am on Monday.” But the models in question already include a sentence representing the claim “It is now Monday.” I will read “It is now 8 am on Monday” as a conjunction of the claim that it’s Monday and the claim that it’s 8 am, and will focus on whether a sentence representing this last claim should be added to the modeling language.
Judging from some of the suggestions in his (2012), Joel Pust may actually be such a person.
In the parlance of CLF, Beauty’s information about the determination of daily waking times would be represented in the model’s extrasystematic constraints.
Thanks to Namjoong Kim for an extended and extremely helpful correspondence about these issues.
References
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Titelbaum, M.G. Reply to Kim’s “Two Versions of Sleeping Beauty”. Erkenn 80, 1237–1243 (2015). https://doi.org/10.1007/s10670-015-9748-8
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DOI: https://doi.org/10.1007/s10670-015-9748-8