A subjectivist’s guide to deterministic chance

Abstract

I present an account of deterministic chance which builds upon the physico-mathematical approach to theorizing about deterministic chance known as the method of arbitrary functions. This approach promisingly yields deterministic probabilities which align with what we take the chances to be-it tells us that there is approximately a 1/2 probability of a spun roulette wheel stopping on black, and approximately a 1/2 probability of a flipped coin landing heads up-but it requires some probabilistic materials to work with. I contend that the right probabilistic materials are found in reasonable initial credence distributions. I note that, with some rather weak normative assumptions, the resulting account entails that deterministic chances obey a variant of Lewis's 'principal principle'. I additionally argue that deterministic chances, so understood, are capable of explaining long-run frequencies.

Be modest: you're living on the edge

“… See White 2006;Titelbaum 2010;Salow 2018. Gallow 2019b and Das 2020b discuss subtly different principles-the result of this paper does not directly apply to them.…”

“…At the very least, this has been the working assumption in most of the literature (e.g. Christensen 2010b;Cresto 2012;Lasonen-Aarnio 2013Horowitz 2014a;Sliwa and Horowitz 2015;Schoenfield 2016a;Roush 2016Roush , 2017Salow 2018;Gallow 2019b;Das 2020a,b).…”

Deference Done Better

“…Since Reflection implies that the frame is immodest, 𝔼 𝑤 (𝑆) = 𝔼 𝑤 (𝑆 𝑤 ) ≥ 𝔼 𝑤 (𝑂), meaning that the above sum is at least as great at Again see Skyrms (1990); Huttegger (2014). 18 This frame also shows that Value does not entail another common weakening of Reflection-namely, that your credence must equal your best estimate of the expert's credence: 𝜋(𝑞) = 𝔼 𝜋 (𝑃(𝑞)) (Ismael 2008(Ismael , 2015Salow 2018Salow , 2019Gallow 2019b). For 𝜋(𝑎) = 0.5, yet 𝔼 𝜋 (𝑃(𝑎)) = 𝜋(𝑎) ⋅ 0.9 + 𝜋(𝑏) ⋅ 0.2 = 0.55.…”

“…those that say that the expert's credence in a given 𝑞 is within a given range [𝑙, ℎ]. This leads to a proposition-level formulation that is only very slightly weaker than Reflection as formulated above: 𝜋(𝑞|𝑃(𝑞) ∈ [𝑙, ℎ]) ∈ [𝑙, ℎ] (see Gallow 2017). Meanwhile, the set of biconvex proposition-level conditions are those of the form 𝑃 ∈ {𝜌 ∶ 𝜌(𝑞) ≥ 𝑡}, i.e.…”

How chance explains

“…One popular version of this view is Quantum Bayesianism, 21 which holds that the quantum state represents the credences of agent, and that the apparent collapse when a measurement occurs is best understood as a form of conditionalization. Finally, the method of arbitrary functions takes objective chances to be subjective credences "objectified" by deterministic processes (Myrvold, 2010;Rosenthal, 2009, Rosenthal 2016, Gallow, 2021.…”