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Representing credal imprecision: from sets of measures to hierarchical Bayesian models

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Abstract

The basic Bayesian model of credence states, where each individual’s belief state is represented by a single probability measure, has been criticized as psychologically implausible, unable to represent the intuitive distinction between precise and imprecise probabilities, and normatively unjustifiable due to a need to adopt arbitrary, unmotivated priors. These arguments are often used to motivate a model on which imprecise credal states are represented by sets of probability measures. I connect this debate with recent work in Bayesian cognitive science, where probabilistic models are typically provided with explicit hierarchical structure. Hierarchical Bayesian models are immune to many classic arguments against single-measure models. They represent grades of imprecision in probability assignments automatically, have strong psychological motivation, and can be normatively justified even when certain arbitrary decisions are required. In addition, hierarchical models show much more plausible learning behavior than flat representations in terms of sets of measures, which—on standard assumptions about update—rule out simple cases of learning from a starting point of total ignorance.

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Notes

  1. See, for example, Halpern (2003) and Joyce (2005, 2010).

  2. Note that the term “imprecise credences” is sometimes used to designate a specific formal model of belief based on sets of probability measures, rather than the phenomenon being modeled. To avoid confusion between model and the thing modeled, I will avoid the term “imprecise credences” altogether, using “credal imprecision” as a name for the phenomenon and “sets-of-measures” for the formal model under discussion.

  3. Representations based on probability intervals or on upper and lower probabilities can for present purposes be treated as a special case of sets-of-measures models.

  4. In addition to references just cited, and among many others: Spirtes et al. (1993), Pearl (2000), Glymour (2001), Woodward (2003), Sloman (2005), Koller and Friedman (2009), Russell and Norvig (2010), Goodman et al. (2016), Danks (2014) and Icard (2017).

  5. This is called “probabilistic dilation”: see Seidenfeld and Wasserman (1993), van Fraassen (2006) and White (2010). While this feature of sets-of-measures models is intuitively bizarre, Pedersen and Wheeler (2014) discuss important subtleties that may help to improve its plausibility.

  6. In general, conditioning a \({\text {Beta}}(a,b)\) prior on n heads/successes/wins and m tails/failures/losses yields a \({\text {Beta}}(a + n, b + m)\) posterior: see e.g. Griffiths et al. (2008) and Hoff (2009).

  7. A fourth way to deal with the inability of sets-of-measures models to allow serious learning from a starting point of ignorance is suggested by Rinard (2013): we can conclude that a precise formal model of belief states is not possible. This might well be correct, but it would be defeatist to draw this conclusion simply because sets-of-measures models cannot account for simple cases of inductive learning. In particular, the hierarchical approach that I will sketch momentarily gives us another reason not to abandon hope for a formal model of belief.

  8. The part that still hits home is the accusation that precise credence models give rise to “very specific inductive policies” which are not justified by evidence. This is closely related to the impossibility of assumption-free learning noted above, as well as the question of whether and how rules like the Principle of Indifference can be used to justify certain choices of priors. We will return to this issue in Sect. 6 below.

  9. For discussion of richer languages based on probabilistic programming principles that can describe hierarchical Bayesian models with uncertainty over individuals, properties, relations, etc., see for example Milch et al. (2007) and Goodman et al. (2008, 2016), Tenenbaum et al. (2011), Goodman and Lassiter (2015), Pfeffer (2016) and Icard (2017).

  10. The model is directly inspired by the Microsoft Trueskill system that is used to rank Xbox Live players in order to ensure engaging match-ups in online games: see Bishop (2013). It is conceptually close to the more complex tug-of-war model, with quantification and inference over individuals and their properties and relations, that is explored by Gerstenberg and Goodman (2012) and Goodman and Lassiter (2015).

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  1. Department of Linguistics, Stanford University, 450 Serra Mall, Building 460, Stanford, CA, 94305, USA

    Daniel Lassiter

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Lassiter, D. Representing credal imprecision: from sets of measures to hierarchical Bayesian models. Philos Stud 177, 1463–1485 (2020). https://doi.org/10.1007/s11098-019-01262-8

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