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Predicting the Unpredictable under Subjective Expected Utility

Burkhard C. Schipper

TL;DR

The paper develops a delabeled subjective expected utility framework to model how a decision maker forms beliefs about novelty under unawareness, using partitions of sampling times and partition exchangeability. It characterizes predictive probabilities that align with the two-parameter Pitman–Zabell model (parameters $\alpha$ and $\theta$) and shows how these parameters can be inferred from choices, with connections to the Ewens sampling formula and De Morgan's rule. It then analyzes reverse Bayesianism, showing that certain partition-exchangeable models exhibit reverse, Bayesian, and extended Bayesian invariance under learning, while others like Kuipers' rule do not. Finally, it demonstrates how these beliefs can be interpreted within unawareness structures and discusses normative, empirical, and extension potential, linking decision theory with species-discovery literature and population genetics.

Abstract

We consider a decision maker who is unaware of objects to be sampled and thus cannot form beliefs about the occurrence of particular objects. Ex ante she can form beliefs about the occurrence of novelty and the frequencies of yet to be known objects. Conditional on any sampled objects, she can also form beliefs about the next object being novel or being one of the previously sampled objects. We characterize behaviorally such beliefs under subjective expected utility. In doing so, we relate "reverse" Bayesianism, a central property in the literature on decision making under growing awareness, with exchangeable random partitions, the central property in the literature on the discovery of species problem and mutations in statistics, combinatorial probability theory, and population genetics. Partition exchangeable beliefs do not necessarily satisfy "reverse" Bayesianism. Yet, the most prominent models of exchangeable random partitions, the model by De Morgan (1838), the one parameter model of Ewens (1972), and the two parameter model of Pitman (1995) and Zabell (1997), do satisfy "reverse" Bayesianism. Our characterization allows us to interpret these models as subjective beliefs of a decision maker and to derive the parameters from choice behavior.

Predicting the Unpredictable under Subjective Expected Utility

TL;DR

The paper develops a delabeled subjective expected utility framework to model how a decision maker forms beliefs about novelty under unawareness, using partitions of sampling times and partition exchangeability. It characterizes predictive probabilities that align with the two-parameter Pitman–Zabell model (parameters and ) and shows how these parameters can be inferred from choices, with connections to the Ewens sampling formula and De Morgan's rule. It then analyzes reverse Bayesianism, showing that certain partition-exchangeable models exhibit reverse, Bayesian, and extended Bayesian invariance under learning, while others like Kuipers' rule do not. Finally, it demonstrates how these beliefs can be interpreted within unawareness structures and discusses normative, empirical, and extension potential, linking decision theory with species-discovery literature and population genetics.

Abstract

We consider a decision maker who is unaware of objects to be sampled and thus cannot form beliefs about the occurrence of particular objects. Ex ante she can form beliefs about the occurrence of novelty and the frequencies of yet to be known objects. Conditional on any sampled objects, she can also form beliefs about the next object being novel or being one of the previously sampled objects. We characterize behaviorally such beliefs under subjective expected utility. In doing so, we relate "reverse" Bayesianism, a central property in the literature on decision making under growing awareness, with exchangeable random partitions, the central property in the literature on the discovery of species problem and mutations in statistics, combinatorial probability theory, and population genetics. Partition exchangeable beliefs do not necessarily satisfy "reverse" Bayesianism. Yet, the most prominent models of exchangeable random partitions, the model by De Morgan (1838), the one parameter model of Ewens (1972), and the two parameter model of Pitman (1995) and Zabell (1997), do satisfy "reverse" Bayesianism. Our characterization allows us to interpret these models as subjective beliefs of a decision maker and to derive the parameters from choice behavior.
Paper Structure (8 sections, 11 theorems, 60 equations, 4 figures)

This paper contains 8 sections, 11 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. "Delabeled" Subjective Expected Utility
  3. Partition Exchangeability
  4. "Reverse" Bayesianism
  5. Application to Unawareness Structures
  6. Discussion
  7. Proofs
  8. SEU Representation

Key Result

Proposition 1

Let $\left\{\succeq_{\bm{\pi}^T}\right\}_{\bm{\pi}^T \in \Pi^T, T \geq 1}$ satisfy Assumption assSEU and denote by $\left(\mu(\cdot \mid [\bm{\pi}^T])\right)_{\bm{\pi}^T \in \Pi^T, T \geq 1}$ the associated partition-conditional probability measures. Preferences $\left\{\succeq_{\bm{\pi}^T}\right\}_

Figures (4)

  • Figure 1: Partition of Sampling Times till $T = 4$
  • Figure 2: Partition Exchangeability Does Not Imply "Extended" Bayesianism
  • Figure 3: Partition Exchangeability Does Not Imply "Reverse" Bayesianism
  • Figure 4: Example of an Unawareness Structure with Predictive Probabilities Implied by the Two-Parameter Model

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Proposition 2
  • Proposition 3
  • Corollary 3
  • Lemma 1: Zabell, 1997
  • Definition 1: Countable-Additive Subjective Expected Utility
  • ...and 3 more