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Sheaves of Probability

Owen D. Biesel

Abstract

What does it mean for multiple agents' credence functions to be consistent with each other, if the agents have distinct but overlapping sets of evidence? Mathematical philosopher Michael Titelbaum's rule, called Generalized Conditionalization (GC), sensibly requires each pair of agents to acquire identical credences if they updated on each other's evidence. However, GC allows for paradoxical arrangements of agent credences that we would not like to call consistent. We interpret GC as a gluing condition in the context of sheaf theory, and show that if we further assume that the agents' evidence is logically consistent then the sheaf condition is satisfied and the paradoxes are resolved.

Sheaves of Probability

Abstract

What does it mean for multiple agents' credence functions to be consistent with each other, if the agents have distinct but overlapping sets of evidence? Mathematical philosopher Michael Titelbaum's rule, called Generalized Conditionalization (GC), sensibly requires each pair of agents to acquire identical credences if they updated on each other's evidence. However, GC allows for paradoxical arrangements of agent credences that we would not like to call consistent. We interpret GC as a gluing condition in the context of sheaf theory, and show that if we further assume that the agents' evidence is logically consistent then the sheaf condition is satisfied and the paradoxes are resolved.
Paper Structure (5 sections, 4 theorems, 12 equations)

This paper contains 5 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Sheaves on Measurable Spaces
  3. Sheaves of Measures
  4. Sheaves of Probability
  5. Conclusion and Further Directions

Key Result

Theorem 4

Let each agents $i\in\{1,\dots,n\}$ have credence given by probability measure $P_i$ on a set $A_i\subseteq X$. Suppose that $P_i|_{A_i\cap A_j} = P_j|_{A_i\cap A_j}$ for all $i,j\in\{1,\dots,n\}$, and that there exists a set $E\subseteq \bigcap_{i=1}^n A_i$ such that each $P_i(E)>0$. Then there exi

Theorems & Definitions (14)

  • Definition 1
  • Example 2
  • Example 3
  • Theorem 4: proven as \ref{['prob_e']}
  • Definition 5
  • Definition 6
  • Remark 7
  • Theorem 8
  • proof
  • Corollary 9
  • ...and 4 more