Empirical distributions of beliefs under imperfect observation

Empirical distributions of beliefs under imperfect observation — Mathematics of Operations Research, 31: 13-30, 2006
Authors

Olivier Gossner

Tristan Tomala

Abstract

Let (x_n) be a process with values in a finite set X and law P, and let y_n = f(x_n) be a function of the process. At stage n, the conditional distribution p_n = P[x_n |x_1 … x_{n−1}], element of Pi = Delta(X), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n. A statistician observing the signals y_1 …, y_n holds a belief e_n = P[p_n |x_1,…,x_n] ∈ Delta(Pi) on the possible predictions of the perfect observer. Given X and f, we characterize the set of limits of expected empirical distributions of the process e_n when P ranges over all possible laws of (x_n).

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Mathematics of Operations Research, 31: 13-30, 2006

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Citation

BibTeX citation:
@article{gossner2006,
  author = {Gossner, Olivier and Tomala, Tristan},
  title = {Empirical Distributions of Beliefs Under Imperfect
    Observation},
  journal = {Mathematics of Operations Research},
  date = {2006},
  url = {https://gossner.me/papers/empirical-distributions-of-beliefs-under-imperfect-observation.html},
  langid = {en},
  abstract = {Let (x\_n) be a process with values in a finite set X and
    law P, and let y\_n = f(x\_n) be a function of the process. At stage
    n, the conditional distribution p\_n = P{[}x\_n \textbar x\_1 …
    x\_\{n−1\}{]}, element of Pi = Delta(X), is the belief that a
    perfect observer, who observes the process online, holds on its
    realization at stage n. A statistician observing the signals y\_1 …,
    y\_n holds a belief e\_n = P{[}p\_n \textbar x\_1,…,x\_n{]} ∈
    Delta(Pi) on the possible predictions of the perfect observer. Given
    X and f, we characterize the set of limits of expected empirical
    distributions of the process e\_n when P ranges over all possible
    laws of (x\_n).}
}
For attribution, please cite this work as:
Gossner, Olivier, and Tristan Tomala. 2006. “Empirical Distributions of Beliefs Under Imperfect Observation.” Mathematics of Operations Research. https://gossner.me/papers/empirical-distributions-of-beliefs-under-imperfect-observation.html.