Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Empirical Content of Bayesian Updating under Misspecification

Pooya Molavi

Abstract

An agent is a misspecified Bayesian if she updates her belief using Bayes' rule given a subjective, possibly misspecified model of her signals. This paper shows that a belief sequence is consistent with misspecified Bayesian updating if and only if the set of posteriors admits a countable partition such that the prior contains a grain of the conditional average posterior on each cell. The condition imposes essentially no restrictions on posteriors given a full-support prior over a finite state space and reduces to a support inclusion condition on compact state spaces under mild regularity assumptions. However, it rules out posterior beliefs with heavier tails than the prior on unbounded state spaces. In Gaussian environments, it implies that posterior uncertainty cannot exceed prior uncertainty. The results delineate the boundary between updating rules that are observationally equivalent to Bayesian updating under misspecification and genuinely non-Bayesian rules. As an application, the paper shows that diagnostic expectations are consistent with misspecified Bayesianism, whereas some parameterizations of smooth diagnostic expectations are not.

The Empirical Content of Bayesian Updating under Misspecification

Abstract

An agent is a misspecified Bayesian if she updates her belief using Bayes' rule given a subjective, possibly misspecified model of her signals. This paper shows that a belief sequence is consistent with misspecified Bayesian updating if and only if the set of posteriors admits a countable partition such that the prior contains a grain of the conditional average posterior on each cell. The condition imposes essentially no restrictions on posteriors given a full-support prior over a finite state space and reduces to a support inclusion condition on compact state spaces under mild regularity assumptions. However, it rules out posterior beliefs with heavier tails than the prior on unbounded state spaces. In Gaussian environments, it implies that posterior uncertainty cannot exceed prior uncertainty. The results delineate the boundary between updating rules that are observationally equivalent to Bayesian updating under misspecification and genuinely non-Bayesian rules. As an application, the paper shows that diagnostic expectations are consistent with misspecified Bayesianism, whereas some parameterizations of smooth diagnostic expectations are not.

Paper Structure

This paper contains 22 sections, 14 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conceptual framework
  3. Setup
  4. Misspecified Bayesianism
  5. Characterization
  6. Main result
  7. Why the partition is necessary
  8. Implications of the grain condition
  9. Finite state spaces
  10. Compact state spaces
  11. Unbounded state spaces and tail restrictions
  12. Gaussian priors and Gaussian posteriors
  13. Application: Overreaction in macroeconomic expectations
  14. Diagnostic expectations
  15. Smooth diagnostic expectations
  16. ...and 7 more sections

Key Result

Proposition 1

For probability distributions $P$ and $Q$ defined over the same measurable space, the following are equivalent:

Figures (2)

  • Figure 1: The prior (left) and posterior (right). Each point mass in the right panel is a realization of the posterior; the dashed line shows the density of the location of those point masses.
  • Figure 2: The prior (left) and the conditional average posterior given a positive-$F_1$-measure cell of the partition of $\Delta(X)$ (right).

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3: *kalai1993rational
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • Definition 4
  • ...and 12 more