Coarse Bayesian Updating

This paper introduces and axiomatically characterizes Coarse Bayesian updating, a generalization of Bayes’ rule designed to accommodate the wide empirical evidence that individuals systematically deviate from standard Bayesian belief revision. The research question is: what is the minimal, tractable, axiomatically grounded generalization of Bayes’ rule that can accommodate heterogeneous non-Bayesian behaviors — including under-reaction, over-reaction, asymmetric updating, limited perception, and motivated reasoning — while remaining portable to standard economic settings?

The paper takes as primitive a finite state space Omega = {1, …, N} and an updating rule mu: S -> Delta assigning posterior beliefs to signals, where signals represent likelihood profiles from stochastic information structures. No data are used; the methodology is axiomatic decision theory combined with analysis of the model’s implications in static, dynamic, and decision-theoretic settings.

A Coarse Bayesian agent is characterized by (i) a partition of the probability simplex Delta into convex cells, and (ii) a representative distribution for each cell, one of which is the prior. Upon observing a signal, the agent determines which cell contains the Bayesian posterior and adopts the representative of that cell as his posterior belief. The agent need not point-identify the Bayesian posterior; he merely approximates it by identifying which cell it belongs to.

The central characterization result (Theorem 1) establishes that an updating rule has a Coarse Bayesian representation if and only if it satisfies three axioms: Homogeneity (beliefs depend only on likelihood ratios of the signal, not its scale), Cognizance (if two signals induce the same belief, then a garbled signal indicating one of them was generated also induces that belief), and Confirmation (if a signal is perfect evidence of some feasible belief, the agent adopts that belief). The representation — partition, representative points, and prior — is unique.

Proposition 1 shows that, under mild regularity conditions, strengthening any of the three axioms to an if-and-only-if form forces the agent to be perfectly Bayesian. This identifies the Coarse Bayesian framework as a qualitatively small but substantively rich departure from Bayes’ rule. The converse statements identify three necessary non-Bayesian behaviors exhibited by any proper Coarse Bayesian: (i) treating some signals as equivalent when a Bayesian would not; (ii) collapsing to a default belief when uncertain between two signals the agent would otherwise distinguish; (iii) false extrapolation — arriving at a belief via signals that are not perfect evidence of it.

In dynamic settings, Pooled Coarse Bayesian rules (which apply the full signal history at each period) are invariant to signal ordering and pooling and converge whenever Bayesian beliefs do, though to the representative point of the cell containing the true state rather than the true state itself. Sequential Signal Distortion rules are invariant to signal ordering but not pooling, and beliefs converge almost surely — but not necessarily to the true state (Example 1 illustrates convergence to the wrong state in a two-state setting). Sequential Coarse Bayesian rules need not satisfy either form of path-independence and need not converge at all.

In the decision-theoretic application (Section 4), a Coarse Bayesian’s value of information is posterior-separable and generally violates the Blackwell (1951) information ordering — more informative experiments need not be valued more highly. Two Coarse Bayesians are shown to be identical (same cells and representative points) if and only if they benefit from the same Blackwell improvements, providing a behavioral identification result. Agents with finer partitions are more sophisticated (higher ex-ante value of information), while agents with larger distortions from Bayesian posteriors are more biased (larger worst-case losses relative to a Bayesian). Neither greater sophistication nor lower bias implies being better off at all menus or signal realizations.

Q: What are the three axioms that characterize Coarse Bayesian updating, and what property of Bayes’ rule does each capture? A: Homogeneity requires that beliefs depend only on likelihood ratios of the signal — if two signals are proportional (s ~ t), they induce the same posterior. Cognizance requires that if two signals induce the same belief, then a garbled signal indicating that one of them was generated also induces that belief (mu_{s+t} = mu_s when mu_s = mu_t). Confirmation requires that if a signal is perfect evidence of some feasible belief — i.e., the Bayesian posterior at that signal equals a candidate belief — then the agent adopts that belief. Each axiom is satisfied by standard Bayesian updating.

Q: In what sense is Coarse Bayesian updating a “small” departure from Bayes’ rule? A: Proposition 1 establishes that strengthening any one of the three axioms to an if-and-only-if form forces the agent to be perfectly Bayesian. The converses are: (i) different likelihood ratios lead to different posteriors; (ii) if a garbled signal does not change beliefs, then the two signals must induce the same belief individually; (iii) if a signal induces the same posterior as another, then it must be perfect evidence of that posterior. Any Coarse Bayesian satisfying any one of these is in fact perfectly Bayesian, meaning the three axioms together come very close to fully characterizing Bayesian rationality.

Q: What non-Bayesian behaviors does the model generate as special cases? A: The framework generates under-reaction (representative points of cells close to the prior boundary), over-reaction (representative points at the far boundary), asymmetric updating (favoring one state, making upward revision easier than downward), limited perception (the agent retains the prior unless the Bayesian posterior is sufficiently far from the prior), extreme-belief aversion (the agent applies Bayes’ rule except when posteriors are near degenerate distributions), and reactions to unexpected news (non-Bayesian behavior only when signals have low prior probability). In each case the Coarse Bayesian Representation provides an axiomatic foundation via Axioms 1–3.

Q: What are the three necessary non-Bayesian behaviors exhibited by any proper (non-Bayesian) Coarse Bayesian? A: These follow from the negations of properties (i)-(iii) in Proposition 1. First, there exist signals s and t that are not proportional yet induce the same posterior — the agent treats informationally distinct signals as equivalent. Second, there exist signals s and t such that mu_s ≠ mu_t but mu_{s+t} = mu_s — signals the agent distinguishes individually collapse to a default when the agent is uncertain which one was generated. Third, there exist signals s and t with mu_s = mu_t where t is not perfect evidence of mu_s — a form of false extrapolation. Together, these three biases account for all non-Bayesian behavior the model generates.

Q: How does the model accommodate globally uniform biases like always-under-reaction, and how common does it predict such behavior to be? A: Global under-reaction requires representative points of cells to sit on their cell boundaries (as close to the prior as possible given the partition). This is a non-generic, hairline case — representative points generically lie in the interior of their cells, so a typical Coarse Bayesian under-reacts to some signals and over-reacts to others depending on which cell the Bayesian posterior falls into. The model additionally predicts local stability: if an agent over-reacts to signal s, nearby signals typically produce the same response; if an agent is Bayesian at s, nearby signals are almost surely also Bayesian.

Q: What does the model imply about dynamic updating under sequential signal-by-signal processing versus pooled processing? A: Pooled Coarse Bayesian rules apply the full signal history at each period, are invariant to both signal ordering and signal pooling, and converge almost surely whenever Bayesian beliefs converge — but to the representative point of the cell containing the true state, not necessarily the true state itself. Sequential Signal Distortion rules are invariant to signal ordering but not signal pooling, and also yield almost-sure convergence though potentially to the wrong state (Example 1 shows this for a two-state setting). Sequential Coarse Bayesian rules need not be invariant to either form of path-dependence and need not converge at all.

Q: How does the paper provide a behavioral identification of the model’s parameters? A: Theorem 1 establishes that the partition, representative points, and prior are uniquely determined by the agent’s updating rule alone — they are identifiable from observable updating behavior without additional assumptions. In the decision-theoretic setting of Section 4, a stronger result holds: two Coarse Bayesians are identical (same cells and same representative points) if and only if they benefit from the same Blackwell improvements across all menus (decision problems). This means the model’s parameters can be uniquely identified from menu-contingent rankings of Blackwell-comparable experiments.

Q: Does the Coarse Bayesian framework respect the Blackwell information ordering, and what characterizes when Blackwell improvements are beneficial? A: Unlike Bayesians, Coarse Bayesians typically violate the Blackwell ordering — they need not assign higher ex-ante value to more informative experiments. The paper characterizes the menus (decision problems) for which a given Coarse Bayesian benefits from Blackwell improvements, and shows this characterization runs deep: the complete set of such menus fully identifies the agent’s representation.

Q: How do the sophistication and bias orderings relate to welfare? A: An agent is more sophisticated if he employs a finer partition; more-sophisticated agents have a higher ex-ante value of information. An agent is more biased if his updating rule exhibits larger distortions from Bayesian posteriors; greater bias is characterized by greater worst-case losses relative to a Bayesian. Crucially, neither greater sophistication nor lower bias implies the agent is better off at all menus or signal realizations — welfare improvements require the agent to be perfectly Bayesian on a strictly larger set of signal realizations, giving rise to a third ordering that jointly refines the other two.

Q: How does the model relate to Wilson (2014) and Ortoleva (2012)? A: Wilson (2014) studies optimal updating for a boundedly rational agent with K memory states over binary decisions: each memory state is associated with a convex set of posteriors and a representative, so the optimal protocol is a dynamic Coarse Bayesian updating procedure. However, Wilson’s parameters are endogenous (determined by signal structure, stakes, and the bound K), whereas Coarse Bayesian updating does not require optimality or a bound on the number of cells — the model can accommodate behavior (e.g., Bayesian updating except at “extreme” signals) that Wilson’s model cannot. Ortoleva’s (2012) Hypothesis Testing model applies Bayes’ rule when the prior probability of a signal exceeds a threshold epsilon and otherwise uses a maximum-likelihood criterion; Coarse Bayesian updating can accommodate similar behavior, and the paper shows that Coarse Bayesian rules can be expressed as Maximum-Likelihood rules when there are only two states, but neither class subsumes the other in general — Maximum-Likelihood rules may violate Confirmation.

Q: What are the main limitations of the Coarse Bayesian framework? A: The paper identifies four. First, only likelihood ratios of the realized signal matter — sensitivity to framing and extraneous environmental features are ruled out. Second, beliefs must be probability distributions, so phenomena like the conjunction fallacy (where subjects assign higher probability to a conjunction than a component event) are outside the model’s scope. Third, the model exhibits discontinuities when signal perturbations move the Bayesian posterior across a cell boundary — a feature shared with Wilson (2014), Ortoleva (2012), and related models. Fourth, cells must be convex (driven by Cognizance); dropping Cognizance allows non-convex cells but removes the normative foundation that agents correctly forecast their own updating behavior.

Coarse Bayesian Representation: A pair consisting of a partition P of the probability simplex Delta into convex cells and a profile of representative distributions (one per cell, including the prior), such that the agent’s posterior after observing signal s equals the representative of the cell containing the Bayesian posterior B(mu_e|s).

Homogeneity: The axiom that if two signals are proportional (s ~ t, meaning s = lambda*t for some lambda > 0), they induce the same posterior belief — updating depends only on likelihood ratios, not signal scale.

Cognizance: The axiom that if signals s and t induce the same posterior, then the garbled signal s+t (indicating that either s or t was generated) also induces that belief — the agent correctly forecasts his own updating behavior.

Confirmation: The axiom that if a signal constitutes perfect evidence of some feasible belief (i.e., the Bayesian posterior equals a candidate belief), the agent adopts that belief — candidate beliefs are adopted when the signal confirms them exactly.

Signal Distortion Representation: An equivalent representation of Coarse Bayesian behavior as a function d: S -> S that distorts signals before Bayesian updating is applied (mu_s = B(mu_e|d(s))), satisfying properties analogous to the three axioms; equivalent to the partition representation in static settings but distinct in dynamic settings.

Blackwell Information Ordering: The partial order on experiments under which sigma is more informative than sigma’ if sigma can be obtained from sigma’ by a garbling; Bayesians always weakly prefer more informative experiments in this ordering, but Coarse Bayesians typically do not.

Sophistication Ordering: The partial order under which one Coarse Bayesian is more sophisticated than another if he employs a finer partition; more-sophisticated agents exhibit greater responsiveness to information as measured by ex-ante value of information.

Bias Ordering: The partial order under which one Coarse Bayesian is more biased than another if his updating rule exhibits larger distortions away from Bayesian posteriors; greater bias is characterized by larger worst-case losses relative to a Bayesian benchmark.