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Tests of Bayesian Rationality

What are the testable implications of the Bayesian rationality hypothesis? This paper argues that the absolute continuity of posteriors with respect to priors constitutes the entirety of the empirical content of this hypothesis. I consider a decision-maker who chooses a sequence of actions and an econometrician who observes the decision-maker's actions, but not her signals. The econometrician is interested in testing the hypothesis that the decision-maker follows Bayes' rule to update her belief. I show that without a priori knowledge of the set of models considered by the decision-maker, there are almost no observations that would lead the econometrician to conclude that the decision-maker is not Bayesian. The absolute continuity of posteriors with respect to priors remains the only implication of Bayesian rationality, even if the set of actions is sufficiently rich that the decision-maker's actions fully reveal her beliefs, and even if the econometrician observes a large number of ex ante identical agents who observe i.i.d. signals and face the same sequence of decision problems.


Introduction
Following the treatise of Savage (1972), the subjective (or Bayesian) theory of probability has become the dominant paradigm in the modeling of decision-making under uncertainty.
The dominance of this paradigm in economics is not unwarranted. It allows one to assign probabilities to unique (or rare) events. It has an elegant foundation in the study of rational choice under uncertainty. And it is appealing from a normative point of view-as Epstein and Le Breton (1993) proclaim, "dynamically consistent beliefs must be Bayesian." What is less clear is whether Bayesian rationality is a good positive model of individual behavior. To settle this question requires one to develop formal tests of Bayesian rationality.
A class of tests commonly used in the literature are those based on the martingale property of beliefs. A Bayesian agent's belief sequence is a martingale given the agent's own prior belief. Conversely, any sequence of beliefs that constitutes a martingale given some probability distribution ℙ can be rationalized as the belief sequence of a Bayesian agent with prior ℙ. This equivalence of Bayesian rationality and the martingale property of beliefs has been known by economists since at least Kamenica and Gentzkow (2009). 1 But although a Bayesian agent's beliefs constitute a martingale with respect to her subjective belief, operationalizing the martingale test requires one to sample individuals whose observations depend on an objective data-generating process (DGP). The martingale test is thus a valid test of Bayesian rationality only if agents have objectively correct beliefs about the DGP. Said differently, a group of agents could fail a martingale-based test of Bayesian rationality for one of two distinct reasons: (i) the agents may not be Bayesian, or (ii) their belief about the distribution of the signals they observe may not coincide with the true, objective DGP. This paper's main contribution is to characterize the empirical content of the theory of Bayesian rationality without predicating it on the a priori assumption that individuals' beliefs about the DGP are accurate. I consider an observer (or econometrician) who observes a sequence of decisions made by a decision-maker (DM). The econometrician (he) is wondering whether the observation of any decision sequence would lead him to conclude, with some certainty, that the decision-maker (she) is not Bayesian. The econometrician observes the DM's actions but does not know the full description of her probabilistic model of the world- 1 The necessity of the martingale property for Bayesian rationality is a trivial consequence of the law of iterated expectation. To the best of my knowledge, Kamenica and Gentzkow (2009) are the first in the economics literature to formalize the sufficiency of the martingale property for Bayesian rationality. Brooks, Frankel, and Kamenica (2021) show that when the agent has access to multiple information sources, beliefs must also satisfy Blackwell monotonicity, but not all beliefs that satisfy the martingale property and Blackwell monotonicity are consistent with Bayesian rationality.
including her belief about the DGP. He attempts to rationalize the sequence of decisions made by the DM by constructing a probabilistic model for the DM and a true DGP, given which the DM's observed choices are consistent with Bayesian rationality. If he is unable to do so, he rejects the DM's Bayesian rationality. The paper's main result is that the econometrician can rationalize almost any sequence of decisions by the DM in the manner just described.
One may object that two obstacles stand in the way of the econometrician's rejecting of the decision-maker's Bayesian rationality. First, the mapping from the DM's beliefs to her actions may not be known to the econometrician or be invertible. So the econometrician may not be able to identify the DM's preferences and her beliefs separately. Second, any seemingly irrational realization of the DM's belief sequence may result from observations by the DM that were unlikely ex ante.
I address these potential objections by giving the econometrician tools to overcome the obstacles outlined in the previous paragraph. First, I assume that the mapping from the DM's beliefs to her actions is invertible and known to the econometrician. This assumption allows the econometrician to identify the DM's beliefs simply by observing her actions. I additionally assume that the econometrician can directly observe the true, objective population distribution of the DM's belief sequence. This assumption represents the limit where the econometrician has access to the belief sequence of a large sample of ex ante identical decision-makers who observe independent signals drawn from a common distribution and who face identical decision problems. The observation of the population distribution allows the econometrician to overcome the second obstacle mentioned above.
The paper establishes that few observations would lead the econometrician to conclude that the DM is not Bayesian-despite the unrealistically powerful tools at his disposal. The only testable implication of the theory of Bayesian rationality is the absolute continuity of the posterior with respect to the prior for any prior-posterior pair that is realized with positive probability. The result holds even when the DM's prior belief about the payoff-relevant state agrees with its objective distribution, and it does not rely on the inapplicability of Bayes' rule after contingencies that were assigned zero probability by the DM's prior.
The result characterizes the empirical content of the theory of Bayesian rationalityabsent additional a priori assumptions on an individual's probabilistic model of the world.
It shows that, in isolation, Bayesian rationality has only weak testable predictions. This finding suggests imposing assumptions on what constitutes a reasonable model as a way of obtaining theories with more predictive power. Doing so leads to hybrid theories in which some aspects of individuals' models conform to objective reasonableness requirements while other dimensions of the models are subjective. I proceed by introducing two such a priori reasonableness assumptions, discussing the contexts in which they are likely to be satisfied, and characterizing their testable implications in conjunction with the assumption of Bayesian rationality. One such reasonableness assumption is that the DM's belief about the DGP conforms to the objective truth. This assumption is likely to be satisfied in lab experiments where the econometrician has complete control over the DGP and can communicate its parameters to the experiment subjects. The martingale property of belief sequences summarizes the empirical content of this theory.
The second reasonableness assumption is that a DM's decisions fully reflect her belief over the entire state space (and not a section of it). This assumption is likely to be satisfied whenever the econometrician can (i) plausibly constrain the set of informative signals observed by the decision-makers and (ii) elicit their beliefs about any event in their model of the world, including their subjective beliefs about the DGP. I show that such a theory imposes sharp restrictions on the prior-posterior pairs consistent with Bayesian rationality. In particular, the decision-maker's prior belief fully pins down her attainable set of Bayesian posteriors.
Related Literature. The paper most closely related to mine is by Shmaya and Yariv (2016), who also study the testable implications of Bayesian rationality. However, the two papers are different in their assumptions on the econometrician's a priori knowledge and their conclusions. Shmaya and Yariv focus on experimental settings where the econometrician has a priori knowledge of the decision-makers' set of signals. They show that for an observed belief sequence to be consistent with Bayesian rationality in such a setting, the prior must fall in the convex hull of the set of posteriors conditional on different signals. 2 In contrast, in this paper the econometrician cannot rule out the possibility that decision-makers observe private signals between the two periods. I show that then a violation of absolute continuity is the only obstruction to rationalizing an observed belief sequence. This paper also contributes to the extensive theoretical literature that considers deviations from rational expectations. This literature can be roughly divided into two strands. The first strand, such as Pouzo (2016, 2021), maintains the assumption of Bayesian updating but admits the possibility that agents hold misspecified priors. 3 The second strand of the literature studies the implications of non-Bayesian models of behavior such as representativeness and availability heuristics (Tversky and Kahneman, 1974), confirmation bias (Rabin and Schrag, 1999), and diagnostic expectations (Bordalo, Gennaioli, and Shleifer, 2018). 4 This paper's results clarify the relationship between these two strands of the literature by showing that almost any non-Bayesian updating rule is observationally equivalent to Bayesian updating given a misspecified prior about the DGP.

The Environment
There are two periods indexed by = 0, 1 and a fixed payoff-relevant state that belongs to the measurable space ( , S). There is a population of ex ante identical decision-makers The second actor is an econometrician who is interested in testing the hypothesis that the DMs are Bayesian. The econometrician knows what is described in the previous paragraph but does not know anything about the signals observed by the DMs between subsequent periods (if any). He observes the panel data {( * 0 , * 1 )} =1,..., , consisting of the actions of a sample of size of DMs in periods = 0, 1. He wants to devise a test that would allow him to reject the hypothesis that the DMs use Bayes' rule to update their beliefs between the two periods. The econometrician's task is challenging for a number of reasons. First, the econometrician's data might be a poor approximation to the true population distribution of actions due to sampling errors or correlated signals. Second, Bayesian rationality is a restriction on the evolution of the DMs' beliefs, but the econometrician only observes their actions, and he may not be able to recover the beliefs by observing the actions.
While important in practice, these issues are not particularly interesting for the purpose of this paper; neither are they unique to the problem of characterizing the empirical content of the theory of Bayesian rationality. I instead consider an ideal setting in which the econometrician has exceptionally rich data which allow him to overcome these challenges.

An Ideal Setting
I make three assumptions to allow the econometrician to overcome the challenges just discussed. First, I assume that the mapping from a DM's belief about the payoff-relevant state, * , to her action, * , is invertible and known by the econometrician. This assumption is equivalent to the assumption that the econometrician directly observes the DMs' beliefs about the payoff-relevant state. It allows me to abstract away from the question of whether preferences and beliefs can be identified separately.
Second, I assume that the signals realized between the two periods are independent and identically distributed across the DMs. This assumption implies that {( * 0 , * 1 )} =1,..., is a representative sample drawn from the true population distribution of the DMs' actions.
Finally, I assume that is large enough for the sampling error to be negligible. This assumption, together with the second assumption above, implies that the empirical distribution of actions well approximates the true population distribution from which the DMs' actions are drawn.
Note that these assumptions all make it easier for the econometrician to reject the DMs' Bayesian rationality. They only strengthen the negative results: if the econometrician cannot rule out the possibility that the DMs are Bayesian under these unrealistically strong assumptions, then a fortiori, he will not be able to rule out their Bayesian rationality in more realistic settings where these assumptions are violated.
Focusing on this ideal setting simplifies the econometrician's problem significantly. I can use the assumption of large and i.i.d. signals to replace the DMs with a single representative DM with random beliefs. Since the DMs are ex ante identical, the prior of the representative DM is drawn from a degenerate distribution with unit mass at some * 0 ∈ Δ . On the other hand, the DMs may observe different realizations of the signal between periods 0 and 1, and so, may end up with different posteriors. I let * 1 ∈ Δ(Δ ) denote the distribution of the representative DM's posterior. The assumption that the mapping from beliefs to actions is invertible and known to the econometrician implies that the econometrician can compute * 0 , * 1 by observing the DM's actions. Henceforth, I refer to the representative DM simply as the DM.

The Main Result
The econometrician's question is then which pairs * 0 , * 1 , consisting of the DM's prior and the probability distribution of her posterior about the payoff-relevant state, are consistent with Bayesian rationality. The paper's main result establishes that any such pair that satisfies the following condition is consistent with Bayesian rationality: Condition AC. For any * 1 in the support of * 1 , the probability distribution * 1 is absolutely continuous with respect to * 0 with an essentially bounded Radon-Nikodym derivative. 5 This is a weak condition. It reduces to absolute continuity if is a finite set. It is easy to see that absolute continuity is often necessary for * 0 , * 1 to be consistent with Bayesian rationality: if the prior of a Bayesian agent assigns zero probability to an event, her posterior must also assign zero probability to the event-regardless of the set of signals, the agent's belief about the distribution of the signals, and the true DGP. 6 More surprisingly, absolute continuity is sufficient for the observed pair * 0 , * 1 to be consistent with Bayesian rationality: Condition AC thus encompasses the entire empirical content of the hypothesis of Bayesian rationality. Absent additional a priori restrictions on what constitutes a reasonable model for the DM or the signals she observes, any belief sequence that satisfies Condition AC is consistent with the DM's Bayesian rationality. Note that Condition AC is only a restriction on the support of the distribution of posteriors, * 1 , and not on the probabilities of observing different posteriors in the support of * 1 . In contrast, rational expectations is a much tighter restriction on * 1 , requiring posteriors to average out to the prior, i.e., * 0 = The Radon-Nikodym derivative ≡ * 1 / * 0 is essentially bounded if there exists a constant < ∞ and a set ∈ S with * 0 ( ) = 1 such that ( ) ≤ for all ∈ . 6 To be precise, the absolute continuity condition is necessary for Bayesian rationality only when the agent's prior assigns positive probability to any signal that is realized with positive probability under the true DGP. In histories where the agent observes a signal that she had assumed to have zero probability, Bayes' rule is inapplicable and Bayesian posteriors are unrestricted. This is why I say that absolute continuity is often necessary.
In Section 4, I introduce the notation needed to formally define what it means for * 0 , * 1 to be consistent with Bayesian rationality. There, I also discuss some generalizations and limitations of the result. But first, it is useful to illustrate the result in the context of an example.

A Simple Example
The payoff-relevant state takes values in the set = { , }. The DM's observed prior about the payoff-relevant state is uniform over . The observed distribution of the posterior, * 1 , is as follows: with a one-quarter probability, the DM's belief that the state is goes up to 0.8; with the complementary probability, the DM becomes certain that the state is .
Should the observation of this belief sequence lead the econometrician to conclude that the DM is not Bayesian? The answer may seem to be yes at first. After all, this belief sequence does not constitute a martingale with respect to the DGP that generates the signals observed by the DM: the DM always becomes more confident in period = 1 that is the true state.
But the martingale test of Bayesian rationality requires the DM's belief to be a martingale only with respect to her subjective belief about the DGP (and not the true DGP).
The observed pair ( * 0 , * 1 ) is indeed consistent with the DM's Bayesian rationality. This conclusion follows Theorem 1 by noting that * 0 has full support over , and so, the pair ( * 0 , * 1 ) satisfies Condition AC. I illustrate how ( * 0 , * 1 ) can be rationalized by specifying the set of signals, the DM's belief about the DGP, and the true DGP in such a way that the belief se- It is easy to verify that this system does not have a solution.
That the DM's observed pair ( * 0 , * 1 ) satisfies Condition AC is sufficient to ensure that this system has a solution. One such solution is as follows:

Tests of Bayesian Rationality
In this section, I generalize the insights of the example discussed in the previous section by characterizing the testable implications of Bayesian rationality. The main challenge is to formally express what it means for a decision-maker to be subjectively rational when there is a true underlying data-generating process that determines the signals observed by the DM. I start by introducing a framework that combines elements of subjective and objective probabilities: the econometrician is interested in testing the hypothesis that the DM has an internally consistent subjective probability system but can only use samples drawn from an objective distribution. I then formally state the question of whether a belief sequence is consistent with Bayesian rationality.

Technical Assumptions
I maintain the following standard technical assumptions throughout the remainder of the paper. Every set is assumed to be a complete separable metric space that is endowed with its corresponding Borel sigma-algebra X. The set of probability distributions over ( , X) is denoted by Δ and is endowed with the topology of weak convergence and the corresponding Borel sigma-algebra, which I denote by B (Δ ). Finally, I assume that the DM's observed prior, * 0 , is a probability distribution over ( , S) and the observed distribution of the DM's posterior, * 1 , is a probability distribution over (Δ , B (Δ )).

Formalism
The underlying uncertainty faced by the DM can be captured by a measure space. Let (Ω, F ) be an abstract measurable space that captures all the uncertainty faced by the DM. Each ∈ Ω is a complete description of all the variables that could potentially affect the DM's decisions. It includes, at a minimum, the value of the payoff-relevant state , but it also includes the description of any signals that might be observed by the DM between the two periods. I refer to as the state of the world to contrast it with , the payoff-relevant state. 7 I let S( ) ∈ denote the value of the payoff-relevant state when the state of the world is given by , with S : Ω → a measurable mapping.
The DM's information can be represented by sigma-algebras F 0 ⊆ F 1 ⊆ F . The sigmaalgebra F captures the DM's information in period -the DM's posteriors (and actions) in period are measurable with respect to F . I assume without loss of generality that F 0 is the trivial sigma-algebra. Therefore, F 1 represents the information content of the signal observed by the DM between periods 0 and 1.
for any arbitrary event ∈ S. 8 The DM is Bayesian if her posterior about the state of the world is obtained from her prior by conditioning on the sigma-algebra F 1 . This requirement can be stated formally using the concept of regular conditional probability. A mapping 1 : Ω × F → [0, 1] is a subfield regular conditional probability for (Ω, F , 0 ) given F 1 if (i) the mapping ↦ → 1 ( , ) is F 1measurable for all ∈ F , (ii) 1 ( , ·) is a probability distribution over (Ω, F ) for every ∈ Ω, and (iii) for any ∈ F and ∈ F 1 . 9 The notion of regular conditional probability is the natural generalization of the elementary notion of conditional probability to probability distributions with uncountable supports. Condition (i) is the measurability requirement: the posteriors need to be the same conditional on any two states that are indistinguishable given F 1 . Condition (ii) is the requirement that posteriors are well-defined probability distributions. And condition (iii) is the appropriate statement of Bayes' rule for distributions with uncountable supports.
It is the internal consistency requirement that underpins any test of Bayesian updating.
The consistency condition (12) is not directly verifiable by the econometrician. The econometrician only observes the DM's prior and posterior over , but Bayesian rationality requires the DM's prior and posterior to satisfy the consistency requirement (12) over Ωand is in general only a slice of Ω. 10 Yet equation (12) induces a consistency requirement for the DM's belief over . More specifically, the regular conditional probability 1 and the random variable S : Ω → define a regular conditional probability 1 : Ω × S → [0, 1] as follows: for any ∈ Ω and ∈ S, Intuitively, 1 ( , ) is the DM's posterior belief that the payoff-relevant state belongs to set conditional on the event that the realized state of the world is .
While the DM's prior is a subjective probability distribution, the distribution of her posterior depends on the objective distribution of the signals she observes. Given a model (Ω, F , F 1 , S) and a prior 0 for the DM, equation (13) defines the DM's posterior belief about the value of the payoff-relevant state. This is a random variable whose realization depends on the signal observed by the DM. Therefore, the distribution of the DM's posterior depends on the objective distribution from which her signal is drawn. I let ℙ ∈ ΔΩ denote the objective probability distribution that determines the distribution of the signal observed by the DM.

Tests of Bayesian Rationality
I can now use the notation introduced in the previous subsection to formally state what it means for the DM's belief sequence to be consistent with Bayesian rationality.
Definition 1. A pair * 0 , * 1 of observations consisting of the DM's prior and the distribution of her posterior about the payoff-relevant state is consistent with Bayesian rationality given a model (Ω, F , F 1 , S), a subjective prior 0 , and an objective distribution ℙ if 0 = * 0 and = * , where 0 is defined in (11) and 1 is defined in (14).
The definition considers one extreme case where the econometrician has a priori knowledge about every aspect of the environment: the state space (specifying, among other things, the set of signals), the DM's prior on the entire state space, and the true DGP. Given this knowledge, the only distribution of posteriors that is consistent with the DM's Bayesian rationality and her subjective prior is the one defined in (14). I next consider the other extreme where the econometrician does not have any a priori knowledge about the environment. The question formalizes an intuitive scenario. The econometrician observes the prior * 0 and distribution * 1 of posteriors but has no a priori knowledge of the DM's model, her prior, or the true DGP. He chooses the tuple (Ω, F , F 1 , S, 0 , ℙ) in an attempt to explain his observation as resulting from Bayesian updating by a DM with model (Ω, F , F 1 , S) and prior 0 and given a true DGP described by ℙ. If he is unable to find a tuple (Ω, F , F 1 , S, 0 , ℙ) that explains his observation, he concludes that the DM is not Bayesian. Theorem 1 establishes that he can rationalize any observation that satisfies Condition AC.
The proof of the theorem is constructive. The econometrician constructs a large enough state space Ω, an objective distribution ℙ, and a subjective belief 0 for the DM under which the DM's prior about the probability of the payoff-relevant state coincides with * 0 and the distribution of her Bayesian posterior about coincides with * 1 . The construction requires the econometrician to postulate an objective probability distribution that is in general different from the subjective prior held by the agent.
But the econometrician observes the DM's prior belief about the payoff-relevant state.
Can requiring the objective distribution to respect the observed prior of the DM restrict the set of observations that are consistent with Bayesian rationality? The next theorem shows that the answer to this question is negative. The econometrician can rationalize any obser-vation that satisfies Condition AC using an objective distribution which agrees with the DM's prior about the distribution of the payoff-relevant state.
Before stating the theorem, I formally define what it means for an objective distribution ℙ to agree with the DM's prior about the payoff-relevant state. Recall that (Ω, F ) denotes the set of states of the world and S is the random variable that determines the value of the payoffrelevant state as a function of the state of the world. Given (Ω, F ) and S, the distribution of the payoff-relevant state implied by the objective distribution ℙ is given by for any arbitrary event ∈ S. Intuitively, if the econometrician only observes the dynamics of the DM's belief about the variables that belong to , then there is always a richer state space Ω that encodes more complex models of the world such that the observed belief dynamic over is rational given some subjective prior over Ω. The next result further refines Theorem 2 by establishing that the DM's model can be chosen independently of the observation * 0 , * 1 , which the econometrician is attempting to rationalize. Theorem 3. Given a set of payoff-relevant states ( , S), there exists a model (Ω, F , F 1 , S) for the DM such that any pair * 0 , * 1 satisfying Condition AC is consistent with Bayesian rationality given (Ω, F , F 1 , S, 0 , ℙ) for some subjective prior 0 and objective distribution ℙ.
The results presented so far cast doubts on the possibility of deciding whether decisionmakers are Bayesian in non-experimental settings. In such settings, an econometrician may be able to impute the DM's beliefs over some set . But there is no guarantee that any such set captures all the uncertainty that the DM believes to be relevant to what she thinks about ∈ . In particular, given any set such that the DM's belief on can be elicited by the econometrician, it may be the case that the relevant uncertainty (from the point of view of the DM) is captured by a larger set Ω. The theorems then show that, for any such set (capturing all that the econometrician can learn about the DM's beliefs), he can postulate a larger set Ω (capturing all the uncertainty that is relevant to the DM) such that almost any belief sequence over can be rationalized by fine-tuning the DM's beliefs over Ω.
Meanwhile, in experimental settings where the econometrician controls the DM's observations, he can elicit her belief over a set that can be plausibly assumed to be large enough to capture all the uncertainty that is relevant to the DM's decisions. In other words, in experimental settings the set Ω can be taken to be fixed and known to the econometrician.
Moreover, the econometrician can elicit the beliefs over the entire set Ω (so that coincides with Ω). Then there are relatively tight restrictions on the belief sequences that are consistent with Bayesian rationality. These restrictions are spelled out in the following proposition for the case where is finite and * 0 has full support. 11 Proposition 1. Suppose (Ω, F ) = ( , S) and S = id Ω , where is a finite set. Given a prior * 0 with full support and a distribution of posteriors * 1 , there exists some F 1 , 0 , and ℙ such that * 0 , * 1 is consistent with Bayesian rationality given (Ω, F , F 1 , S, 0 , ℙ) if and only if (i) supp ∩ supp = ∅ for any distinct , ∈ supp * 1 ; (ii) ( ) = * 0 ( | supp ) for any ∈ S and any ∈ supp * 1 .
The proposition clarifies the scope and logic of Theorems 1-3. The theorems rely on the assumption that the econometrician does not have any a priori knowledge about the DM's model of the world. He is thus free to postulate a model and a prior for the DM and a true DGP that jointly rationalize his observations. Proposition 1 shows how knowing the DM's model and observing what she believes about the DGP tie the econometrician's hand. 11 The extensions to the cases where is not finite or * 0 does not have full support are straightforward. But such extensions are awkward to state since they require care when dealing with zero probability events. I do not pursue those extensions here for the sake of exposition.

Concluding Remarks
The paper's main result is that the only testable prediction of Bayesian rationality is the absolute continuity of the posterior with respect to the prior for any prior-posterior pair that is realized with positive probability. The paper also shows that this negative conclusion can be overturned if the beliefs about the DGP itself can be elicited.
I conclude by discussing two modifications of the setup studied in the paper, which do not overturn the negative results. Suppose first that the DM's actions or beliefs are observed over a horizon that is longer than two periods. Theorems 1-3 trivially generalize to such a setting: as long as the econometrician does not have any a priori knowledge of the true DGP and does not observe the DM's subjective beliefs about the DGP, the only observations that are not consistent with Bayesian rationality are those that violate absolute continuity.
A perhaps more interesting modification of the setup of the paper is to assume that the econometrician observes some-but not all-of the signals observed by the DM and knows the conditional distributions of the observed signals. It remains the case that absolute continuity is the only testable prediction of Bayesian rationality. As long as the econometrician cannot rule out the possibility that the DM observes private signals unbeknown to him, he can reject her Bayesian rationality only if he observes a violation of absolute continuity.

Proof of Theorems 1-3
Since Theorems 1 and 2 are corollaries of Theorem 3, I only prove Theorem 3. Let Ω = × Δ × {+, −}, and let F denote the product sigma-algebra. A generic element of Ω is denoted by ( , ⋄ ), where is an element of , is a probability distribution over , and ⋄ ∈ {+, −}. Let S : Ω → be the canonical projection onto , that is, the mapping that maps ( , ⋄ ) to . Let F 1 be the smallest sub-sigma-algebra of F that makes all sets of the form × { } × {⋄} for ∈ Δ and ⋄ ∈ {+, −} measurable. That is, given the sigma-algebra F 1 , the DM learns the realized value of ⋄ between periods zero and one but learns nothing else about .
In the remainder of the proof, I fix an observed tuple ( * 0 , * 1 ) that satisfies Condition AC and show how it can be rationalized by the appropriate choice of the subjective prior 0 and the objective distribution ℙ (together with the tuple (Ω, F , S, F 1 ) chosen above). Note that while Ω, F , S, and F 1 are independent of ( * 0 , * 1 ), the probability distributions 0 and ℙ do depend on it. I specify 0 such that the DM's posterior belief over conditional on observing signal + is given by . I then choose ℙ such that the probability of signal + is consistent with the distribution of in the observed distribution of posteriors * 1 . I start by constructing the regular conditional probability (rcp) * 1 : Ω × S → [0, 1] that represents the DM's posterior belief about the payoff relevant state ∈ conditional on F 1 .
To complete the proof, I need to show that 0 = * 0 and that 1 is distributed according to * 1 given the objective prior ℙ, where 0 and 1 are defined in equations (11) and (13), respectively. By the law of total probability, for any set ∈ S, where in the third equality I am using the fact that * 1 ( ⋄ , ) = * 0 ( ) for all ∉ supp * 1 and in the fourth equality I am using (16) and the fact that * 1 ( + , ) = ( ) = * 0 ( ) for any set ∈ S whenever = 1. I next characterize the subfield rcp, 1 , for (Ω, F , 0 ) given F 1 , defined in (12). I do so by guessing some 1 and verifying that my guess is indeed the subfield rcp. For any = ( , ⋄ ) ∈ Ω and measurable sets ⊆ and Δ ⊆ Δ , let Since the sigma algebra F is generated by the sets of the form × Δ × {⋄} for ⋄ ∈ {+, −}, the above equations uniquely determine the probability measure 1 ( , ·) over (Ω, F ). Moreover, since the mapping ↦ → * 1 ( , ) is F 1 -measurable for any ∈ S, the mapping ↦ → 1 ( , ) is also F 1 -measurable for any ∈ F . Thus, to show that 1 is the subfield rcp for (Ω, F , 0 ) given F 1 , I only need to show that it satisfies (12). Since F is generated by sets of the form × Δ × {⋄} for ⋄ ∈ {+, −} and F 1 is generated by sets of the form × Δ × {⋄} for ⋄ ∈ {+, −}, I only need to check (12) for sets of the form = × Δ ×{ ⋄} and of the form = × Δ ×{ ⋄}.
If ⋄ ≠ ⋄, then both sides of (12) are trivially zero. So I assume that ⋄ = ⋄. I only present the argument for the case ⋄ = ⋄ = +; the case ⋄ = ⋄ = − is similar. By construction, ∫ where the first equality is by the definition of 1 , the third and fifth equalities are by the definition of 0 , and the fourth equality is using the fact that * 1 ( , ) = 1 for all ∈ Ω.
Finally, I show that 1 = * 1 and 1 = * 1 , where 1 is defined as in equation (13) and 1 is defined as in equation (14). That 1 = * 1 trivially follows the definition of 1 . On the other hand, under the objective prior ℙ, slices × { + } are distributed according to * 1 ( ) while × Δ × {−} has zero probability. Therefore, 1 as defined in (14) is equal to the observed distribution of posteriors * 1 . The proof is complete once I argue that ℙ(S −1 ( )) = * 0 ( ) for all sets ∈ S. But this is trivially true by construction.
In other words, the agent learns whether the state belongs to set 0 given the information revealed by F 1 in between periods 0 and 1. On the other hand, since (i) is a finite set, (ii) neither of 0 and ( ) is empty, and (iii) supp = 0 ∪ ( ), we have that ( 0 ) ∈ (0, 1). Thus, equation (17) implies that cannot be a realization of 1 , contradicting the assumption that ∈ supp * 1 = 1 .