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A Bayesian agent is observing a sequence of outcomes in {\{S,F\}}. The agent does not know the outcomes in advance, so he forms some belief {\mu} over sequences of outcomes. Suppose that the agent believes that the number {d} of successes in {k} consecutive outcomes is distributed uniformly in {\{0,1,\dots k\}} and that all configuration with {d} successes are equally likely:

\displaystyle \mu\left(a_0,a_1,\dots,a_{k-1} \right)=\frac{1}{(k+1)\cdot {\binom{k}{d}}}

for every {a_0,a_1,\dots,a_{k-1}\in \{S,F\}} where {d=\#\{0\le i<k|a_i=S\}}.

You have seen this belief {\mu} already though maybe not in this form. It is a belief of an agent who tosses an i.i.d. coin and has some uncertainty over the parameter of the coin, given by a uniform distribution over {[0,1]}.

In this post I am gonna make a fuss about the fact that as time goes by the agent learns the parameter of the coin. The word `learning’ has several legitimate formalizations and today I am talking about the oldest and probably the most important one — consistency of posterior beliefs. My focus is somewhat different from that of textbooks because 1) As in the first paragraph, my starting point is the belief {\mu} about outcome sequences, before there are any parameters and 2) I emphasize some aspects of consistency which are unsatisfactory in the sense that they don’t really capture our intuition about learning. Of course this is all part of the grand marketing campaign for my paper with Nabil, which uses a different notion of learning, so this discussion of consistency is a bit of a sidetrack. But I have already came across some VIP who i suspect was unaware of the distinction between different formulations of learning, and it wasn’t easy to treat his cocky blabbering in a respectful way. So it’s better to start with the basics.

Let {A} be a finite set of outcomes. Let {\mu\in\Delta(A^\mathbb{N})} be a belief over the set {A^\mathbb{N}} of infinite sequences of outcomes, also called realizations. A decomposition of {\mu} is given by a set {\Theta} of parameters, a belief {\lambda} over {\Theta}, and, for every {\theta\in\Theta} a belief {\mu_\theta} over {A^\mathbb{N}} such that {\mu=\int \mu_\theta~\lambda(\mathrm{d}\theta)}. The integral in the definition means that the agent can think about the process as a two stages randomization: First a parameter {\theta} is drawn according to {\lambda} and then a realization {\omega} is drawn according to {\mu_\theta}. Thus, a decomposition captures a certain way in which a Bayesian agent arranges his belief. Of course every belief {\mu} admits many decompositions. The extreme decompositions are:

  • The Trivial Decomposition. Take {\Theta=\{\bar\theta\}} and {\mu_{\bar\theta}=\mu}.
  • Dirac’s Decomposition. Take {\Theta=A^\mathbb{N}} and {\lambda=\mu}. A “parameter” in this case is a measure {\delta_\omega} that assigns probability 1 to the realization {\omega}.

Not all decompositions are equally exciting. We are looking for decompositions in which the parameter {\theta} captures some `fundamental property’ of the process. The two extreme cases mentioned above are usually unsatisfactory in this sense. In Dirac’s decomposition, there are as many parameters as there are realizations; parameters simply copy realizations. In the trivial decomposition, there is a single parameter and thus cannot discriminate between different interesting properties. For stationary process, there is a natural decomposition in which the parameters distinguish between fundamental properties of the process. This is the ergodic decomposition, according to which the parameters are the ergodic beliefs. Recall that in this decomposition, a parameter captures the empirical distribution of blocks of outcomes in the infinite realization.

So what about learning ? While observing a process, a Bayesian agent will update his belief about the parameter. We denote by {\lambda_n\left(a_0,\dots,a_{n-1}\right)\in\Delta(\Theta)} the posterior belief about the parameter {\theta} at the beginning of period {n} after observing the outcome sequence {a_0,\dots,a_{n-1}}. The notion of learning I want to talk about in this post is that this belief converges to a belief that is concentrated on the true parameter {\theta}. The example you should have in mind is the coin toss example I started with: while observing the outcomes of the coin the agent becomes more and more certain about the true parameter of the coin, which means his posterior belief becomes concentrated around a belief that gives probability {1} to the true parameter.

Definition 1 A decomposition of {\mu} is consistent if for {\lambda}-almost every {\theta} it holds that

\displaystyle \lambda_n\left(a_0,\dots,a_{n-1}\right)\xrightarrow[n\rightarrow\infty]{w}\delta_\theta

for {\mu_\theta}-almost every realization {\omega=(a_0,a_1,\dots)}.

In this definition, {\delta_\theta} is Dirac atomic measure on {\theta} and the convergence is weak convergence of probability measures. No big deal if you don’t know what it means since it is exactly what you expect.

So, we have a notion of learning, and a seminal theorem of L.J. Doob (more on that below) implies that the ergodic decomposition of every stationary process is consistent. While this is not something that you will read in most textbooks (more on that below too), it is still well known. Why do Nabil and I dig further into the issue of learnability of the ergodic decomposition ? Two reasons. First, one has to write papers. Second, there is something unsatisfactory about the concept of consistency as a formalization of learning. To see why, consider the belief {\mu} that outcomes are i.i.d. with probability {1/2} for success. This belief is ergodic, so from the perspective of the ergodic decomposition the agent `knows the process’ and there is nothing else to learn. But let’s look at Dirac’s decomposition instead of the ergodic decomposition. Then the parameter space equals the space of all realizations. Suppose the true parameter (=realization) is {\omega^\ast=(a_0,a_1,\dots)}, then after observing the first {n} outcomes of the process the agent’s posterior belief about the parameter is concentrated on all {\omega} that agrees with {\omega^\ast} on the first {n} coordinates. These posterior beliefs converge to {\delta_{\omega^\ast}}, so that Dirac decomposition is also consistent ! We may say that we learn the parameter, but “learning the parameter” in this environment is just recording the past. The agent does not gain any new insight about the future of the process from learning the parameter.

In my next post I will talk about other notions of learning, originating in a seminal paper of Blackwell and Dubins, which capture the idea that an agent who learns a parameter can make predictions as if he new the parameter. Let me also say here that this post and the following ones are much influenced by a paper of Jackson, Kalai, Smorodinsky. I will talk more on that paper in another post.

For the rest of this post I am going to make some comments about Bayesian consistency which, though again standard, I don’t usually see them in textbooks. Especially I don’t know of a reference for the version of Doob’s Theorem which I give below, so if any reader can give me such a reference it will be helpful.

First, you may wonder whether every decomposition is consistent. The answer is no. For a trivial example, take a situation where {\mu_\theta} are the same for every {\theta}. More generally troubles arrise when the realization does not pin down the parameter. Formally, let us say that function {f:\Omega\rightarrow \Theta} pins down or identifies the parameter if

\displaystyle \mu_\theta\left(\{\omega: f(\omega)=\theta\}\right)=1

for {\lambda}-almost every {\theta}. If such {f} exists then the decomposition is identifiable.

We have the following

Theorem 2 (Doob’s Theorem) A decomposition is identifiable if and only if it is consistent.

The `if’ part follows immediately from the definitions. The `only if’ part is deep, but not difficult: it follows immediately from the martingale convergence theorem. Indeed, Doob’s Theorem is usually cited as the first application of martingales theory.

Statisticians rarely work with this abstract formulation of decomposition which I use. For this reason, the theorem is usually formulated only for the case that {\Theta=\Delta(A)} and {\mu_\theta} is i.i.d. {\theta}. In this case the fact that the decomposition is identifiable follows from the strong law of large numbers. Doob’s Theorem then implies the standard consistency of Bayesian estimator of the parameter {\theta}.

The abstract of a 2005 paper by Itti and Baldi begins with these words:

The concept of surprise is central to sensory processing, adaptation, learning, and attention. Yet, no widely-accepted mathematical theory currently exists to quantitatively characterize surprise elicited by a stimulus or event, for observers that range from single neurons to complex natural or engineered systems. We describe a formal Bayesian definition of surprise that is the only consistent formulation under minimal axiomatic assumptions.

They propose that surprise be measured by the Kullback-Liebler divergence between the prior and the posterior. As with many good ideas, Itti and Baldi are not the first to propose this. C. L. Martin and G. Meeden did so in 1984 in an unpublished paper entitled: `The distance between the prior and the posterior distributions as a measure of surprise.’ Itti and Baldi go further and provide experimental support that this notion of surprise comports with human notions of surprise. Recently, Ely, Frankel and Kamenica in Economics, have also considered the issue of surprise, focusing instead on how best to release information so as to maximize interest.

Surprise now being defined, one might go on to define novelty, interestingness, beauty and humor. Indeed, Jurgen Schmidhuber has done just that (and more). A paper on the optimal design of jokes cannot be far behind. Odd as this may seem, it is a part of a venerable tradition. Kant defined humor as the sudden transformation of a strained expectation into nothing. Birkhoff himself wrote an entire treatise on Aesthetic Measure (see the review by Garabedian). But, I digress.

Returning to the subject of surprise, the Kulback-Liebler divergence is not the first measure of surprise or even the most wide spread. I think that prize goes to the venerable p-value. Orthodox Bayesians, those who tremble in the sight of measure zero events, look in horror upon the p-value because it does not require one to articulate a model of the alternative. Even they would own, I think, to the convenience of having to avoid listing all alternative models and carefully evaluating them. Indeed I. J. Good  writing in 1981 notes the following:

The evolutionary value of surprise is that it causes us to check our assumptions. Hence if an experiment gives rise to a surprising result given some null hypothesis H it might cause us to wonder whether H is true even in the absence of a vague alternative to H.

Good, by the way, described himself as a cross between Bayesian and Frequentist, called a Doogian. One can tell from this label that he had an irrepressible sense of humor. Born Isadore Guldak Joseph of a Polish family in London, he changed his name to Ian Jack Good, close enough one supposes. At Bletchley park he and Turing came up with the scheme that eventually broke the German Navy’s enigma code. This led to the Good-Turing estimator. Imagine a sequence so symbols chosen from a finite alphabet. How would one estimate the probability of observing a letter from the alphabet that has not yet appeared in the sequence thus far? But, I digress.

Warren Weaver was, I think, the first to propose a measure of surpirse. Weaver is most well known as a popularizer of Science. Some may recall him as the Weaver on the slim volume by Shannon and Weaver on the Mathematical Theory of Communication. Well before that, Weaver played an important role at the Rockefeller foundation, where he used their resources to provide fellowships to many promising scholars and jump start molecular biology. The following is from page 238 of my edition Jonas’ book `The Circuit Riders':

Given the unreliability of such sources, the conscientious philanthropoid has no choice but to become a circuit rider. To do it right, a circuit rider must be more than a scientifically literate ‘tape recorder on legs.’ In order to win the confidence of their informants, circuit riders for Weaver’s Division of Natural Sciences were called upon the offer a high level of ‘intellectual companionship – without becoming ‘too chummy’ with people whose work they had, ultimately, to judge.

But, I digress.

To define Weaver’s notion, suppose a discrete random variable X that takes values in the set \{1, 2, \ldots, m\}. Let p_i be the probability that X = i. The surprise index of outcome k is \frac{\sum_i i p^2_i}{p_k}. Good himself jumped into the fray with some generalizations of Weaver’s index. Here is one \frac{[\sum_iip_i^t]^{1/t}}{p_k}. Others involve the use of logs, leading to measures that are related to notions of entropy as well probability scoring rules. Good also proposed axioms that a good measure to satisfy, but I cannot recall if anyone followed up to derive axiomatic characterizations.

G. L. S. Shackle, who would count as one of the earliest decision theorists, also got into the act. Shackle departed from subjective probability and proposed to order degrees of beliefs by their potential degrees of surprise. Shackle also proposed, I think, that an action be judged interesting by its best possible payoff and its potential for surprise. Shackle, has already passed beyond the ken of men. One can get a sense of his style and vigor from the following response to an invitation to write a piece on Rational Expectations:

Rational expectations’ remains for me a sort of monster living in a cave. I have never ventured into the cave to see what he is like, but I am always uneasily aware that he may come out and eat me. If you will allow me to stir the cauldron of mixed metaphors with a real flourish, I shall suggest that ‘rational expectations’ is neo-classical theory clutching at the last straw. Observable circumstances offer us suggestions as to what may be the sequel of this act or that one. How can we know what invisible circumstances may take effect in time-to come, of which no hint can now be gained? I take it that ‘rational expectations’ assumes that we can work out what will happen as a consequence of this or that course of action. I should rather say that at most we can hope to set bounds to what can happen, at best and at worst, within a stated length of time from ‘the present’, and can invent an endless diversity of possibilities lying between them. I fear that for your purpose I am a broken reed.

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