How one thinks about probability affects how one thinks about economics. Consider the use of the word “probability” in each of the following sentences:1. What is the probability that when a fair coin is flipped it will come up heads?
2. What is the probability that exactly two number-one seeds will make it to the final four in the March Madness basketball tournament?
3. What is the probability that New York City will rank higher relative to other cities five years from now in terms of college graduates?

We would answer the first question by saying that the probability is 50 percent, based on the very definition of a fair coin. This is an axiomatic interpretation of probability. The axiomatic view treats probability as a matter of pure logic, with statements that do not require any empirical testing.

We would answer the second question by looking up historical records for the NCAA basketball tournament. This is the frequentist account of probability, which treats probability as counting outcomes from repeated trials. A frequentist would claim that the only way we can know that a coin has a 50 percent probability of coming up heads is by actually flipping a coin enough times to verify this empirically.

The third question cannot be answered on the basis of axioms or observed frequencies. The probability estimate is purely subjective. The subjective account of probability is that it reflects an individual belief that cannot be proven either logically or empirically.

Sometimes, these three accounts or probability are presented as mutually exclusive. Supposedly, there is a “debate” among adherents of the three views. Instead, I prefer to look at this issue as one that is caused by the same word (“probability”) being used with multiple meanings. Asking me which theory of probability I believe is asking me to face a false choice. I believe in axiomatic probability, frequentist probability, and subjective probability. Each account of probability is true in its own context. My resolution of the issue is to argue that the “debate” arises only because of linguistic imprecision. If we used three words instead of one, then the question of which is the proper account of probability would never arise. We would use word A to refer to axiomatic probability, word B to refer to frequentist probability, and word C to refer to subjective probability.

John Maynard Keynes’ first book was A Treatise on Probability. I have never read it. What I know about it is that Keynes is regarded as a leading exponent of the subjectivist account of probability. I also know that Robert Skidelsky, Keynes’ biographer, argues that Keynes’ outlook on probability is linked to his views of economics. I think this is true, and I think I am naturally inclined to follow Keynes in his views of probability and economics.

The way I see it, the “rational expectations” school of economic theory is linked to the axiomatic treatment of probability. The economist derives predictions about a model from a set of axioms, one of which is that the people whose behavior is modeled are assumed to fully understand the model.

This axiomatic approach is, in my view, absurd. Economist X has one model of the economy. Economist Y has another model of the economy. In X’s model, people believe in X’s theory. In Y’s model, people believe in Y’s theory. It is logically impossible for economist X and economist Y to inhabit the same universe! Yet they do. This tells me that the axiom of rational expectations is too strong.

The frequentist approach is, in my view, too hubristic. When I think of frequentists, I think of Bob Shiller, who is confident that he knows the true prices of assets and is certain that the variation of stock prices and house prices around their true values is unjustifiably high. Shiller’s first paper on asset prices, published in 1979, made the claim that one can compute bounds for the volatility of, say, stock prices. Violation of such variance bounds, he argued, proves that markets are not rational.

When Shiller’s paper came out in the Journal of Political Economy in 1979, I submitted a critical comment, which he rejected, so it was never published. In retrospect, I think that our dispute was over his frequentist interpretation of probability. I was arguing, in effect, that under a subjective interpretation of probability there are no variance bounds on asset prices.

Consider the value of stock prices today. Typical Shiller calculations would suggest that stock prices may be too low relative to fundamentals. However, subjectively, it might be rational to believe that stock prices are reasonable. By the time the overall economy recovers, many of today’s companies will have failed or shrunk. Entirely new industries may emerge. If this is the case, it is possible that owning a representative portfolio of current companies will be a losing strategy. (I am not saying that I personally believe in this pessimistic stock market scenario, but it is rational to regard it as plausible.)

More generally, at any point in time, structural change may be under way that affects the entire future path of share prices. The past is one path out of many possible paths that could have occurred had structural change evolved differently. It is necessarily much more tightly bounded than the future, where the path of structural change is unknown. Shiller inevitably confuses the tight variance bounds that look backwards with a forward-looking variance that need not be so bounded.

From a subjective perspective, the future distribution of stock prices is much more uncertain than anything that is calculated based on past data. In my view, the question of what the stock market will be priced at five years from now is one that belongs in the realm of subjective probability, not frequentist probability. If one takes that position, then Shiller’s frequentist calculation of variance bounds is not valid.

I think that Keynes would have taken my side on this. His use of the term “animal spirits” reflected his view that future probabilities could not be known objectively. Instead, entrepreneurs’ subjective estimates would determine their behavior. Keynes was not alone in this view, of course. Frank Knight was famous for describing a notion of uncertainty that could not be reduced to an objective probability distribution.

In Shiller’s frequentist interpretation, I was merely concerned with what came to be called “the Peso problem,” or in more recent terminology, a Black Swan. In his view, the only way that asset prices can vary more than historical “fundamentals” is if investors were expecting a particular event that did not occur during the time frame. This is called “the Peso problem” because an example that is often used is a currency devaluation that was considered probable within a time frame but which in fact did not occur.

However, the “Peso problem” formulation is the frequentist’s attempt to recast pure Knightian uncertainty in terms of an objective, empirical distribution. In my view, the frequentists just don’t “get” how probability factors into asset prices.

Asset prices reflect subjective probability distributions about the future. The future differs fundamentally from the past in terms of uncertainty. The past followed one path. The future has many paths. Our ignorance of the future cannot be measured on the same scale as the variation of the past.

I guess there is something about the notion of subjective probability that makes it very hard for economists to accept. Many economists prefer the objective axiomatic approach of rational expectations. Still others, like Shiller, see the world through frequentist eyes. My own view, however, is that if you want to really understand the economy, make an effort to grasp the concept of subjective probability.