Risk Latte - Like you, we are all Bayesians, Mr Greenspan!

Like you, we are all Bayesians, Mr Greenspan!

Rahul Bhattacharya
May 13, 2009

Long before it became a fashion to talk about that elusive concept called “productivity” Alan Greenspan, the erstwhile Chairman of the Federal Reserve Bank in the U.S. for two decades, talked about it incessantly in his speeches and testimony to Congress. He believed in deregulation of banks, financial derivatives, the self-regulating aspect of the markets, low interest rates but above all, he believed in the concept of Bayesian probabilities.

Today, you can throw all the rocks at him. In the end, we are all Bayesians.

What are the chances (read probability) that in a family of two children, both the children are boys? Probability questions cannot get any simpler than this. Of course, it is 1/4 (25%). Why? Simply because the total probability space (all possible combinations of a girl and a boy) comprising birth orders in a two children family is (girl, girl), (boy, girl), (girl, boy) and (boy, boy) and out of all these four possible states only one has two boys and therefore the probability is 1/4

Now let’s say we have recently learnt that in the same family one of the children is a boy. Therefore, our estimation problem becomes: what are the chances (once again read probability) that in a family of two children, if one of the children is a boy, both children are boys.

This is where you invoke the ghost of Thomas Bayes. This is the kind of question that perhaps kept Alan Greenspan awake at night and when he spoke during the day it became translated as Greenspanspeak. This is also the kind of question with which we grapple day in and day out, no matter what situation we are in. There is always a prior probability and then there is new information which creates conditional probability and which in turn alters our beliefs and decision making. Probability measure, the way it gets applied in human decision making, is always a conditional measure.

In the above two children problem, if we know that one of the children is a boy then the probability that both children are boys is 1/3 and not 1/4. How? If we know that one of the children is a boy then the sample space (the probability space) gets reduced to (boy, girl), (girl, boy) and (boy, boy) and there is only state with two boys. Therefore, the probability is simply 1/3 (33.33%).

A similar problem could be a central banker’s forecast of interest rates and inflation. Say, in a highly simplistic state of economy (assumed only for argument’s sake) there are only two economic variables, interest rates and inflation and both can either go up or down in any particular regime. The central banker could ask himself the question: what is the probability that there would be a situation when both interest rates and inflation go down (let’s call this, out of deference for Mr Greenspan, the “productivity scenario”? The sample space is (up, up), (up, down), (down, up) and (down, down), with the first word inside the bracket representing interest rates and the second word representing inflation. From this sample space we can see that there is 1/4 (25%) probability (chance) that both interest rates and inflation will go down.

However, if say, the Central Banker has a just received some information about the economy which shows that one of the two variables – interest rates and inflation – is certainly down (ignore any economic rationale for this argument), then what will be the probability that both interest rates and inflation are down? With this new information the sample space shrinks to (up, down), (down, up) and (down, down) and the probability that both these variables will be down would be 1/3 (33.33%).

The above is a highly stylized and a highly simplistic economic scenario. But you can see where such methods of reasoning, the Bayesian thought process, will take you. This is the Bayesian model of probability and is, at all times, applied to complex decision making within the Federal Reserve Bank in the U.S. and as a matter of fact in Central Banks, commercial banks and corporations all over the world. This is the way we all think.

There is a prior probability and there is this new information and based on that information we change our view of the world. Isn’t this obvious? It is not just Alan Greenspan, all of us are Bayesians.

Reference: See the excellent book on probability and randomness titled The Drunkard’s Walk by Leonard Mlodinow.

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