Problem of Thinking Too Much

"The Problem of Thinking Too Much", a 2003 talk by Persi Diaconis

Intro: Persi Diaconis is a pal of mine. He's also someone who, by his work and interests, demonstrates the unity of intellectual life-that you can have the broadest range and still engage in the deepest projects. Persi is a leading researcher in statistics, probability theory, and Bayesian inference. He's done wonderful work in pure math as well, most notably in group representation theory. He has the gift of being able to ask the simplest of questions. Those are the questions that educate you about a subject just because they're asked. And Persi's research is always illuminated by a story, as he calls it-that is, a thread that ties the pure intellectual question to a wider world.

writing on the general concept of coincidence; and working on historical treatises about probability and magic. As is well known, Persi is also a stage magician, credited with, as Martin Gardner once wrote, "inventing and performing some of the best magic tricks ever."

Persi Diaconis

Consider the predicament of a centipede who starts thinking about which leg to move and winds up going nowhere. It is a familiar problem: Any action we take has so many unforeseen consequences, how can we possibly choose?

I'm not brazen enough to attempt a careful definition of "thinking" in the face of a reasonably well-posed problem.

The problem is this: We can spend endless time thinking and wind up doing nothing-or, worse, getting involved in the minutiae of a partially baked idea and believing that pursuing it is the same as making progress on the original problem. The study of what to do given limited resources has many tendrils

An Example

One of the most satisfying parts of the subjective approach to statistics is Bruno de Finetti's solution of common inferential problems through exchangeability. Some of us think de Finetti has solved Hume's Problem: When is it reasonable to think that the future will be like the past? I want to present the simplest example and show how thinking too much can make a mess of something beautiful

In many situations, the order of the outcomes is judged irrelevant

Such probability assignments are called "exchangeable."

Bruno de Finetti proved that an exchangeable probability assignment for a long series of outcomes can be represented as a mixture of coin tossing: For any sequence a, b, ..., z of potential outcomes,

The right side of this formula has been used since Thomas Bayes (1764) and Pierre-Simon Laplace (1774) introduced Bayesian statistics

Subjectivists such as de Finetti, Ramsey, and Savage (as well as Diaconis) prefer not to speak about nonobservable things such as "p, the long-term frequency of heads." They are willing to assign probabilities to potentially observable things such as "one head in the next ten tosses

Enter Physics

Our analysis of coin tossing thus far has made no contact with the physical act of tossing a coin. We now put in a bit of physics and stir; I promise, a mess will emerge. When a coin is flipped and leaves the hand, it has a definite velocity in the upward direction and a rate of spin (revolutions per second). If we know these parameters, Newton's Laws allow us to calculate how long the coin will take before returning to its starting height and, thus, how many times it will turn over. If the coin is caught without bouncing, we can predict whether it will land heads or tails.

A neat analysis by Joe Keller appeared in a 1986 issue of American Mathematical Monthly.

Most human coni flippers do not have this kind of control and are in the range of 5Y2 mph and 35 to 40 rps. Where is this on Figure 1? In the units of Figure 1, the velocity is about %--very close to the zero. However, the spin coordinate is about 40-way off the graph. Thus, the picture says nothing about real flips. However, the math behind the picture determines how close the regions are in the appropriate zone. Using this and the observed spread of the measured data allows us to conclude that coin tossing is fair to two decimals but not to three. That is, typical flips show biases such as .495 or .503.

Blending Subjective Probability and Physics

What's the Point

The analysis led to introspection about opinions on which we have small hold and to a focus on technical issues far from the original problem. I hope the details of the example do not obscure what I regard as its nearly universal quality. In every area of academic and more practical study, we can find simple examples that on introspection grow into unspeakable "creatures." The technical details take over, and practitioners are fooled into thinking they are doing serious work. Contact with the original problem is lost.

Thinking About Thinking Too Much

second-century attempts to balance between rationalist and empiricist physicians ring true today. In his Three Treatises on the Nature of Science (trans. R. Walzer and M. Frode), Galen noted that an opponent of the new theories claimed "there was a simple way in which mankind actually had made enormous progress in medicine.

In my own field of statistics, the rationalists are called decision theorists and the empiricists are called exploratory data analysts. The modern debaters make many of the same rhetorical moves that Galen chronicled.

Economists use Herbert Simon's ideas of "satisficing" and "bounded rationality," along with more theoretical tools associated with John Harsanyi's "value of information." Psychologists such as Daniel Kahneman and Amos Tversky accept the value of the heuristics that we use when we abandon calculation and go with our gut. They have created theories of framing and support that allow adjustment for the inevitable biases. These give a framework for balancing the decision to keep thinking versus getting on with deciding.

An agglomeration of economics, psychology, decision theory, and a bit of complexity theory is the current dominant paradigm. It advises roughly quantifying our uncertainty, costs, and benefits (utility) and then choosing the course that maximizes expected utility per unit of time. A lively account can be found in I. J. Good's book Good Thinking (don't miss his essay on "How Rational Should a Manager Be?").

To be honest, the academic discussion doesn't shed much light on the practical problem. Here's an illustration: Some years ago I was trying to decide whether or not to move to Harvard from Stanford. I had bored my friends silly with endless discussion. Finally, one of them said, "You're one of our leading decision theorists. Maybe you should make a list of the costs and benefits and try to roughly calculate your expected utility." Without thinking, I blurted out, "Come on, Sandy, this is serious."

Some Rules of Thumb

One of the most useful things to come out of my study is a collection of the rules of thumb my friends use in their decision making.

There are many ways we avoid thinking

Clearly, we have a wealth of experience, gathered over millennia, coded into our gut responses. Surely, we all hope to call on this. A rule of thumb in this direction is "Trust your gut reaction when dealing with natural tasks such as raising children."

In retrospect, I think I should have followed my friend's advice and made a list of costs and benefits-if only so that I could tap into what I was really after, along the lines of the following "grook" by Piet Hein: A Psychological Tip. Whenever you're called on to make up your mind, and you're hampered by not having any, the best way to solve the dilemma, you'll find, is simply by spinning a penny. No-not so that chance shall decide the affair while you're passively standing there moping; but the moment the penny is up in the air, you suddenly know what you're hoping.