Tuesday, July 26, 2011

Thomas Bayes

I have started a book by Sharon Bertsch McGrayne entitled The theory that would not die. It is about Bayesian statistics, about which I know next to nothing, but which I am told were important to crack the Enigma code during the second world war, hunted down Russian submarines and have recently become respectable following two centuries of controversy.

My son works in central London close to the Dissenters Cemetery where the Rev Thomas Beyes, Fellow of the Royal Society and amateur mathematician, is buried. He also lives in Tunbridge Wells where Bayes was a Presbyterian minister, so my interest is raised. Since my son's job is to do with statistics I am ensnared.

As far as I can tell Bayesian statistics might be a short cut for those statistical imponderables endemic in very large clinical trials; they allow you to change your assumptions as you accumulate data. The purists insist that you don't look at your data until you cross a pre-determined threshold and I remember orthodox statisticians dismissing Bayes with contempt. So I will be interested to read the book. I am only on page 21 and already Bayes has been replaced as the hero by Frenchman Pierre Simon Laplace.

1 comment:

Grateful said...

This is a late reply because it took me a while to get through the book.

What the Bayesians need is someone like my wife. Early in our marriage, I made the mistake of starting an argument with her because she was too spontaneous and we did not accomplish the tasks on the detailed To-Do lists I painstakingly wrote. She asked to see my To-Do list for that day, studied it for a moment, then told me that I could not argue with her because arguing was not on my To-Do list. She had me trapped in my own logic. To continue the argument would be to indulge in the type of behavior I was arguing against, so being a logical person I had no choice but to agree with her.

The classical statistical approach to clinical trials is not like the law of gravity. Rather, the classical approach is one of many possible methods. After much discussion, this classical approach was decided upon as a standard consensus method capable of yielding reproducible and reliable results with measures of expected errors. There is an easy way for Bayesians to convince classical statisticians of the value of the Bayesian approach. Let Treatment X be the classical approach, let Treatment Y be the Bayesian approach, and conduct a classical statistical experiment comparing the two methods – how would different clinical trials be done under the two different methods? But instead of using the consensus classical logic to prove their point, the Bayesians change the logic.

In my opinion, the book’s discussion of the Enigma machine could have been better. A good resource to read is the Time-Life series of books on World War II, the volume titled The Secret War. Many aspects of the Enigma were of course classified during World War II, but they were also classified and indeed were subject to a campaign of dis-information during the Cold War, so this is a topic that has been hard to research. The book did not mention that, in the early 1930’s, the Poles had intercepted a shipment of an Enigma, carefully opened the box, examined the machine, then sealed the box back up so the Germans did not know the box had been opened. The Poles then made working copies of the Enigma, and Polish mathematicians began with the knowledge of how the Enigma worked. The Poles shared their findings and their machines with the British but not with the Russians, who at that time had not yet signed their pact with Hitler. When Stalin took over Poland after WWII, the Allies supposedly launched a campaign of dis-information about Enigma so that Stalin did not punish the Poles who kept their findings from him.

Briefly, quoting from The Secret War volume, “Deciphering an Enigma message required an identical machine with identically wired rotors. The receiver had to know which rotor to insert, in what order to insert them and at what position to set each one. This required a list of so-called key settings shared by sender and receiver.”

The Enigma machine is an example of one type of random process – one where the underlying process is non-random (aka deterministic) but there are so many unknown variables that it seems like the process is random. The other type of random process is where the underlying process is intrinsically random. The Bayesian approach is good at finding unknown parameters in a non-random process.

A brief digression: classical physics said that the sub-atomic world is deterministic but there are so many unknown variables that it seems random. Quantum mechanics says the sub-atomic world is intrinsically random. Einstein felt that determinism was correct and reportedly said, “God does not play dice with the universe.” Niels Bohr reportedly replied to Einstein, “Stop telling God what to do.”