Update 8/14/2011: I wrote this article more than a year ago. There is a book out on the very topic of Bayesian reasoning. Times published an article on the book. The article gives a very similar coin-toss problem. You can find the solution here.
A street smart guy called Fat Tony, a by-the-book numbers guy called Dr.John (PhD) and Rev. Thomas Bayes walk into a bar. There they meet a man, Mr. NNT, who shows them a coin and tells them to assume it is fair (equal probability of getting head or tail). Mr. NNT flips the coin 99 times and gets heads each time. He then asks them,
“What are the chances of getting tails in my next toss?”
While you think about your answer, here is a background on these three people. Fat Tony and Dr.John are two imaginary people described in the book The Black Swan (pp124) by Nassim Nicholas Taleb (NNT) who describes them as follows
Fat Tony?Fat Tony’s motto is, “Finding who the sucker is”. He is remarkably gifted in getting free upgrades, getting unlisted phone numbers through his forceful charm. (Fat Tony reminds me of Soprano and his methods)
There is no Thomas Bayes in NNT’s story, he died in the 18th century. His methods, however, are very relevant here.
Back to NNT’s question, which one of the two answers will you agree with?
Dr. John answers 50% because that was the assumption and each toss is independent of the other.
Fat Tony answers no more 1% and believes the coin must be loaded and it can’t be a fair game.
Dr. John follows the science of marketing by numbers to the letter. He applies hypothesis testing, sampling and statistical significance all the time. But, he confuses assumptions with facts. When he starts with an assumption he refuses to look beyond the obvious and refine his knowledge with new data. He sticks to the prior knowledge as given and dismisses events stating otherwise.
Fat Tony has no system. He shoots from the hip. He is the big picture visionary guy. He has been there, done that. He has no prior knowledge nor does he care about analyzing whether data fits theory. He is simply convinced that getting heads 99 times in a row means funny business. He believes plural of anecdotes is data, worse, irrefutable evidence.
In this specific example, Fat Tony is most likely to be correct and he gets it right not because of his superior street skills, gut feel, “Blink” but because of how he always makes decisions. He is correct in questioning the assumption but his methods are not repeatable or teachable.
Until now, I did not say what Thomas Bayes said. His answer was
“I am all but certain (almost 100%) that the next toss will not be tails*”
There is a better way between the street smart, gut-feel ways of Fat Tony who goes by just what what he sees (and he has seen enough) and the rigid number crunching of Dr.John who believes that the assumption holds despite data. That’s the Bayesian way.
Bayesian Marketer does not “assume” the very thing he is trying to prove and does not mistake statistical significance for economic significance. For a Bayesian, assumptions are just that, not irrefutable facts. He does not let the gut decide but lets it guide and effectively uses data to make more informed decisions. He knows he is making decisions under uncertainty and wisely uses experimentation and information combined with his mind to reduce this uncertainty. (see below for the math)
That is the way to confidently pursue strategy in the presence of uncertainties!
How do you make your decisions?
*For those mathematically inclined:
Bayesian does not look at probabilities as ratio of count of events but as a measure of certainty. Bayesian also accounts for uncertainty and does not take the hypothesis as given (i.e., assuming the coin is fair). In this case instead of stating, ” probability of getting heads with a fair coin is 50%”, he states his prior as, ” i am 50% confident I will get tails in a coin toss”
P(C) = 0.5 where C is for his confidence level in getting tail.
The 99 heads we saw are the data D. Bayesian asks given the observed data D, how does my estimate change. That is he asks what is P(C|D)?
P(C|D) = P(D|C) * P(C) / P(D)
P(D) = P(D|C)*P(C) + P(D|Not C)*P(Not C)
P(D|C) is the the chance of getting 99 heads in a row given that coin was fair. That is (1/2) raised to the power of 99, a very low number.
P(D|Not C) is 1- P(D|C) , i.e, you can get this data in every possible scenario except with a fair coin. P(D| Not C) is almost close to 1. Here is a simpler explanation, the coin can be fair in only one way but it can be unfair in any number of ways.
Plugging in the numbers we compute P(C|D) to be close to 0 and hence the refined estimate of confidence.
Hence his answer that says how confident he is about getting tails in the 100th toss. To reiterate unlike Fat Tony or Dr.John, Bayesian does not say what the chances are but how confident he is about the outcome.
For those raising the valid point that the coin could still be fair if you continue to toss it for 10,000 or 10 million times. Yes, you are correct and Bayesian will continue to refine his uncertainty as new information comes in. He does not stop with initial information.