Statistics is sometimes described as inverse probability.

In a typical probability problem, one starts by positing that a certain quantity has some given probability distribution, say the number of people entering a bank branch follows a Poisson distribution, and then goes on to compute probabilities such as the chance that more than 100 people (max capacity) require service at the same time. In a typical statistical problem, one observes the distribution, that is to say, the number of patrons over a period of time, and then finds a model to best represent the observed pattern.

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In Charles Wheelan's Naked Statistics (I'll post a proper review when I finish the book), he offered readers the following:

If you flip a fair coin 1,000,000 times and get 1,000,000 heads in a row, the probability of getting a tail on the next flip is still 1/2. (p. 102)

I'm here to tell you why it isn't 1/2. And it has to do with the difference between how a probabilist thinks and how a statistician thinks.

A probabilist starts with a couple of truths:

- the coin is fair
- each coin toss is an independent event

and from there, he or she computes the probability of different outcomes: one such outcome is to obtain 1 million heads followed by 1 tail in 1 million and one tosses. The probabilist tells us that that outcome is extremely rare but possible.

Now, if I find that 1,000,000 coin tosses produced 1,000,000 heads, I reject the notion that the coin is fair! Based on the observed data, I am more comfortable believing that the coin is severely biased towards heads. Therefore, my expectation of the next throw would be a very high chance of heads -- I'd certainly not conclude that the chance of heads is 50%, as Wheelan said there.

What Wheelan said is probably very commonly taught in statistics classes. This is unfortunate because in statistics, we start with the data, and figure out which probability distributions would be most consistent with the data. This is the inverse of probability modeling, in which one starts with the probability distribution.

It would be doubly unfortunate if that kind of statement shows up in a Bayesian textbook but I suspect you can find examples of that too in the section on probability.

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If you want to understand how statisticans think, this is a great place to start.

...and yes, that's a reference to a Dominican beer.

Posted by: Jordan | 05/01/2013 at 03:00 PM

Well, it depends on how many coins you started off with. If we started off with some very, very, very large number of coins (to Gelman's point, an impossible number), it would not be unusual that we'd get one fair coin that landed heads 1,000,000 times in a row.

But we can easily move this problem into the space of the possible. If we look at how advisory (not index) mutual funds are marketed, a strategy is to start a large number of funds with small seed money. Some strategies will perform well, possibly by accident. These get heavily promoted and the rest get quietly folded.

We're told that the fund has a glowing track record (for instance, it has always gone up -- the equivalent in this space of always coming up heads) but not given full information about the sampling space.

So, the best conclusion in these situations isn't that those who run the fund are geniuses for having a run of X excellent years, but people who've flipped a whole bunch of coins and will return to coin-flip accuracy as soon as you invest in the fund.

(I note the sum of coin flips is a random walk, and there's a whole literature on stock prices as random walks)

Posted by: zbicyclist | 05/02/2013 at 10:00 AM

The example seems related to Gambler's fallacy http://en.wikipedia.org/wiki/Gambler%27s_fallacy

Posted by: Max | 05/19/2013 at 05:37 PM