Statistics as inverse probability

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.