Today is the quietest day, a time when all is still, a moment when nary a voice is raised and, quite suddenly, appointments are remembered, people have to be seen, the room empties. Because this is the day I introduce the classical confidence interval, a creation so curious that I have yet to have a frequentist stick around to defend it.
Up until now we have specified the evidence, or premises, we used (“All Martians wear hats…”) and this evidence has let us deduce the probabilities of the conclusions (which we have also specified, and always will, and always must; e.g. “George wears a hat”).
But sometimes we are not able to use the premises (data, evidence) in a direct way. We still follows the rules and dictates of logic, of course, but sometimes the evidence is not as clear as it was when we learned that “Most Martians wear hats.”
The game of petanque is played by drawing a small circle into which one steps, keeping both feet firmly planted. A small wooden ball called a cochonette is tossed 6 to 10 meters downstream. Then opposing teams take turns throwing manly steel balls, or boules, towards the cochonette trying to get as close as possible to it. It is not unlike the Italian game of bocce, which uses meek wooden balls.
Now I am interested in the distance the boule will be from the cochonette. I do not know, before I throw, what this distance will be. I therefore want to use probability to quantify my uncertainty in this distance. I needn’t do this in any formal way. I can, as all people do, use my experience in playing and make rough guesses. “It’s pretty likely, given all the games I have seen, the boule will be within 1 meter of the cochonette.” Notice the clause “given all the games I have seen”, a clause which must always appear in any judgment of certainty or uncertainty, as we have already seen.
But I can do this more formally and use a store-bought probability distribution to quantify my uncertainty. How about the normal? Well, why not. Everybody else uses it, despite its many manifest flaws. So we’ll use it too. That I’m using it and accepting it as a true representation of my uncertainty is just another premise which I list. Since we always must list such premises, there is nothing wrong so far.
The normal distribution requires two parameters, two numbers which must be plugged in else we cannot do any calculations. These are the “m = central parameter” and the “s = spread parameter.” Sometimes these are mistakenly called the “mean” and “standard deviation.” These latter two objects are not parameters, but are functions of other numbers. For example, everybody knows how to calculate a numerical mean; that is just a function of numbers.
Now I can add to my list of premises values for m and s. Why not? I already, quite arbitrarily, added the normal distribution to the list. Might as well just plug in values for m and s, too. That is certainly legitimate. Or you can act like a classical statistician and go out and “collect data.”
This would be in the form of actual measurements of actual distances. Suppose I collect three such measurements: 4cm, -15cm, 1cm. This list of measurement is just another premise, added to the list. A frequentist statistician would say to himself, “Well, why don’t I use the mean of these numbers as my guess for m?” And of course he may do this. This becomes another premise. He will then say, “As long as I’m at it, why don’t I use the standard deviation of these numbers as my guess for s?” Yet another premise. And why, I insist, not.
We at least see how the mistake arises from calling the parameters by the names of their guesses. Understandable. Anyway, once we have these guesses (and any will do) we can plug them into our normal distribution and calculate probabilities. Well, only some probabilities. The normal always—as in always—gives 0 probabilities for actual observable (singular) events. But skip that. We have our guesses and we can calculate.
The frequntist statistician then begins to have pangs of (let us say) conscience. He doubts whether m really does equal -3.3cm (as it does here) and whether s really does equal 10.2cm (as it does here). After all, three data points isn’t very many. Collecting more data would probably (given his experience) change these guesses. But he hasn’t collected more data: he just has these three. So he derives a statement of the “uncertainty” he has in the guesses as estimates of the real m and s. He calls this statement a “95% confidence interval.” That 95% has been dictated by God Himself. It cannot be questioned.
Now the confidence interval is just another function of the data, the form of which is utterly uninteresting. In this example, it gives us (-10cm to 3.3cm). What you must never say, what is forbidden by frequentist theory, is to say anything like this, “There is a 95% chance (or so) that the true value of m lies in this confidence interval.” No, no, no. This is disallowed. It is anathema. The reason for this proscription has to do with the frequentist definition of probability, which always involves limits.
The real definition of the CI is this: if I were to repeat this experiment (where I measured three numbers) an infinite number of times, and for each repetition I calculated a guess for m and a confidence interval for this guess, and then I kept track of all these confidence intervals (all of them), then 95% of them (after I got to infinity) would “cover”, or contain, the real value of m. Stop short of infinity, then I can say nothing.
The only thing I am allowed to say about the confidence interval I actually do have (that -10cm to 3.3cm) is this: “Either the real value of m is in this interval or it isn’t.” That, dear reader, is known as a tautology. It is always true. It is true even (in this case) for the interval (100 million cm, 10 billion cm). It is true for any interval.
The interval we have then, at least according to strict frequentist theory, has no meaning. It cannot be used to say anything about the uncertainty for the real m we have in front of us. Any move in this direction is verboten. Including finite experiments to measure the “width” of these intervals (let he who readth understand).
Still, people do make these moves. They cannot help but say something like, “There is (about) a 95% chance that m lies in the interval.” My dear ones, these are all Bayesian interpretation. This is why I often say that everybody is a Bayesian, even frequentists.
And of course they must be.
Typo patrol away!
Find, in real-life, instances where the normal has been used with confidence intervals. Just you see if whoever used the interval interpreed it wrong.