It is often difficult to keep in mind all the links in a chain of argument when that chain is long.^{1} This is especially so when one expects that chain to lead to a familiar place, and instead it veers in an unexpected direction. One sits “at home” waiting for the argument to reach them, but it never does. The (incorrect) conclusion is that the chain is faulty.

The argument I gave these past two weeks lands in an entirely different spot from that offered by the classical school of statistics. Not a new spot, just a different one. The theory I have been calling “predictive” is actually old. Philosophically old, not mathematically. The philosophy is positivism or logical probability (see below about “positivism”).

This view of probability is the one given by John Maynard Keyenes (*A Treatise on Probability*), Harold Jeffereys (*Theory of Probability*), Rudolph Carnap (*Logical Foundations of Probability*), Edwin Jaynes (*Probability Theory: The Logic of Science*), Bruno de Finetti (*Theory of Probability*), Seymour Geisser (*Predictive Inference: An Introduction*), David Stove (*The Rationality of Induction*), and other similar authors. And don’t forget Pierre-Simon, Marquis de Laplace (*A Philosophical Essay On Probabilities *).

Few of these books, or none, is ever read or taught in ordinary university courses in probability and statistics, so it is not surprising they are unfamiliar to most statisticians. Certainly these authors never show up in undergraduate classes. And when they appear in graduate courses—a rare thing—it is the mathematics which is emphasized, not the philosophy.

I have tried to be careful to show that, even if the classical frequentist theory is accepted, the language used to describe results is often wrong or exaggerated. Few recall the precise definitions of p-values and confidence intervals, for example. Even text books are schizophrenic: different interpretations of these creatures often appear even within individual books! This is partly because the definitions of these are tangled, long, and non-intuitive.

But even if the language is proper, the results themselves—for which, I hasten to add, the mathematics are flawless—are not what anybody wants; worse, the results produce over-certainty. If presented with the choice of two drugs, what you want to know is the chance that one is better than another. You are not interested in the probability of seeing a larger value of an obscure test statistic assuming that you repeat some trial an infinite number of times and assuming the truth of some probability model, given that the parameters of which are set equal or set equal to zero (this is, of course, the p-value).

What exacerbates the disparity is that the evidence given by the p-value is guaranteed to be exaggerated with respect to the chance that one drug is better than another (this is a provable, and well proved, mathematical claim). The p-value can be as small as you like (“highly statistically significant”), but that does not imply that the chance that one drug is better than another is high.

The situation is improved when one moves to considering posterior distributions of parameters of models. But not by much. Language used to describe posterior distributions is certainly more intuitive and memorable than that used in hypothesis testing, but it is still the case that knowledge of the value of a parameter (given some data and accepting the truth of some model) still exaggerates the evidence that the chance that one drug is better than another. Once again, one can have almost certain knowledge of the value of some parameter or parameters (as in estimation or hypothesis testing), but this does not translate into high probability that one drug is better than another.

The situation is improved once more by moving to predictive statistics. Improved, not solved—for there is no solution. Uncertainty in contingent events and theories will always be with us. An event or theory being contingent makes this so. But at least we can say the chance that one drug is better than another is X%—*given* we accept the truth of some model and given our observations.

Hypothesis testing and Bayesian parameter estimation cannot give direct evidence whether the models we have been assuming true really are. This too is a provable, and well proved, mathematical and philosophical statement. Predictive methods can and do give us this evidence. Further, this evidence is expressed in natural language in statements of probabilities of observables and in terms of decisions we make with these observables.

I’ll show more examples as time goes on. But for those who want a head start, you cannot go wrong by reading the books given above.

**Update** When I say positivism, I not mean the Vienna Circle logical positivism. I do mean positivistic. See comments below.

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^{1}Another difficulty is when that chain is offered in 800-word, non-contiguous chunks. And when those words are unclear, the situation is not made any easier. But I’m working on fixing that.

You can get the Keynes book from the Gutenberg Project: http://www.gutenberg.org/ebooks/32625

I’ll just note as a philosopher that we all agreed to abandon logical positivism (described as the view that statements are meaningful only if they’re empirically falsifiable) a few decades ago.

Indeed, when A J Ayer was asked what was wrong with Logical Positivism he replied, “Almost all of it was false.”

So there.

Jaynes is the easiest read in the list, IMO. It used to be available on-line until somebody decided there was money in it. Too bad, the on-line version was searchable.

blockquote>JH, Subjectivity, eh? Tell me: how do you pick your models? Objectively?

Does this imply that you admit there is no such thing called â€œobjective Bayesian statisticsâ€ then?

I examine how the data set is collected and the data structure, employ various graphs, and then construct or pick a model accordingly depending whether there is an appropriate existing model. And hopefully, my explanations of the chosen model make sense, at least, statistically and mathematically.

Yes, itâ€™s subjective because that I only try to appropriately extract information from the data the best I can and that everyone is free to disagree with me.

So why make it worse by piling one subjectivity on top of another?

Ooop… commented on the wrong post.

For ET Jayne checkout

http://bayes.wustl.edu/

and

bayes.wustl.edu/etj/prob/book.pdf

Rich, Jason,

He did and he was right. I do not mean AJ Ayer’s “system” of logical positivism, nor do I mean Carnap’s “system” of logical probability. I do mean, using Stove’s language, a neo-positivism. The twist is that, unlike the old logical positivism, metaphysics is not only entertained but is essential to neo-positivism.

DAV,

True, true. But the book isn’t bad, as long as you marry it to the several websites that correct the index and so forth.

Drat,

The only probability books I have are:

von Mises, 1957

Parzen, 1960

Papoulis, 1991

and

Briggs ~2009.

And none of these guys are on the approved list ;>)

Oi Brigs! Just two “e”s in Harold Jeffreys.

P.S. He used to be a neighbour of ours. There’s a claim to fame, eh?