prior probability

Note: this is the third installment of our review of the paper “Bayesian reasoning in science” by Colin Howson and Peter Urbach.

Thus far, we have seen how Howson and Urbach briefly consider the relation between probability and truth (see previous blog post), and we also compared and contrasted ordinary gambling odds, like the familiar system of odds used in sports betting, with the more elegant probability scale (ranging from 0 to 1, inclusive) used in probability theory. In the next part of their paper (pp. 371-372), Howson and Urbach summarize the seminal contributions of two important probability theorists — Frank Ramsey and Bruno de Finetti — and then derive Bayes’ theorem from the standard axioms of probability theory for good measure. We won’t rehash all the technical mathematical details of Bayes’s rule here, or of Ramsey and de Finetti’s theoretical work either. Instead, we shall summarize (in words) two fundamental qualitative insights from this part of their paper:

Insight #1: beliefs are like gambles — The task of assigning a probability value from 0 to 1 to a future event (or to the truth value of a hypothesis) is ultimately based on subjective personal beliefs and can thus be quantified by a wager, the same way sports bettors wager on the outcome of sporting events. Why? Because a person’s degree of belief in something, although entirely subjective and personal, can be measured objectively or “translated” (so to speak) by the amount of money he is actually willing to bet on his beliefs. (For those of you keeping “intellectual score” at home, we owe this important insight to Frank Ramsey and Bruno de Finetti.)

Insight #2: gambles must be updated — What is Bayes’s rule really all about? Stated informally, it’s ultimately about “updating” (as we like to say) your gambles, i.e. revising your subjective prior beliefs regarding the truth value of a given hypothesis h (cf. insight #1 above) after you are able to review some amount of evidence e relevant to h. That is, would you be willing to bet more money, or less money, on your beliefs after evaluating e? The evidence may consist of an empirical test of h (as in science), or testimony from a witness (as in law), or a scouting report (as in sports). Whatever the case might be, a good Bayesian should assign some weight to e and update his priors in light of e. (Thank you, Rev. Bayes!)

But now we must contend with a new and perhaps insurmountable problem posed by the Bayesian approach to truth (in addition to the pesky problem we mentioned in our previous post regarding the subjectivity of Bayesian methods): where do you get your priors from? It is no exaggeration to say that this question re: priors has stirred up the most controversy in scientific and philosophical circles. (Note the motto of this blog.) For their part, Howson and Urbach devote most of their paper responding to this question (and the related problem re: subjectivity in science), so rest assured, we shall continue our review of their paper “Bayesian reasoning in science” in future posts …

Don’t we all!

That is the title of this commentary by Colin Howson and Peter Urbach published in the journal Nature on 4 April 1991. (Howson and Urbach also published a book with the title “Scientific Reasoning: The Bayesian Approach”; see the image of their book cover below.) Their dense 1991 paper offers a concise overview of Bayesian methods, provides a powerful critique of alternative statistical methods, and has shaped our work as well (in which we apply Bayesian methods to litigation). We shall thus review the main points and insights of “Bayesian reasoning in science” in this and the next few blog posts.

Let’s begin at the beginning, shall we? Howson and Urbach start out by acknowledging that “ours is uncertain world” and by noting how gamblers use odds to measure numerically the likely outcomes of future events. (This method of expressing probabilities is especially common in sports betting. For example, prior to the running of this year’s Kentucky Derby, the odds that California Chrome would win the race were 5 to 2, meaning that a $2 wager on this horse finishing in first place would pay out a total of $7 in winnings–i.e. a profit of $5, plus the bettor’s original $2 wager.) The authors also identify a major problem with this familiar system of gambling odds: “Because odds are ratios the odds scale starts at 0 and is unbounded to the right (infinite odds).” The solution to this problem is to transform the odds scale into a finite probability scale (ranging from 0 to 1) by restating the probability of an event p using the formula p = odds divided by 1 + odds. In other words, we want to express probabilities using a uniform and finite space (i.e. from 0 to 1) in order to make probability problems tractable and easier to solve. “Stay tuned” … for we shall review the remaining parts of this important paper over the next few days.