Essai philosophique sur les probabilitiés was published a couple of years after Théorie analytique des probabilitiés, as a popular introduction to that earlier work. Objecting that Essai is not in fact a very original work, given that Théorie was the summary of more than a century and a half of mathematical research together with important developments by the author, is a bit absurd.

An editor should have brought this dissonance to Gillies' attention. I don't quite know what editors do these days, beyond deciding whether a given work may be expected to sell.

As noted earlier, I've been reading Subjective Probability: The Real Thing by Richard C. Jeffrey. It's a short book, but I've been distracted by other things, and I've also been slowed by the condition of the book; it's full of errors. For example,

It seems evident that black ravens confirm (H) All ravens are black and that nonblack nonravens do not. Yet H is equivalent to All nonravens are nonblack.

Uhm, no: (X ⇒ Y) ≡ (¬X ∨ Y) = (Y ∨ ¬X) = (¬¬Y ∨ ¬X) = [¬(¬Y) ∨ ¬X] ≡ (¬Y ⇒ ¬X) In words, that all ravens are black is equivalent to that all non-black things are non-ravens.[1]

The bobbled expressions and at least one expositional omission sometimes had me wondering if he and his felllows were barking mad. Some of the notational errors have really thrown me, as my first reäction was to wonder if I'd missed something.

Authors make mistakes. That's principally why there are editors. But it appears that Cambridge University Press did little or no real editting of this book. (A link to a PDF file of the manuscript may be found at Jeffrey's website, and used for comparison.) Granted that the book is posthumous, and that Jeffrey was dead more than a year before publication, so they couldn't ask him about various things. But someone should have read this thing carefully enough to spot all these errors. In most of the cases that I've seen, I can identify the appropriate correction. Perhaps in some cases the best that could be done would be to alert the reader that there was a problem. In any case, it seems that Cambridge University Press wouldn't be bothered.

[1]The question, then, is of why, say, a red flower (a non-black non-raven) isn't taken as confirmation that all ravens are black. The answer, of course, lies principally in the difference between reasoning from plausibility versus reasoning from certainty.

The Classic Monty Hall Problem

Andy is a contestant in a game. In this game, each contestant makes a choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments.

After Andy makes his choice, it is revealed to him that a specific tag that he did not choose is committed to an undesirable outcome. Andy is offered a chance to change his selection. Should he change?

Three Contestants

Andy, Barb, and Pat are contestants in a game. In this game, each contestant makes an independent choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments. In the event that multiple players select the same tag, outcomes are duplicated.

After all contestants make their choices, it is revealed that Andy, Barb, and Pat have selected tags each different from those of the other two contestants. And it is revealed that Pat's tag is associated with an undesirable outcome. Andy and Barb are each offerd a chance to change their selections. What should each do?

3-Player Monty Hall

• Neither should change.
• Andy should stick, but Barb should change.
• Andy should change, but Barb should stick.
• Andy and Barb should exchange choices.

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I've been taking another run at Subjective Probability: The Real Thing (2004) by Richard C. Jeffrey. I'd started reading it a while back, but got distracted. Anyway, Jeffrey was an important subjectivist — someone who argued that probability is a measure of belief, and that any degree of belief that does not violate certain rationality constraints is permitted. (As I have noted earlier, the subjectivism here is in the assignment of quantities not specifically required by objective criteria. The subjectivists believe either that quantity by reason must be assigned, albeït often arbitrarily, or that Ockham's Razor is not a binding constraint.) And the posthumous Subjective Probability was his final statement.

At some point, I encountered the following entry in the index:

Nozick, Robert, 119, 123

which entry was almost immediately annoying. Page 119 is in the References section, and indeed has the references for Nozick, but that's a pretty punk thing to drop in an index. Even more punk would be an index entry that refers to itself; and, indeed, page 123 is in the index, and it is on that page that one finds Nozick, Robert, 119, 123.

Well, actually, I'd forgot something about this book, which is probably an artefact of its being posthumous: Most or all of the index entries are off by ten pages, such that one ought to translate Nozick, Robert, 119, 123 to Nozick, Robert, 109, 113. And, yes, there are references to Nozick on those pages (which are part of a discussion of Newcomb's Problem and of related puzzles). It was just chance-coïncidence that ten pages later one found the listings in the references and in the index.

In decision theory, there are propositions call independence axiomata. The first such proposition to be explicitly advanced for discussion (in an article by Paul Anthony Samuelson) is the Strong Independence Axiom, the gist of which is that the value of a reälized outcome is independent of the probability that it had before it was reälized. Say that we had a lottery of possible outcomes X₁, X₂,… X_n, each X_i having associated probability p_i. If we assert that the expected value of this lottery were

Σ[p_i · u(X_i)]

where u( ) is some utility function, then (amongst other things) we've accepted an independence proposition. Otherwise, we may have to assert something such as that the expect value were

Σ[p_i · u(X_i,p_i)]

to account for such things as people taking an unlikely million dollars to be somehow better than a likely million dollars.

Anyway, there's another proposition which to most of us doesn't look like the Strong Independence Axiom, and yet is pretty much the same thing, the Sure Thing Principle, which is associated with Leonard Jimmie Savage (an important subjectivist, whom I much admire, and with whom I markèdly disagree). Formally, it's thus:

{[(A ∧ B) pref C] ∧ [(A ∧ ¬B) pref C]} ⇒ (A pref C)

Less formally,

If the combination of A and B is preferred to C, and the combination of A without B is preferred to C, then A is just plain preferred to C, regardless of B.

Savage gives us the example of a businessman trying to decide whether to buy a piece of property with an election coming-up. He thinks-through whether he would be better off with the property if a Democrat is elected, and decides that he would prefer that he had bought the property in that case. He thinks-through whether he would be better off with the property if a Republican is elected, and decides that he would prefer that he had bought the property in that case. So he buys the property. This seems very reasonable.

But there is a famous class of counter-examples, presented by Jeffrey in the form of the case of the Marlboro Man. The hypothetical Marlboro Man is trying to decide whether to smoke. He considers that, if he should live a long life, he would wish at its end that he had enjoyed the pleasure of smoking. He considers that, if he should live a short life, he would wish at its end that he had enjoyed the pleasure of smoking. So he smokes. That doesn't seem nearly so reasonable.

There is an underlying difference between our two examples. The businessman would not normally expect his choice to affect the outcome of the election; the Marlboro Man ought to expect his choice to affect the length of his life. Jeffrey asserts that Savage only meant the Sure Thing Principle to hold in cases where the probability of B were independent of A.

But what makes the discussion poignant is this: Jeffrey, dying of surfeit of Pall Malls, wrote this book as his last, and passed-away from lung cancer on 9 November 2002.

All these experiments, however, are thrown completely into the shade by the enormously extensive investigations of the Swiss astronomer Wolf, the earliest of which were published in 1850 and the latest in 1893. In his first set of experiments Wolf completed 1000 sets of tosses with two dice, each set continuing until every one of the 21 possible combinations had occurred at least once. This involved altogether 97,899 tosses, and he then completed a total of 100,000. These data enabled him to work out a great number of calculations, of which Czuber quotes the following, namely a proportion of .83533 of unlike pairs, as against the theoretical value .83333, i.e. 5/6. In his second set of experiments Wolf used two dice, one white and one red (in the first set the dice were indistinguishable), and complete 20,000 tosses, the details of each result being recorded in the Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich. He studied particularly the number of sequences with each die, and the relative frequency of each of the 36 possible combinations of the two dice. The sequences were somewhat fewer than they ought to have been, and the relative frequency of the different combinations very different indeed from what theory would predict. The explanation is easily found; for the records of the relative frequency of each face show that the dice must have been very irregular, the six face of the white die, for example, falling 38 percent more often than the four face of the same die. This, then, is the sole conclusion of these immensely laborious experiments,—that Wolf's dice were very ill made. Indeed the experiments could have had no bearing except upon the accuracy of his dice. But ten years later Wolf embarked upon one more series of experiments, using four distinguishable dice,—white, yellow, red, and blue,—and tossing this set of four 10,000 times. Wolf recorded altogether, therefore, in the course of his life 280,000 results of tossing individual dice. It is not clear that Wolf had any well-defined object in view in making these records, which are published in curious conjunction with various astronomical results, and they afford a wonderful example of the pure love of experiment and observation.

While I was doing some research to-day, I ran across yet another article that classified John Maynard Keynes as a subjectivist when it came to probability theory. I feel moved to explain why this is incorrect.

First, let me explain something about the general issue. There is an outstanding question about just what a probability is. One could take many courses about probability without ever being alerted to the question. The textbook and lecturer might not ever touch on that basic question, or might present a definition of probability as if it is universally accepted by all Wise People. But Wise People are not in agreement. When it comes to answers to the basic question, the two dominant answers are very different one from another.

One answer is provided by the frequentists, who say that a probability is some sort of frequency of occurrence. They don't agree amongst themselves as to the precise answer, but the gist of their answers is that if a process is repeated m times, where m is satisfactorily large, and results in some particular outcome n of those times, then the probability of that outcome is n/m.

One problem with this notion of probability is that it is only useful in cases where we are concerned with a sufficiently large sample. If one is concerned only with a single instance, then there is actually no logic to get us from a mere probability to a course of action. A single patient won't have average mortality; she will either live or die.

Another answer is provided by the subjectivists, who assert that a probability is a degree of belief, formed subject to certain rationality constraints. These constraints can be largely motivated in terms of avoiding probability assignments under which believers would accept gambles that they are sure to lose. The rationality constraints themselves are ostensibly objective — rules that should hold for everyone; amongst other things, these rules are to constrain the evolution of one's degrees of belief, as new information is introduced. The subjectivism is present in that one supposedly gets to start with any degrees of belief that don't violate these rules.

One immediate consequence of this notion of probability is that probabilities become largely unarguable. There is no real contradiction in Tim claiming that there is an 80% chance of rain and Bob claiming that there is a 20% chance; each is describing his respective belief per se. (The rationality constraints force a convergence of belief at the limit, but that could take forever.)

The subjectivist notion is often defined in terms such as degree of rational belief or rational degree of belief; it's best to be wary of such terms. The rationality constraints themselves only preclude certain sorts of irrationality; aspects of the degrees of belief permitted are at best not irrational. And if we are not somehow required to assign some quantity to that belief, then the assignment violates Ockham's Razor.

Now, Keynes's position is that we can make meaningful statements about the plausibility of uncertain outcomes for which frequencies are unknown or otherwise inapplicable. And he certainly wants to impose rationality constraints much like those of the subjectivists. But he sees no requirement that one always assign a quantity to belief. Indeed, he sees no reason to treat the set of possible outcomes as even necessarily totally ordered; that is to say that he holds that, when asked to compare the likelihood of two events, sometimes one can only shrug, rather than making claims that one event is more likely or that the two are equally likely.

Under Keynes's theory, a rational person says no more about the probability of an event than the application of objective rules to the information set yields, and any other rational person with the same information set would reach exactly the same conclusions about probabilities (except, perhaps, where one person halted consideration where the other continued). Keynes rejects the very thing that is subjective in the subjectivist framework.