Posts Tagged ‘probability’

Just a Note

Thursday, 12 June 2014

Years ago, I planned to write a paper on decision-making under uncertainty when possible outcomes were completely ordered neither by desirability nor by plausibility.

On the way to writing that paper, I was impressed by Mark Machina with the need for a paper that would explain how an incompleteness of preferences would operationalize, so I wrote that article before exploring the logic of the dual incompleteness that interested me.

Returning to the previously planned paper, I did not find existing work on qualitative probability that was adequate to my purposes, so I began trying to formulating just that as a part of the paper, and found that the work was growing large and cumbersome. I have enough trouble getting my hyper-modernistic work read without delivering it in large quantities! So I began developing a paper concerned only with qualitative probability as such.

In the course of writing that spin-off paper, I noticed that a rather well-established proposition concerning the axiomata of probability contains an unnecessary restriction; and that, over the course of more than 80 years, the proposition has repeatedly been discussed without the excessiveness of the restriction being noted. Yet it's one of those points that will be taken as obvious once it has been made. I originally planned to note that dispensibility in the paper on qualitative probability, but I have to be concerned about increasing clutter in that paper. Yester-day, I decided to write a note — a very brief paper — that draws attention to the needlessness of the restriction. The note didn't take very long to write; I spent more time with the process of submission than with that of writing.

So, yes, a spin-off of a spin-off; but at least it is spun-off, instead of being one more thing pending. Meanwhile, as well as there now being three papers developed or being developed prior to that originally planned, I long ago saw that the original paper ought to have at least two sequels. If I complete the whole project, what was to be one paper will have become at least six.

The note has been submitted to a journal of logic, rather than of economics; likewise, I plan to submit the paper on qualitative probability to such a journal. While economics draws upon theories of probability, work that does not itself go beyond such theories would not typically be seen as economics. The body of the note just submitted is only about a hundred words and three formulæ. On top of the usual reasons for not knowing whether a paper will be accepted, a problem in this case is exactly that the point made by the paper will seem obvious, in spite of being repeatedly overlooked.

As to the remainder of the paper on qualitative probability, I'm working to get its axiomata into a presentable state. At present, it has more of them than I'd like.

Notions of Probability

Wednesday, 26 March 2014

I've previously touched on the matter of there being markèdly differing notions all associated with the word probability. Various attempts have been made by various writers to catalogue and to coördinate these notions; this will be one of my own attempts.

[an attempt to discuss conceptions of probability]

Quantifying Evidence

Friday, 12 August 2011
The only novel thing [in the Dark Ages] concerning probability is the following remarkable text, which appears in the False Decretals, an influential mixture of old papal letters, quotations taken out of context, and outright forgeries put together somewhere in Western Europe about 850. The passage itself may be much older. A bishop should not be condemned except with seventy-two witnesses … a cardinal priest should not be condemned except with forty-four witnesses, a cardinal deacon of the city of Rome without thirty-six witnesses, a subdeacon, acolyte, exorcist, lector, or doorkeeper except with seven witnesses.⁹ It is the world's first quantitative theory of probability. Which shows why being quantitative about probability is not necessarily a good thing.
James Franklin
The Science of Conjecture: Evidence and Probability before Pascal
Chapter 2

(Actually, there is some evidence that a quantitative theory of probability developed and then disappeared in ancient India.[10] But Franklin's essential point here is none-the-less well-taken.)

⁹ Foot-note in the original, citing Decretales Pseudo-Isidorianae, et Capitula Angilramni edited by Paul Hinschius, and recommending comparison with The Collection in Seventy-Four Titles: A Canon Law Manual of the Gregorian Reform edited by John Gilchrist.

[10] In The Story of Nala and Damayanti within the Mahābhārata, there is a character Rtuparna (aka Rituparna, and mistakenly as Rtupama and as Ritupama) who seems to have a marvelous understanding of sampling and is a master of dice-play. I learned about Rtuparna by way of Ian Hacking's outstanding The Emergence of Probability; Hacking seems to have learned of it by way of V.P. Godambe, who noted the apparent implication in A historical perspective of the recent developments in the theory of sampling from actual populations, Journal of the Indian Society of Agricultural Statistics v. 38 #1 (Apr 1976) pp 1-12.

Disappointment and Disgust

Sunday, 21 March 2010

In his Philosophical Theories of Probability, Donald Gillies proposes what he calls an intersubjective theory of probability. A better name for it would be group-strategy model of probability.

Subjectivists such as Bruno di Finetti ask the reader to consider the following sort of game:

  • Some potential event is identified.
  • Our hero must choose a real number (negative or positive) q, a betting quotient.
  • The nemesis, who is rational, must choose a stake S, which is a positive or negative sum of money or zero.
  • Our hero must, under any circumstance, pay the nemesis q·S. (If the product q·S is negative, then this amounts to the nemesis paying money to our hero.)
  • If the identified event occurs, then the nemesis must pay our hero S (which, if S is negative, then amounts to taking money from our hero). If it does not occur, then our hero gets nothing.
Di Finetti argues that a rational betting quotient will capture a rational degree of personal belief, and that a probability is exactly and only a degree of personal belief.

Gillies asks us to consider games of the very same sort, except that the betting quotients must be chosen jointly amongst a team of players. Such betting quotients would be at least examples of what Gillies calls intersubjective probabilities. Gillies tells us that these are the probabilities of rational consensus. For example, these are ostensibly the probabilities of scientific consensus.

Opponents of subjectivists such as di Finetti have long argued that the sort of game that he proposes fails in one way or another to be formally identical to the general problem for the application of personal degrees of belief. Gillies doesn't even try to show how the game, if played by a team, is formally identical to the general problem of group commitment to propositions. He instead belabors a different point, which should already be obvious to all of his readers, that teamwork is sometimes in the interest of the individual.

Amongst other things, scientific method is about best approximation of the truth. There are some genuine, difficult questions about just what makes one approximation better than another, but an approximation isn't relevantly better for promoting such things as the social standing as such or material wealth as such of a particular clique. It isn't at all clear who or what, in the formation of genuinely scientific consensus, would play a rôle that corresponds to that of the nemesis in the betting game.

Karl Popper, who proposed to explain probabilities in terms of objective propensities (rather than in terms of judgmental orderings or in terms of frequencies), asserted that

Causation is just a special case of propensity: the case of propensity equal to 1, a determining demand, or force, for realization.
Gillies joins others in taking him to task for the simple reason that probabilities can be inverted — one can talk both about the probability of A given B and that of B given A, whereäs presumably if A caused B then B cannot have caused A.

Later, for his own propensity theory, Gillies proposes to define probability to apply only to events that display a sort of independence. Thus, flips of coins might be described by probabilities, but the value of a random-walk process (where changes are independent but present value is a sum of past changes) would not itself have a probability. None-the-less, while the value of a random walk and similar processes would not themselves have probabilities, they'd still be subject to compositions of probabilities which we would previously have called probabilities.

In other words, Gillies has basically taken the liberty of employing a foundational notion of probability, and permitting its extension; he chooses not to call the extension probability, but that's just notation. Well, Popper had a foundational notion of propensity, which is a generalization of causality. He identified this notion with probability, and implicitly extended the notion to include inversions.

Later, Gillies offers dreadful criticism of Keynes. Keynes's judgmental theory of probability implies that every rational person with sufficient intellect and the same information set would ascribe exactly the same probability to a proposition. Gillies asserts

[…] different individuals may come to quite different conclusions even though they have the same background knowledge and expertise in the relevant area, and even though they are all quite rational. A single rational degree of belief on which all rational being should agree seems to be a myth.

So much for the logical interpretation of probability, […].
No two human beings have or could have the same information set. (I am reminded of infuriating claims that monozygotic children raised by the same parents have both the same heredity and the same environment.) Gillies writes of the relevant area, but in the formation of judgments about uncertain matters, we may and as I believe should be informed by a very extensive body of knowledge. Awareness that others might dismiss as irrelevant can provide support for general relationships. And I don't recall Keynes ever suggesting that there would be real-world cases of two people having the same information set and hence not disagreeing unless one of them were of inferior intellect.

After objecting that the traditional subjective theory doesn't satisfactorily cover all manner of judgmental probability, and claiming that his intersubjective notion can describe probabilities imputed by groups, Gillies takes another shot at Keynes:

When Keynes propounded his logical theory of probability, he was a member of an elite group of logically minded Cambridge intellectuals (the Apostles). In these circumstances, what he regarded as a single rational degree of belief valid for the whole of humanity may have been no more than the consensus belief of the Apostles. However admirable the Apostles, their consensus beliefs were very far from being shared by the rest of humanity. This became obvious in the 1930s when the Apostles developed a consensus belief in Soviet communism, a belief which was certainly not shared by everyone else.
Note the insinuation that Keynes thought that there were a single rational degree of belief valid for the whole of humanity, whereäs there is no indication that Keynes felt that everyone did, should, or could have the same information set. Rather than becoming obvious to him in the 1930s, it would have been evident to Keynes much earlier that many of his own beliefs and those of the other Apostles were at odds with those of most of mankind. Gillies' reference to embrace of Marxism in the '30s by most of the Apostles simply looks like irrelevant, Red-baiting ad hominem to me. One doesn't have to like Keynes (as I don't), Marxism (as I don't) or the Apostles (as I don't) to be appalled by this passage (as I am).

A Note to the Other Five

Sunday, 14 March 2010

Probability is one elephant, not two or more formally identical or formally similar elephants.

Dear Sir or Madam, will you read my book?

Saturday, 6 February 2010

Despite the fame of Laplace's Philosophical Essay on Probabilities, it is not in fact a very original work. The classical interpretation of probability emerged from discussion in the period roughly from 1650 to 1800, which saw the introduction which saw the introduction and development of the mathematical theory of probability. Most of the ideas of the classical theory are to be found in Part IV of Jacob Bernoulli's Ars Conjectandi, published in 1713, and Bernoulli had discussed these ideas in correspondence with Leibniz. Nonethless, it was Laplace's essay which introduced the ideas of the classical interpretation of probability to mathematicians and philosophers in the nineteenth century. This may simply have been because Laplace's essay was written in French and Bernoulli's's Ars Conjectandi in Latin, a language which was becoming increasingly unreadable by scientists and mathematicians in the nineteenth century.

Donald Gillies
Philosophical Theories of Probability
Ch 1 §1 (p3)

[…] Laplace generalised and improved the results of his predecessors — particularly those of Bernoulli, De Moivre and Bayes. His massive Théorie analytique des Probabilitiés, published in 1812, was the summary of more than a century and a half of mathematical research together with important developments by the author. This book established probability theory as no longer a minority interest but rather a major branch of mathematics.

Donald Gillies
Philosophical Theories of Probability
Ch 1 §2 (p8)

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.

this ebony bird beguiling

Tuesday, 14 April 2009

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.

We Don't Need No Stinkin' Bayesian Up-Dating!

Wednesday, 1 April 2009

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

View Results

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Marlboro Man

Wednesday, 18 March 2009

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,… Xn, each Xi having associated probability pi. If we assert that the expected value of this lottery were

Σ[pi · u(Xi)]
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[pi · u(Xi,pi)]
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:

{[(AB) 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.

another runner in the night

Saturday, 14 March 2009

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.

John Maynard Keynes
A Treatise on Probability (1921)
Part V Ch XXIX §18 ¶2 (pp 362-3)