tag:blogger.com,1999:blog-15272751260384083372011-11-13T03:08:05.981-08:00This blog is related to the book "The Search for Certainty. On the Clash of Science and Philosophy of Probability" by Krzysztof Burdzy (World Scientific, 2009). See the <a href="http://www.worldscibooks.com/mathematics/7312.html">book description and sample chapters, and buy the book online</a>.Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.comBlogger311100tag:blogger.com,1999:blog-1527275126038408337.post-44233379741585748992011-09-06T22:17:00.000-07:002011-09-06T22:17:58.139-07:00

The main thesis of this essay is that practitioners of the so-called frequency approach to probability apply probabilistic ideas in ways that are exactly the opposite to the theoretical claims of the philosophical theory known as the frequency theory of probability. Similarly, practitioners of the so-called Bayesian approach to probability apply probabilistic ideas in ways that are exactly the

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-54665592927764280492011-08-11T22:17:00.001-07:002011-08-11T22:19:29.012-07:00

My scientific laws of probability (L1)-(L5) are based on a philosophical idea of Karl Popper. Specifically, (L5) is an embodiment of Popper's falsifiability idea. A number of people pointed out to me that Popper is not popular any more. Mathematical truths and scientific laws are forever. The Pythagorean Theorem was proved 2,500 years ago and it is still true today. Archimedes' Principle was

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-59714720869419723432011-07-07T11:09:00.000-07:002011-07-07T11:12:44.591-07:00

One should stress events rather than random variables in the foundations of probability. This is because, most of the time, events are less controversial than random variables. For example, the data are usually uncontroversial from the philosophical point of view. If the blood pressure of a patient was measured to be 110, then this is accepted as a fact and does not lead to philosophical disputes

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-34961177992341313522011-06-01T20:11:00.000-07:002011-06-01T20:17:17.890-07:00

There are many philosophical theories of science. I will outline a particular view of science to provide a context for an explanation of why I am unhappy with the division of statistics into frequency and Bayesian branches.I consider a direction in intellectual activity a hard-core science if scientists active in this area either achieved consensus on all essential issues or have a reasonable

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-82133919618018616552011-05-07T12:52:00.000-07:002011-05-07T12:57:02.489-07:00

Logic is used by mathematicians and scientists at different levels. At the most elementary level, we have the "propositional calculus" which represents the standard logic applied by practically all mathematicians and scientists in the bulk of their research. At an intermediate level lie the induction principle and the diagonal argument of Cantor. A much more sophisticated example is the "

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-48185311361932750272011-04-05T11:35:00.000-07:002011-04-05T11:40:22.737-07:00

A subjective (personal) utility function has no predictive power because subjective preferences can change in an arbitrary way. This does not mean that they do. But if they do, there is nothing irrational about an arbitrary change of subjective (personal) utility.To see the significance of the above remarks, we should consider two related but substantially (perhaps fundamentally) different ways

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-3633859989864059582011-03-07T20:39:00.000-08:002011-03-07T20:43:44.399-08:00

A utility function is interesting from the philosophical point of view only if it is subjective. If the utility function can be effectively measured in an objective and scientific way then it constitutes, in part, an objective rescaling of the real line, from currency units to utility units. An objective utility function also assigns value to each non-monetary reward, such as friendship. On the

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-35173067924958483692011-02-08T18:27:00.000-08:002011-02-08T18:31:41.651-08:00

I guess that there are many unrelated reasons why mathematics is useful. I will discuss only one of them in relation to the frequency philosophy of probability.Consider a scientific law that involves a mathematical formula, such as F = ma, one of Newton's laws of motion. Here F is the force, m is the mass and a is the acceleration. Suppose that we verified that the law is correct with sufficient

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-24286374161715508302011-01-08T12:02:00.000-08:002011-01-08T12:10:47.122-08:00

Recall the St Petersburg paradox from an earlier blog entry. Consider the following variant of St Petersburg game. In the new game, the payoffs are 1 dollar with probability 1/2, 2 dollars with probability 1/4, 4 dollars with probability 1/8, 100 times (2 to power 1,000,000,000,000,000,000,000,000) dollars with probability 2 to power -1,000,000,000,000,000,000,000,000,and with the remaining

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com2tag:blogger.com,1999:blog-1527275126038408337.post-44002362664472698542010-12-09T12:01:00.000-08:002010-12-09T15:59:33.195-08:00

Please see a description of the St Petersburg paradox in an earlier blog entry. Many of the philosophical questions related to probability can be illustrated with a game much simpler than the St Petersburg paradox. Let G be a game which pays a reward 50 times (2 to power 1,000,000,000,000,000,000,000,000) dollars with probability 2 to power -1,000,000,000,000,000,000,000,000.With the remaining

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-80923191603598116492010-11-06T14:57:00.000-07:002010-11-07T07:50:39.485-08:00

The rules that apply to ordinary followers of a religion do not necessarily apply to high priests. Ordinary Catholic men must remove their hats when they enter a church. This "universal" rule does not apply to the Pope and the cardinals. Suppose that a company wants to introduce a new drug. Before the drug is approved, the authorities, doctors and patients expect the company to perform clinical

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com3tag:blogger.com,1999:blog-1527275126038408337.post-58119129755078131332010-10-09T13:38:00.000-07:002010-10-09T13:48:34.709-07:00

A.I. Dale wrote a Math Review (MR2510150) of my book. I find his style somewhat strange and I do not think that the aphorisms quoted by Dale contribute much to our understanding of probability or to the discussioninitiated in my book (see On aphorisms). However, I consider the review to be fair and reasonably accurate. I have multiple complaints about only one paragraph in Dale's review. The

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-77424680101294661422010-09-10T04:50:00.000-07:002010-09-10T05:10:09.642-07:00

A book review gives an opportunity to its author to express an idea without supplying any justification - the necessary brevity of a review provides a good excuse. The result is that an idea expressed by the reviewer may be a dead end - it cannot be further discussed because there is no foundation for any further discussion. Readers who like the general tone of the review will applaud the

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com2tag:blogger.com,1999:blog-1527275126038408337.post-90089910475545301822010-08-11T20:31:00.000-07:002010-08-11T20:40:16.684-07:00

I used to admire aphorisms and their authors. My admiration faded when I read the Math Review of my book (MR2510150) by A.I. Dale. Let me start with a legal disclaimer - I do not blame Dale himself for all the problems with aphorisms that I complain about below, just like an atheist should not blame a single religious person for all the alleged problems with religion.After reading of Dale's

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com4tag:blogger.com,1999:blog-1527275126038408337.post-82367471426346756522010-07-09T11:05:00.000-07:002010-07-09T11:10:15.563-07:00

Stephen Unwin applied the Bayes formula to calculate the probability of God in his book, "The Probability of God: A Simple Calculation That Proves the Ultimate Truth".I believe that the question of probability of God has much more to do with philosophy of God than philosophy of probability. Hence, personally, I do not find it particularly interesting. But Unwin's calculation does inspire some

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-56294133796557699992010-06-03T10:11:00.000-07:002010-06-03T10:30:07.676-07:00

One expects articles that appear in professional journals to be more polished and accurate than blog posts. So I waited with interest to see how Andrew Gelman's comments on my book published in his blog would change in the "final" version published in a scientific journal Bayesian Analysis. They appeared in vol. 5 (2010) as Comment on Article by Robert pp. 229 - 232. In what follows, I will refer

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-38186624068965086812010-05-08T22:22:00.000-07:002010-05-08T22:27:52.756-07:00

Subjective utility of very large numbers is not bounded but undefined. Let N = 2 to power 1,000,000,000,000,000,000,000,000.Some people may say that the utility of having 2N dollars is not smaller than the utility of N dollars. The standard explanation for this position is that if you happen to prefer N dollars rather than 2N dollars then you can give away N dollars. When was it last time that

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-81198391957911351612010-04-10T14:27:00.000-07:002010-04-10T14:36:57.105-07:00

Statisticians may help their colleagues working in the distant future evaluate currently used statistical methods. One way to do that would be to include explicit isolated predictions in published results of statistical analysis. Typical results of a statistical analysis include a large number of probabilistic statements, implicit or explicit. A good example is the posterior distribution in

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com2tag:blogger.com,1999:blog-1527275126038408337.post-50293962224159694942010-03-09T12:02:00.000-08:002010-03-09T12:06:48.873-08:00

The area of foundations of probability is awash with interesting ideas. The first memorable ideas were proposed at least three centuries ago by Bernoulli in relation to St. Petersburg paradox. The Achilles heel of foundations of probability is a construction of a complete philosophical theory of probability. A number of philosophers (Carnap, de Finetti, Gillies, Hacking, von Mises, to name just a

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-58478689456449253152010-02-13T15:42:00.000-08:002010-02-13T15:57:11.938-08:00

This is an answer to Andrew Gelman's blog entries in which he disagrees with my statement that "standard textbooks on chemistry do not discuss subjectivity in their introductions, and so statistical textbooks need not to do that either." (See here and here.)Words have multiple meanings. Consider, for example, the following statements about "communism". (i) Communism is a system free of alienation

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com2tag:blogger.com,1999:blog-1527275126038408337.post-53747959989241220722010-02-11T10:31:00.000-08:002010-07-10T14:18:56.948-07:00

Here is a simple illustration of how the concepts of subjectivity used in de Finetti's theory and Bayesian statistics differ. People have sensations of temperature, beauty and probability. All of these sensations are subjective in more than one sense: (i) They are all experienced by humans. (ii) There are differences between how different people experience these sensations. (iii) The sensations

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-81952463989026146502010-01-09T17:03:00.000-08:002010-01-09T17:08:25.656-08:00

There is only one practical situation where the notion of unbiasedness has a natural frequency interpretation. This is estimation of the current time by owners of various clocks and watches. In this example, estimates, that is, time readings by different clocks, are (hopefully) unbiased. With some stretch of reality, we may say that the estimates are not very far from identically distributed and

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-32775056764935476222009-12-10T13:03:00.000-08:002010-11-08T19:15:50.393-08:00

The following comes from the Wikipedia article on St Petersburg paradox"In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a tail first appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-53149356190973672672009-11-09T20:57:00.000-08:002009-11-11T11:15:49.882-08:00

A good way to illustrate the complete intellectual failure of the frequency and subjective philosophies of probability is to compare them to the behaviorist movement in psychology.The following comes from the Wikipedia article on "Cognitive revolution" (accessed on October 30, 2009):"... [Behaviorists] proposed that psychology could only become an objective science were it based on observable

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-43224640205079840692009-10-10T09:43:00.000-07:002009-10-10T13:36:10.828-07:00

A probabilistic prediction in the sense of (L5) is a crude approximation of real applications of probability, a "black and white" theoretical concept. Classical logic and set theory are based on similarly crude truth-falsehood dichotomy - an extreme oversimplification of reality. Nevertheless both fields of mathematics are very useful in real life. When needed, scientists and ordinary people

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-76209640505660591702009-09-09T20:37:00.000-07:002009-09-09T20:40:51.878-07:00

De Finetti's justification for using probability is bogus. If mathematical rules for calculating probabilities are used only to make sure that decisions are consistent, one has to check which of the two methods of attaining consistency is more efficient: (i) checking consistency of decisions in the direct way, using only decision-theoretic axioms and deductions from them, or (ii) via

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-24652185384135008342009-08-11T03:10:00.000-07:002010-07-10T14:15:35.211-07:00

Von Mises' collectives were not intended to be interpreted as fixed populations. We can see this from two features of von Mises' theory. (i) The frequency interpretation of probability holds in all fixed populations, of any size. Von Mises clearly thought about collectives as large sets. (ii) The order of events in a fixed population does not matter - the usual probabilistic formulas hold for

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-57910616572007876202009-06-16T08:15:00.000-07:002009-06-16T08:21:03.631-07:00

First, let me explain what I mean by "necessary". I consider the statement "11 is a prime number" necessary because I cannot imagine a universe in which one could arrange 11 apples in some number of rows, with the same number of apples in each row, except 1 row or 11 rows.I do not consider (L1)-(L5) necessary in the same sense. I can imagine a universe in whichsome laws of nature are

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-4256658859707064172009-06-08T08:55:00.000-07:002009-06-08T09:01:12.042-07:00

De Finetti and von Mises were geniuses because they realized that one cannot avoid making some inconvenient or counterintuitive philosophical choices in the area of foundations of probability. This puts them far above many other authors who tried to fix the most striking problems with the frequency and subjective theories by taking compromise positions on some issues. Such fixes inevitably

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-52504551609899405492009-06-02T22:12:00.000-07:002009-06-08T17:15:54.217-07:00

Logical, subjective and frequency philosophies, and Kolmogorov's axioms contain a misleading implicit message that one has to know probabilities of all events in the sample space to be able to proceed. In fact, knowledge of some restrictions on probability values is enough to generate useful predictions. For example, it is not necessary to know the probability of heads for a deformed coin to be

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0tag:blogger.com,1999:blog-1527275126038408337.post-10234845572318610222009-06-02T15:56:00.000-07:002009-06-08T17:19:26.631-07:00

Page 78, line 10 from the bottom:`Only in rare practical situations one decision maker deals with a sequence of independent and identical decision problems.'should be`Only in some practical situations one decision maker deals with a sequence of independent and identical decision problems.'See Sec. 6.3 on page 121 for a more detailed discussion of situations when repeated decisions may or may not

Krzysztof Burdzyhttp://www.blogger.com/profile/14066066834084162583noreply@blogger.com0