Posts Tagged With: Jerzy Neyman

Guest Blog: ARIS SPANOS: The Enduring Legacy of R. A. Fisher

By Aris Spanos

One of R. A. Fisher’s (17 February 1890 — 29 July 1962) most re­markable, but least recognized, achievement was to initiate the recast­ing of statistical induction. Fisher (1922) pioneered modern frequentist statistics as a model-based approach to statistical induction anchored on the notion of a statistical model, formalized by:

Mθ(x)={f(x;θ); θ∈Θ}; x∈Rn ;Θ⊂Rm; m < n; (1)

where the distribution of the sample f(x;θ) ‘encapsulates’ the proba­bilistic information in the statistical model.

Before Fisher, the notion of a statistical model was vague and often implicit, and its role was primarily confined to the description of the distributional features of the data in hand using the histogram and the first few sample moments; implicitly imposing random (IID) samples. The problem was that statisticians at the time would use descriptive summaries of the data to claim generality beyond the data in hand x0:=(x1,x2,…,xn) As late as the 1920s, the problem of statistical induction was understood by Karl Pearson in terms of invoking (i) the ‘stability’ of empirical results for subsequent samples and (ii) a prior distribution for θ.

Fisher was able to recast statistical inference by turning Karl Pear­son’s approach, proceeding from data x0 in search of a frequency curve f(x;ϑ) to describe its histogram, on its head. He proposed to begin with a prespecified Mθ(x) (a ‘hypothetical infinite population’), and view x0 as a ‘typical’ realization thereof; see Spanos (1999). Continue reading

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“Statistical Concepts in Their Relation to Reality” by E.S. Pearson

To complete the last post, here’s Pearson’s portion of the “triad” 

E.S.Pearson on Gate

E.S.Pearson on Gate (sketch by D. Mayo)

“Statistical Concepts in Their Relation to Reality”

by E.S. PEARSON (1955)

SUMMARY: This paper contains a reply to some criticisms made by Sir Ronald Fisher in his recent article on “Scientific Methods and Scientific Induction”.

Controversies in the field of mathematical statistics seem largely to have arisen because statisticians have been unable to agree upon how theory is to provide, in terms of probability statements, the numerical measures most helpful to those who have to draw conclusions from observational data.  We are concerned here with the ways in which mathematical theory may be put, as it were, into gear with the common processes of rational thought, and there seems no reason to suppose that there is one best way in which this can be done.  If, therefore, Sir Ronald Fisher recapitulates and enlarges on his views upon statistical methods and scientific induction we can all only be grateful, but when he takes this opportunity to criticize the work of others through misapprehension of their views as he has done in his recent contribution to this Journal (Fisher 1955), it is impossible to leave him altogether unanswered.

In the first place it seems unfortunate that much of Fisher’s criticism of Neyman and Pearson’s approach to the testing of statistical hypotheses should be built upon a “penetrating observation” ascribed to Professor G.A. Barnard, the assumption involved in which happens to be historically incorrect.  There was no question of a difference in point of view having “originated” when Neyman “reinterpreted” Fisher’s early work on tests of significance “in terms of that technological and commercial apparatus which is known as an acceptance procedure”.  There was no sudden descent upon British soil of Russian ideas regarding the function of science in relation to technology and to five-year plans.  It was really much simpler–or worse.  The original heresy, as we shall see, was a Pearson one!

TO CONTINUE READING E.S. PEARSON’S PAPER CLICK HERE.

Categories: E.S. Pearson, phil/history of stat, Statistics | Tags: , , | Leave a comment

Aris Spanos: The Enduring Legacy of R. A. Fisher

spanos 2014

More Fisher insights from A. Spanos, this from 2 years ago:

One of R. A. Fisher’s (17 February 1890 — 29 July 1962) most re­markable, but least recognized, achievement was to initiate the recast­ing of statistical induction. Fisher (1922) pioneered modern frequentist statistics as a model-based approach to statistical induction anchored on the notion of a statistical model, formalized by:

Mθ(x)={f(x;θ); θ∈Θ}; x∈Rn ;Θ⊂Rm; m < n; (1)

where the distribution of the sample f(x;θ) ‘encapsulates’ the proba­bilistic information in the statistical model.

Before Fisher, the notion of a statistical model was vague and often implicit, and its role was primarily confined to the description of the distributional features of the data in hand using the histogram and the first few sample moments; implicitly imposing random (IID) samples. The problem was that statisticians at the time would use descriptive summaries of the data to claim generality beyond the data in hand x0:=(x1,x2,…,xn). As late as the 1920s, the problem of statistical induction was understood by Karl Pearson in terms of invoking (i) the ‘stability’ of empirical results for subsequent samples and (ii) a prior distribution for θ.

Fisher was able to recast statistical inference by turning Karl Pear­son’s approach, proceeding from data x0 in search of a frequency curve f(x;ϑ) to describe its histogram, on its head. He proposed to begin with a prespecified Mθ(x) (a ‘hypothetical infinite population’), and view x0 as a ‘typical’ realization thereof; see Spanos (1999).

In my mind, Fisher’s most enduring contribution is his devising a general way to ‘operationalize’ errors by embedding the material ex­periment into Mθ(x), and taming errors via probabilification, i.e. to define frequentist error probabilities in the context of a statistical model. These error probabilities are (a) deductively derived from the statistical model, and (b) provide a measure of the ‘effectiviness’ of the inference procedure: how often a certain method will give rise to correct in­ferences concerning the underlying ‘true’ Data Generating Mechanism (DGM). This cast aside the need for a prior. Both of these key elements, the statistical model and the error probabilities, have been refined and extended by Mayo’s error statistical approach (EGEK 1996). Learning from data is achieved when an inference is reached by an inductive procedure which, with high probability, will yield true conclusions from valid inductive premises (a statistical model); Mayo and Spanos (2011). Continue reading

Categories: Fisher, phil/history of stat, Statistics | Tags: , , , , , , | 2 Comments

Was Janina Hosiasson pulling Harold Jeffreys’ leg?

images

Hosiasson 1899-1942

The very fact that Jerzy Neyman considers she might have been playing a “mischievous joke” on Harold Jeffreys (concerning probability) is enough to intrigue and impress me (with Hosiasson!). I’ve long been curious about what really happened. Eleonore Stump, a leading medieval philosopher and friend (and one-time colleague), and I pledged to travel to Vilnius to research Hosiasson. I first heard her name from Neyman’s dedication of Lectures and Conferences in Mathematical Statistics and Probability: “To the memory of: Janina Hosiasson, murdered by the Gestapo” along with around 9 other “colleagues and friends lost during World War II.” (He doesn’t mention her husband Lindenbaum, shot alongside her*.)  Hosiasson is responsible for Hempel’s Raven Paradox, and I definitely think we should be calling it Hosiasson’s (Raven) Paradox for much of the lost credit to her contributions to Carnapian confirmation theory[i].

questionmark pink

But what about this mischievous joke she might have pulled off with Harold Jeffreys? Or did Jeffreys misunderstand what she intended to say about this howler, or?  Since it’s a weekend and all of the U.S. monuments and parks are shut down, you might read this snippet and share your speculations…. The following is from Neyman 1952:

“Example 6.—The inclusion of the present example is occasioned by certain statements of Harold Jeffreys (1939, 300) which suggest that, in spite of my 
insistence on the phrase, “probability that an object A will possess the 
property B,” and in spite of the five foregoing examples, the definition of 
probability given above may be misunderstood.
 Jeffreys is an important proponent of the subjective theory of probability
 designed to measure the “degree of reasonable belief.” His ideas on the
 subject are quite radical. He claims (1939, 303) that no consistent theory of probability is possible without the basic notion of degrees of reasonable belief. 
His further contention is that proponents of theories of probabilities alternative to his own forget their definitions “before the ink is dry.”  In 
Jeffreys’ opinion, they use the notion of reasonable belief without ever
 noticing that they are using it and, by so doing, contradict the principles 
which they have laid down at the outset.

The necessity of any given axiom in a mathematical theory is something
 which is subject to proof. …
However, Dr. Jeffreys’ contention that the notion of degrees of reasonable
 belief and his Axiom 1are necessary for the development of the theory 
of probability is not backed by any attempt at proof. Instead, he considers
 definitions of probability alternative to his own and attempts to show by
 example that, if these definitions are adhered to, the results of their application would be totally unreasonable and unacceptable to anyone. Some 
of the examples are striking. On page 300, Jeffreys refers to an article of
 mine in which probability is defined exactly as it is in the present volume.
 Jeffreys writes:

The first definition is sometimes called the “classical” one, and is stated in much 
modern work, notably that of J. Neyman.

However, Jeffreys does not quote the definition that I use but chooses 
to reword it as follows:

If there are n possible alternatives, for m of which p is true, then the probability of 
p is defined to be m/n.


He goes on to say:

The first definition appears at the beginning of De Moivre’s book (Doctrine of 
Chances, 1738). It often gives a definite value to a probability; the trouble is that the 
value is one that its user immediately rejects. Thus suppose that we are considering 
two boxes, one containing one white and one black ball, and the other one white and 
two black. A box is to be selected at random and then a ball at random from that box.
 What is the probability that the ball will be white? There are five balls, two of which 
are white. Therefore, according to the definition, the probability is 2/5. But most
 statistical writers, including, I think, most of those that professedly accept the definition,
would give (1/2)•(1/2) + (1/2)•(1/3) = 5/12. This follows at once on the present theory, 
the terms representing two applications of the product rule to give the probability of 
drawing each of the two white balls. These are then added by the addition rule. But 
the proposition cannot be expressed as the disjunction of five alternatives out of twelve.
 My attention was called to this point by Miss J. Hosiasson.


The solution, 2/5, suggested by Jeffreys as the result of an allegedly 
strict application of my definition of probability is obviously wrong. The 
mistake seems to be due to Jeffreys’ apparently harmless rewording of the 
definition. If we adhere to the original wording (p. 4) and, in particular, to the 
phrase “probability of an object A having the property B,” then, prior to
attempting a solution, we would probably ask ourselves the questions: 
”What are the ‘objects A’ in this particular case?” and “What is the
’property B,’ the probability of which it is desired to compute?” Once 
these questions have been asked, the answer to them usually follows and
 determines the solution.

In the particular example of Dr. Jeffreys, the objects A are obviously 
not balls, but pairs of random selections, the first of a box and the second 
of a ball. If we like to state the problem without dangerous abbreviations, 
the probability sought is that of a pair of selections ending with a white 
ball. All the conditions of there being two boxes, the first with two balls 
only and the second with three, etc., must be interpreted as picturesque 
descriptions of the F.P.S. of pairs of selections. The elements of this set 
fall into four categories, conveniently described by pairs of symbols (1,w),
(1,b), (2,w), (2,b), so that, for example, (2,w) stands for a pair of 
selections in which the second box was selected in the first instance, and
 then this was followed by the selection of the white ball. Denote by n1,w, n1,b, n2,w, and n2,b the (unknown) numbers of the elements of the F.P.S. belonging to each of the above categories, and by n their sum. Then the probability sought is “(Neyman 1952, 10-11).

Then there are the detailed computations from which Neyman gets the right answer (entered 10/9/13):

P{w|pair of selections} = (n1,w + n2,w)/n.

The conditions of the problem imply

P{1|pair of selections} = (n1,w + n1,b)/n = ½,

P{2|pair of selections} = (n2,w + n2,b)/n = ½,

P{w| pair of selections beginning with box No. 1} = n1,w/(n1,w + n1,b) = ½,

P{w| pair of selections beginning with box No. 2} = n2,w/(n2,w + n2,b) = 1/3.

It follows

n1,w = 1/2(n1,w + n1,b) = n/4,

n2,w = 1/3(n2,w + n2,b)  = n/6,

P{w|pair of selections} = 5/12.

The method of computing probability used here is a direct enumeration 
of elements of the F.P.S. For this reason it is called the “direct method.”
 As we can see from this particular example, the direct method is occasionally cumbersome and the correct solution is more easily reached through
 the application of certain theorems basic in the theory of probability. These theorems, the addition theorem and the multiplication theorem, are very 
easy to apply, with the result that students frequently manage to learn the
 machinery of application without understanding the theorems. To check
 whether or not a student does understand the theorems, it is advisable to 
ask him to solve problems by the direct method. If he cannot, then he
 does not understand what he is doing.

Checks of this kind were part of the regular program of instruction in
 Warsaw where Miss Hosiasson was one of my assistants. Miss Hosiasson
 was a very talented lady who has written several interesting contributions 
to the theory of probability. One of these papers deals specifically with
 various misunderstandings which, under the high sounding name of paradoxes, still litter the scientific books and journals. Most of these paradoxes originate from lack of precision in stating the conditions of the 
problems studied. In these circumstances, it is most unlikely that Miss 
Hosiasson could fail in the application of the direct method to a simple 
problem like the one described by Dr. Jeffreys. On the other hand, I can 
well imagine Miss Hosiasson making a somewhat mischievous joke.


Some of the paradoxes solved by Miss Hosiasson are quite amusing…….”
 (Neyman 1952, 10-13)

What think you? I will offer a first speculation in a comment.

The entire book Neyman (1952) may be found here, in plain text, here.

*June, 2017: I read somewhere today that her husband was killed in 41, so before she was, but all refs I know are sketchy.

[i]Of course there are many good, recent sources on the philosophy and history of Carnap, some of which mention her, but obviously do not touch on this matter. I read that Hosiasson was trying to build a Carnapian-style inductive logic setting out axioms (which to my knowledge Carnap never did). That was what some of my fledgling graduate school attempts had tried, but the axioms always seemed to admit counterexamples (if non-trivial). So much for the purely syntactic approach. But I wish I’d known of her attempts back then, and especially her treatment of paradoxes of confirmation. {I’m sometimes tempted to give a logic for severity, but I fight the temptation.)

REFERENCES

Hosiasson, J. (1931) Why do we prefer probabilities relative to many data? Mind 40 (157): 23-36 (1931)

Hosiasson-Lindenbaum, J. (1940) On confirmation Journal of Symbolic Logic 5 (4): 133-148 (1940)

Hosiasson, J. (1941) Induction et analogie: Comparaison de leur fondement Mind 50 (200): 351-365 (1941)

Hosiasson-Lindenbaum, J. (1948) Theoretical Aspects of the Advancement of Knowledge Synthese 7 (4/5):253 – 261 (1948)

Jeffreys, H. (1939) Theory of Probability (1st ed.). Oxford: The Clarendon Press

Neyman, J. (1952) Lectures and Conferences in Mathematical Statistics and Probability. Graduate School, U.S. Dept. of Agriculture

Categories: Hosiasson, phil/history of stat, Statistics | Tags: , | 19 Comments

A. Spanos: Jerzy Neyman and his Enduring Legacy

A Statistical Model as a Chance Mechanism

Aris Spanos

Jerzy Neyman (April 16, 1894 – August 5, 1981), was a Polish/American statistician[i] who spent most of his professional career at the University of California, Berkeley. Neyman is best known in statistics for his pioneering contributions in framing the Neyman-Pearson (N-P) optimal theory of hypothesis testing and his theory of Confidence Intervals.

One of Neyman’s most remarkable, but least recognized, achievements was his adapting of Fisher’s (1922) notion of a statistical model to render it pertinent for  non-random samples. Fisher’s original parametric statistical model Mθ(x) was based on the idea of ‘a hypothetical infinite population’, chosen so as to ensure that the observed data x0:=(x1,x2,…,xn) can be viewed as a ‘truly representative sample’ from that ‘population’:

“The postulate of randomness thus resolves itself into the question, Of what population is this a random sample? (ibid., p. 313), underscoring that: the adequacy of our choice may be tested a posteriori.’’ (p. 314)

In cases where data x0 come from sample surveys or it can be viewed as a typical realization of a random sample X:=(X1,X2,…,Xn), i.e. Independent and Identically Distributed (IID) random variables, the ‘population’ metaphor can be helpful in adding some intuitive appeal to the inductive dimension of statistical inference, because one can imagine using a subset of a population (the sample) to draw inferences pertaining to the whole population.

This ‘infinite population’ metaphor, however, is of limited value in most applied disciplines relying on observational data. To see how inept this metaphor is consider the question: what is the hypothetical ‘population’ when modeling the gyrations of stock market prices? More generally, what is observed in such cases is a certain on-going process and not a fixed population from which we can select a representative sample. For that very reason, most economists in the 1930s considered Fisher’s statistical modeling irrelevant for economic data! Continue reading

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That Promissory Note From Lehmann’s Letter; Schmidt to Speak

Juliet Shaffer and Erich Lehmann

Monday, April 16, is Jerzy Neyman’s birthday, but this post is not about Neyman (that comes later, I hope). But in thinking of Neyman, I’m reminded of Erich Lehmann, Neyman’s first student, and a promissory note I gave in a post on September 15, 2011.  I wrote:

“One day (in 1997), I received a bulging, six-page, handwritten letter from him in tiny, extremely neat scrawl (and many more after that).  …. I remember it contained two especially noteworthy pieces of information, one intriguing, the other quite surprising.  The intriguing one (I’ll come back to the surprising one another time, if reminded) was this:  He told me he was sitting in a very large room at an ASA meeting where they were shutting down the conference book display (or maybe they were setting it up), and on a very long, dark table sat just one book, all alone, shiny red.  He said he wondered if it might be of interest to him!  So he walked up to it….  It turned out to be my Error and the Growth of Experimental Knowledge (1996, Chicago), which he reviewed soon after.”

But what about the “surprising one” that I was to come back to “if reminded”? (yes, one person did remind me last month). The surprising one is that Lehmann’s letter—this is his first letter to me– asked me to please read a paper by Frank Schmidt to appear in his wife Juliet Shaffer’s new (at the time) journal, Psychological Methods, as he wondered if I had any ideas as to what may be done to answer such criticisms of frequentist tests!   But, clearly, few people could have been in a better position than Lehmann to “do something about” these arguments …hence my surprise.  But I think he was reluctant…. Continue reading

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Guest Blogger. ARIS SPANOS: The Enduring Legacy of R. A. Fisher

By Aris Spanos

One of R. A. Fisher’s (17 February 1890 — 29 July 1962) most re­markable, but least recognized, achievement was to initiate the recast­ing of statistical induction. Fisher (1922) pioneered modern frequentist statistics as a model-based approach to statistical induction anchored on the notion of a statistical model, formalized by:

Mθ(x)={f(x;θ); θ∈Θ}; x∈Rn ;Θ⊂Rm; m < n; (1)

where the distribution of the sample f(x;θ) ‘encapsulates’ the proba­bilistic information in the statistical model.

Before Fisher, the notion of a statistical model was vague and often implicit, and its role was primarily confined to the description of the distributional features of the data in hand using the histogram and the first few sample moments; implicitly imposing random (IID) samples. The problem was that statisticians at the time would use descriptive summaries of the data to claim generality beyond the data in hand x0:=(x1,x2,…,xn) As late as the 1920s, the problem of statistical induction was understood by Karl Pearson in terms of invoking (i) the ‘stability’ of empirical results for subsequent samples and (ii) a prior distribution for θ.

Continue reading

Categories: Statistics | Tags: , , , , , , | 5 Comments

E.S. PEARSON: Statistical Concepts in Their Relation to Reality

by E.S. PEARSON (1955)

SUMMARY: This paper contains a reply to some criticisms made by Sir Ronald Fisher in his recent article on “Scientific Methods and Scientific Induction”.

Controversies in the field of mathematical statistics seem largely to have arisen because statisticians have been unable to agree upon how theory is to provide, in terms of probability statements, the numerical measures most helpful to those who have to draw conclusions from observational data.  We are concerned here with the ways in which mathematical theory may be put, as it were, into gear with the common processes of rational thought, and there seems no reason to suppose that there is one best way in which this can be done.  If, therefore, Sir Ronald Fisher recapitulates and enlarges on his views upon statistical methods and scientific induction we can all only be grateful, but when he takes this opportunity to criticize the work of others through misapprehension of their views as he has done in his recent contribution to this Journal (Fisher 1955), it is impossible to leave him altogether unanswered.

Continue reading

Categories: Statistics | Tags: , , , , , , | 2 Comments

R.A.FISHER: Statistical Methods and Scientific Inference

In honor of R.A. Fisher’s birthday this week (Feb 17), in a year that will mark 50 years since his death, we will post the “Triad” exchange between  Fisher, Pearson and Neyman, and other guest contributions*

by Sir Ronald Fisher (1955)

SUMMARY

The attempt to reinterpret the common tests of significance used in scientific research as though they constituted some kind of  acceptance procedure and led to “decisions” in Wald’s sense, originated in several misapprehensions and has led, apparently, to several more.

The three phrases examined here, with a view to elucidating they fallacies they embody, are:

  1. “Repeated sampling from the same population”,
  2. Errors of the “second kind”,
  3. “Inductive behavior”.

Mathematicians without personal contact with the Natural Sciences have often been misled by such phrases. The errors to which they lead are not only numerical.

TO CONTINUE READING R. A. FISHER’S  PAPER, CLICK HERE.

*If you wish to contribute something in connection to Fisher, send to error@vt.edu

Categories: Statistics | Tags: , , , , , , | 5 Comments

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