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Publisher:

Dover Publications

Publication Date:

1981

Number of Pages:

244

Format:

Paperback

Edition:

2

Price:

12.95

ISBN:

0486242145

Category:

Monograph

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Robert W. Hayden

11/14/2010

Today this work is generally thought of as the classic statement of the frequentist position on the foundations of probability as contrasted with a Bayesian approach. In its own day, and even more in the earlier years of von Mises research, the principal alternative to the frequentist position was not the Bayesian outlook of the past 50 years but rather the “equiprobable” outlook of many of the pioneers of probability theory.

These pioneers often tacitly assumed that one could list the possible outcomes and define the probability of an event as the number of “favorable” outcomes divided by the total number of outcomes. This works for one toss of one coin, but if you toss two, and list the outcomes as 0, 1 or 2 heads, then the results do not agree well with the real world behavior of coins. One head comes up about twice as often as none or two. What von Mises did to deal with this was to assert that probability pertained to limiting values of relative frequencies of repeated events. This made probability more empirical and less *a priori*. In addition, von Mises contributed ideas on determining what situations in the real world might reasonably be treated via probability. In particular, he had some interesting things to say on the nature of randomness.

Like many of the older books, this one is best understood in the context of its own time and still earlier times. Transposing it into issues of our own day may be risky. The author does mention subjective probability here and there, but barely mentions the ideas that today would be labelled “Bayesian.” Today that approach is most common in business. As an example of the difference in the two approaches, consider a manager who believes that a new product will appeal to 40% of its market. A survey might later indicate the appeal is closer to 20%. A frequentist would probably go with the 20% figure while the Bayesian would try to combine the two estimates into a new and intermediate one.

While the frequentist and Bayesian approaches are often treated as mutually exclusive, it might be more accurate to say that Bayesians allow frequentist probabilities but allow subjective probabilities as well. Mathematicians have a very different view of foundations, and some of us might simply say that anything that satisfies the Kolmogorov axioms is probability, while allowing that some situations are a better match than others, and that the proof of the pudding should be how well the results of applying probability theory match reality.

It may also be true that the two approaches suit different problems. A classic 1960 paper by John Tukey distinguishes between conclusions and decisions. Decisions are unique situations such as our example of marketing a new product. A decision has to be made once and for all. It that case the Bayesian approach has appeal. Conclusions are scientific beliefs based on an accumulation of evidence, and in science including the researcher’s prior opinions is not considered cricket. It is clear that von Mises’ interest is in the role of probability in science, so though he takes a dim view of subjective probabilities, he is evaluating them as a basis for scientific conclusions rather than practical decisions.

In light of all this, it might be better to read Tukey and more recent authors on frequentist versus Bayesian statistics. One might read von Mises to learn more about the history of probability and statistics and mathematics, and the connection between these disciplines and science. In its time, the frequentist outlook resolved some of the paradoxes and inconsistencies in earlier writings. As with any authentic source from an earlier age, the reader of this book will find some unfamiliar terminology, probably further complicated by the translation from German. However, for anyone interested in history of the issues it covers, this book is *the* book to read.

**Reference**

Tukey, J. W. (1960) “Conclusions versus Decisions” *Technometrics* **2** 424-433.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA’s Teaching Statistics.

PREFACE

PREFACE TO THE THIRD GERMAN EDITION

FIRST LECTURE The Definition of Probability

Amendment of Popular Terminology

Explanation of Words

Synthetic Definitions

Terminology

The Concept of Work in Mechanics

An Historical Interlude

The Purpose of Rational Concepts

The Inadequacy of Theories

Limitation of Scope

Unlimited Repetition

The Collective

The First Step towards a Definition

Two Different Pairs of Dice

Limiting Value of Relative Frequency

The Experimental Basis of the Theory of Games

The Probability of Death

First the Collective-then the Probability

Probability in the Gas Theory

An Historical Remark

Randomness

Definition of Randomness: Place Selection

The Principle of the Impossibility of a Gambling System

Example of Randomness

Summary of the Definition

SECOND LECTURE The Elements of the Theory of Probability

The Theory of Probability is a Science Similar to Others

The Purpose of the Theory of Probability

The Beginning and the End of Each Problem must be Probabilities

Distribution in a Collective

Probability of a Hit; Continuous Distribution

Probability Density

The Four Fundamental Operations

First Fundamental Operation: Selection

Second Fundamental Operation: Mixing

Inexact Statement of the Addition Rule

Uniform Distribution

Summary of the Mixing Rule

Third Fundamental Operation: Partition

Probabilities after Partition

Initial and Final Probability of an Attribute

The So-called Probability of Causes

Formulation of the rule of Partition

Fourth Fundamental Operation: Combination

A New Method of Forming Partial Sequences: Correlated Sampling

Mutually Independent Collectives

Derivation of the Multiplication Rule

Test of Independence

Combination of Dependent Collectives

Example of Noncombinable Collectives

Summary of the Four Fundamental Operations

A Problem of Chevalier de Mâ€šrâ€š

Solution of the Problem of Chevalier de Mâ€šrâ€š

Discussion of the Solution

Some Final Conclusions

Short Review

THIRD LECTURE Critical Discussion of the Foundations of Probability

The Classical Definition of Probability

Equally Likely Cases ...

... Do Not Always Exist

A Geometrical Analogy

How to Recognize Equally Likely Cases

Are Equally Likely Cases of Exceptional Significance?

The Subjective Conception of Probability

Bertrand's Paradox

The Suggested Link between the Classical and the New Definitions of Probability

Summary of Objections to the Classical Definition

Objections to My Theory

Finite Collectives

Testing Probability Statements

An Objection to the First Postulate

Objections to the Condition of Randomness

Restricted Randomness

Meaning of the Condition of Randomness

Consistency of the Randomness Axiom

A Problem of Terminology

Objections to the Frequency Concept

Theory of the Plausibility of Statements

The Nihilists

Restriction to One Single Initial Collective

Probability as Part of the Theory of Sets

Development of the Frequency Theory

Summary and Conclusion

FOURTH LECTURE The Laws of Large Numbers

Poisson's Two Different Propositions

Equally Likely Events

Arithmetical Explanation

Subsequent Frequency Definition

The Content of Poisson's Theorem

Example of a Sequence to which Poisson's Theorem does not Apply

Bernoulli and non-Bernoulli Sequences

Derivation of the Bernoulli-Poison Theorem

Summary

Inference

Bayes's Problem

Initial and Inferred Probability

Longer Sequences of Trials

Independence of the Initial Distribution

The Relation of Bayes's Theorem to Poisson's Theorem

The Three Propositions

Generalization of the Laws of Large Numbers

The Strong Law of Large Numbers

The Statistical Functions

The First Law of Large Numbers for Statistical Functions

The Second Law of Large Numbers for Statistical Functions

Closing Remarks

FIFTH LECTURE Application Statistics and the Theory of Errors

What is Statistics?

Games of Chance and Games of Skill

Marbe's Uniformity in the World'

Answer to Marbe's Problem

Theory of Accumulation and the Law of Series

Linked Events

The General Purpose of Statistics

Lexis' Theory of Dispersion

The Mean and the Dispersion

Comparison between the Observed and the Expected Variance

Lexis' Theory and the Laws of Large Numbers

Normal and Nonnormal Dispersion

Sex Distribution of Infants

Statistics of Deaths with Supernormal Dispersion

Solidarity of Cases

Testing Hypotheses

R. A. Fisher's Likelihood'

Small Sample Theory

Social and Biological Statistics

Mendel's Theory of Heredity

Industrial and Technological Statistics

An Example of Faulty Statistics

Correction

Some Results Summarized

Descriptive Statistics

Foundations of the Theory of Errors

Galton's Board

Normal Curve

Laplace's Law

The Application of the Theory of Errors

SIXTH LECTURE Statistical Problems in Physics

The Second Law of Thermodynamics

Determinism and Probability

Chance Mechanisms

Random Fluctuations

Small Causes and Large Effects

Kinetic Theory of Gases

Order of Magnitude of 'Improbability'

Criticism ofthe Gas Theory

Brownian Motion

Evolution of Phenomena in Time

Probability 'After Effects'

Residence Time and Its Prediction

Entropy Theorem and Markoff Chains

Svedberg's Experiments

Radioactivity

Prediction of Time Intervals

Marsden's and Barratt's Experiments

Recent Development in the Theory of Gases

Degeneration of Gases: Electron Theory of Metals

Quantum Theory

Statistics and Causality

Causal Explanation Newton's Sense

Limitations of Newtonian Mechanics

Simplicity as a Criterion of Causality

Giving up the Concept of Causality

The Law of Causality

New Quantum Statistics

Are Exact Measurements Possible?

Position and Velocity of a Material Particle

Heisenberg's Uncertainty Principle

Consequences for our Physical Concept of the World

Final Considerations

SUMMARY OF THE SIX LECTURES IN SIXTEEN PROPOSITIONS

NOTES AND ADDENDA

SUBJECT INDEX

NAME INDEX

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