Grinstead and Snell's Introduction to Probability
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Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective
HM 23 REVIEWS 203 The reviewer hopes this book will be widely read and enjoyed, and that it will be followed by other volumes telling even more of the fascinating story of Soviet mathematics. It should also be followed in a few years by an update, so that we can know if this great accumulation of talent will have survived the economic and political crisis that is just now robbing it of many of its most brilliant stars (see the article, ``To guard the future of Soviet mathematics,'' by A. M. Vershik, O. Ya. Viro, and L. A. Bokut' in Vol. 14 (1992) of The Mathematical Intelligencer). Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective. By Jan von Plato. Cambridge/New York/Melbourne (Cambridge Univ. Press). 1994. 323 pp. View metadata, citation and similar papers at core.ac.uk brought to you by CORE Reviewed by THOMAS HOCHKIRCHEN* provided by Elsevier - Publisher Connector Fachbereich Mathematik, Bergische UniversitaÈt Wuppertal, 42097 Wuppertal, Germany Aside from the role probabilistic concepts play in modern science, the history of the axiomatic foundation of probability theory is interesting from at least two more points of view. Probability as it is understood nowadays, probability in the sense of Kolmogorov (see [3]), is not easy to grasp, since the de®nition of probability as a normalized measure on a s-algebra of ``events'' is not a very obvious one. Furthermore, the discussion of different concepts of probability might help in under- standing the philosophy and role of ``applied mathematics.'' So the exploration of the creation of axiomatic probability should be interesting not only for historians of science but also for people concerned with didactics of mathematics and for those concerned with philosophical questions. -
A History of Evidence in Medical Decisions: from the Diagnostic Sign to Bayesian Inference
BRIEF REPORTS A History of Evidence in Medical Decisions: From the Diagnostic Sign to Bayesian Inference Dennis J. Mazur, MD, PhD Bayesian inference in medical decision making is a concept through the development of probability theory based on con- that has a long history with 3 essential developments: 1) the siderations of games of chance, and ending with the work of recognition of the need for data (demonstrable scientific Jakob Bernoulli, Laplace, and others, we will examine how evidence), 2) the development of probability, and 3) the Bayesian inference developed. Key words: evidence; games development of inverse probability. Beginning with the of chance; history of evidence; inference; probability. (Med demonstrative evidence of the physician’s sign, continuing Decis Making 2012;32:227–231) here are many senses of the term evidence in the Hacking argues that when humans began to examine Thistory of medicine other than scientifically things pointing beyond themselves to other things, gathered (collected) evidence. Historically, evidence we are examining a new type of evidence to support was first based on testimony and opinion by those the truth or falsehood of a scientific proposition respected for medical expertise. independent of testimony or opinion. This form of evidence can also be used to challenge the data themselves in supporting the truth or falsehood of SIGN AS EVIDENCE IN MEDICINE a proposition. In this treatment of the evidence contained in the Ian Hacking singles out the physician’s sign as the 1 physician’s diagnostic sign, Hacking is examining first example of demonstrable scientific evidence. evidence as a phenomenon of the ‘‘low sciences’’ Once the physician’s sign was recognized as of the Renaissance, which included alchemy, astrol- a form of evidence, numbers could be attached to ogy, geology, and medicine.1 The low sciences were these signs. -
There Is No Pure Empirical Reasoning
There Is No Pure Empirical Reasoning 1. Empiricism and the Question of Empirical Reasons Empiricism may be defined as the view there is no a priori justification for any synthetic claim. Critics object that empiricism cannot account for all the kinds of knowledge we seem to possess, such as moral knowledge, metaphysical knowledge, mathematical knowledge, and modal knowledge.1 In some cases, empiricists try to account for these types of knowledge; in other cases, they shrug off the objections, happily concluding, for example, that there is no moral knowledge, or that there is no metaphysical knowledge.2 But empiricism cannot shrug off just any type of knowledge; to be minimally plausible, empiricism must, for example, at least be able to account for paradigm instances of empirical knowledge, including especially scientific knowledge. Empirical knowledge can be divided into three categories: (a) knowledge by direct observation; (b) knowledge that is deductively inferred from observations; and (c) knowledge that is non-deductively inferred from observations, including knowledge arrived at by induction and inference to the best explanation. Category (c) includes all scientific knowledge. This category is of particular import to empiricists, many of whom take scientific knowledge as a sort of paradigm for knowledge in general; indeed, this forms a central source of motivation for empiricism.3 Thus, if there is any kind of knowledge that empiricists need to be able to account for, it is knowledge of type (c). I use the term “empirical reasoning” to refer to the reasoning involved in acquiring this type of knowledge – that is, to any instance of reasoning in which (i) the premises are justified directly by observation, (ii) the reasoning is non- deductive, and (iii) the reasoning provides adequate justification for the conclusion. -
The Interpretation of Probability: Still an Open Issue? 1
philosophies Article The Interpretation of Probability: Still an Open Issue? 1 Maria Carla Galavotti Department of Philosophy and Communication, University of Bologna, Via Zamboni 38, 40126 Bologna, Italy; [email protected] Received: 19 July 2017; Accepted: 19 August 2017; Published: 29 August 2017 Abstract: Probability as understood today, namely as a quantitative notion expressible by means of a function ranging in the interval between 0–1, took shape in the mid-17th century, and presents both a mathematical and a philosophical aspect. Of these two sides, the second is by far the most controversial, and fuels a heated debate, still ongoing. After a short historical sketch of the birth and developments of probability, its major interpretations are outlined, by referring to the work of their most prominent representatives. The final section addresses the question of whether any of such interpretations can presently be considered predominant, which is answered in the negative. Keywords: probability; classical theory; frequentism; logicism; subjectivism; propensity 1. A Long Story Made Short Probability, taken as a quantitative notion whose value ranges in the interval between 0 and 1, emerged around the middle of the 17th century thanks to the work of two leading French mathematicians: Blaise Pascal and Pierre Fermat. According to a well-known anecdote: “a problem about games of chance proposed to an austere Jansenist by a man of the world was the origin of the calculus of probabilities”2. The ‘man of the world’ was the French gentleman Chevalier de Méré, a conspicuous figure at the court of Louis XIV, who asked Pascal—the ‘austere Jansenist’—the solution to some questions regarding gambling, such as how many dice tosses are needed to have a fair chance to obtain a double-six, or how the players should divide the stakes if a game is interrupted. -
1 Stochastic Processes and Their Classification
1 1 STOCHASTIC PROCESSES AND THEIR CLASSIFICATION 1.1 DEFINITION AND EXAMPLES Definition 1. Stochastic process or random process is a collection of random variables ordered by an index set. ☛ Example 1. Random variables X0;X1;X2;::: form a stochastic process ordered by the discrete index set f0; 1; 2;::: g: Notation: fXn : n = 0; 1; 2;::: g: ☛ Example 2. Stochastic process fYt : t ¸ 0g: with continuous index set ft : t ¸ 0g: The indices n and t are often referred to as "time", so that Xn is a descrete-time process and Yt is a continuous-time process. Convention: the index set of a stochastic process is always infinite. The range (possible values) of the random variables in a stochastic process is called the state space of the process. We consider both discrete-state and continuous-state processes. Further examples: ☛ Example 3. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f0; 1; 2; 3; 4g representing which of four types of transactions a person submits to an on-line data- base service, and time n corresponds to the number of transactions submitted. ☛ Example 4. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f1; 2g re- presenting whether an electronic component is acceptable or defective, and time n corresponds to the number of components produced. ☛ Example 5. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2;::: g representing the number of accidents that have occurred at an intersection, and time t corresponds to weeks. ☛ Example 6. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2; : : : ; sg representing the number of copies of a software product in inventory, and time t corresponds to days. -
1 History of Probability (Part 2)
History of Probability (Part 2) - 17 th Century France The Problem of Points: Pascal, Fermat, and Huygens Every history of probability emphasizes the Figure 1 correspondence between two 17 th century French scholars, Blaise Pascal and Pierre de Fermat . In 1654 they exchanged letters where they discussed how to solve a particular gambling problem, now referred to as The Problem of Points (also called the problem of division of the stakes) . Simply stated, the problem is how to split the pot if the game is interrupted before someone has won. Pascal and Fermat’s work on this problem led to the development of formal rules for probability calculations. Blaise Pascal 1623 -1662 Pierre de Fermat 1601 -1665 The problem of points concerns a game of chance with two competing players who have equal chances of winning each round. [Imagine that one round is a toss of a coin. Player A has heads, B has tails.] The winner is the first one who wins a certain number of pre-agreed upon rounds. [Suppose, for example, they agree that the first person to win 3 tosses wins the game.] Whoever wins the game gets the whole pot. [Say, each player puts in $6, so the total pot is $12 . If A gets three heads before B gets three tails, A wins $12.] Now suppose that the game is interrupted before either player has won. How does one then divide the pot fairly? [Suppose they have to stop early and at that point A has won two tosses and B has won one.] Should each get $6, or should A get more since A was ahead when they stopped? What is fair? Historically, it is important to note that all of these gambling problems were framed in terms of odds for winning a game and not in terms of probabilities of individual events. -
Chapter 1: the Objective and the Subjective in Mid-Nineteenth-Century British Probability Theory
UvA-DARE (Digital Academic Repository) "A terrible piece of bad metaphysics"? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Verburgt, L.M. Publication date 2015 Document Version Final published version Link to publication Citation for published version (APA): Verburgt, L. M. (2015). "A terrible piece of bad metaphysics"? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:27 Sep 2021 chapter 1 The objective and the subjective in mid-nineteenth-century British probability theory 1. Introduction 1.1 The emergence of ‘objective’ and ‘subjective’ probability Approximately at the middle of the nineteenth century at least six authors – Simeon-Denis Poisson, Bernard Bolzano, Robert Leslie Ellis, Jakob Friedrich Fries, John Stuart Mill and A.A. -
Introduction to Bayesian Inference and Modeling Edps 590BAY
Introduction to Bayesian Inference and Modeling Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2019 Introduction What Why Probability Steps Example History Practice Overview ◮ What is Bayes theorem ◮ Why Bayesian analysis ◮ What is probability? ◮ Basic Steps ◮ An little example ◮ History (not all of the 705+ people that influenced development of Bayesian approach) ◮ In class work with probabilities Depending on the book that you select for this course, read either Gelman et al. Chapter 1 or Kruschke Chapters 1 & 2. C.J. Anderson (Illinois) Introduction Fall 2019 2.2/ 29 Introduction What Why Probability Steps Example History Practice Main References for Course Throughout the coures, I will take material from ◮ Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. (20114). Bayesian Data Analysis, 3rd Edition. Boco Raton, FL, CRC/Taylor & Francis.** ◮ Hoff, P.D., (2009). A First Course in Bayesian Statistical Methods. NY: Sringer.** ◮ McElreath, R.M. (2016). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Boco Raton, FL, CRC/Taylor & Francis. ◮ Kruschke, J.K. (2015). Doing Bayesian Data Analysis: A Tutorial with JAGS and Stan. NY: Academic Press.** ** There are e-versions these of from the UofI library. There is a verson of McElreath, but I couldn’t get if from UofI e-collection. C.J. Anderson (Illinois) Introduction Fall 2019 3.3/ 29 Introduction What Why Probability Steps Example History Practice Bayes Theorem A whole semester on this? p(y|θ)p(θ) p(θ|y)= p(y) where ◮ y is data, sample from some population. -
Harold Jeffreys's Theory of Probability Revisited
Statistical Science 2009, Vol. 24, No. 2, 141–172 DOI: 10.1214/09-STS284 c Institute of Mathematical Statistics, 2009 Harold Jeffreys’s Theory of Probability Revisited Christian P. Robert, Nicolas Chopin and Judith Rousseau Abstract. Published exactly seventy years ago, Jeffreys’s Theory of Probability (1939) has had a unique impact on the Bayesian commu- nity and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In par- ticular, its advances on the derivation of noninformative priors as well as on the scaling of Bayes factors have had a lasting impact on the field. However, the book reflects the characteristics of the time, especially in terms of mathematical rigor. In this paper we point out the fundamen- tal aspects of this reference work, especially the thorough coverage of testing problems and the construction of both estimation and testing noninformative priors based on functional divergences. Our major aim here is to help modern readers in navigating in this difficult text and in concentrating on passages that are still relevant today. Key words and phrases: Bayesian foundations, noninformative prior, σ-finite measure, Jeffreys’s prior, Kullback divergence, tests, Bayes fac- tor, p-values, goodness of fit. 1. INTRODUCTION Christian P. Robert is Professor of Statistics, Applied Mathematics Department, Universit´eParis Dauphine The theory of probability makes it possible to and Head of the Statistics Laboratory, Center for respect the great men on whose shoulders we stand. Research in Economics and Statistics (CREST), H. Jeffreys, Theory of Probability, Section 1.6. -
Probability and Logic
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Applied Logic 1 (2003) 151–165 www.elsevier.com/locate/jal Probability and logic Colin Howson London School of Economics, Houghton Street, London WC2A 2AE, UK Abstract The paper is an attempt to show that the formalism of subjective probability has a logical in- terpretation of the sort proposed by Frank Ramsey: as a complete set of constraints for consistent distributions of partial belief. Though Ramsey proposed this view, he did not actually establish it in a way that showed an authentically logical character for the probability axioms (he started the current fashion for generating probabilities from suitably constrained preferences over uncertain options). Other people have also sought to provide the probability calculus with a logical character, though also unsuccessfully. The present paper gives a completeness and soundness theorem supporting a logical interpretation: the syntax is the probability axioms, and the semantics is that of fairness (for bets). 2003 Elsevier B.V. All rights reserved. Keywords: Probability; Logic; Fair betting quotients; Soundness; Completeness 1. Anticipations The connection between deductive logic and probability, at any rate epistemic probabil- ity, has been the subject of exploration and controversy for a long time: over three centuries, in fact. Both disciplines specify rules of valid non-domain-specific reasoning, and it would seem therefore a reasonable question why one should be distinguished as logic and the other not. I will argue that there is no good reason for this difference, and that both are indeed equally logic. -
Pascal's and Huygens's Game-Theoretic Foundations For
Pascal's and Huygens's game-theoretic foundations for probability Glenn Shafer Rutgers University [email protected] The Game-Theoretic Probability and Finance Project Working Paper #53 First posted December 28, 2018. Last revised December 28, 2018. Project web site: http://www.probabilityandfinance.com Abstract Blaise Pascal and Christiaan Huygens developed game-theoretic foundations for the calculus of chances | foundations that replaced appeals to frequency with arguments based on a game's temporal structure. Pascal argued for equal division when chances are equal. Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. These game-theoretic foundations were disregarded by Pascal's and Huy- gens's 18th century successors, who found the already established foundation of equally frequent cases more conceptually relevant and mathematically fruit- ful. But the game-theoretic foundations can be developed in ways that merit attention in the 21st century. 1 The calculus of chances before Pascal and Fermat 1 1.1 Counting chances . .2 1.2 Fixing stakes and bets . .3 2 The division problem 5 2.1 Pascal's solution of the division problem . .6 2.2 Published antecedents . .7 2.3 Unpublished antecedents . .8 3 Pascal's game-theoretic foundation 9 3.1 Enter the Chevalier de M´er´e. .9 3.2 Carrying its demonstration in itself . 11 4 Huygens's game-theoretic foundation 12 4.1 What did Huygens learn in Paris? . 13 4.2 Only games of pure chance? . 15 4.3 Using algebra . 16 5 Back to frequency 18 5.1 Montmort . -
Discrete and Continuous: a Fundamental Dichotomy in Mathematics
Journal of Humanistic Mathematics Volume 7 | Issue 2 July 2017 Discrete and Continuous: A Fundamental Dichotomy in Mathematics James Franklin University of New South Wales Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Other Mathematics Commons Recommended Citation Franklin, J. "Discrete and Continuous: A Fundamental Dichotomy in Mathematics," Journal of Humanistic Mathematics, Volume 7 Issue 2 (July 2017), pages 355-378. DOI: 10.5642/jhummath.201702.18 . Available at: https://scholarship.claremont.edu/jhm/vol7/iss2/18 ©2017 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Discrete and Continuous: A Fundamental Dichotomy in Mathematics James Franklin1 School of Mathematics & Statistics, University of New South Wales, Sydney, AUSTRALIA [email protected] Synopsis The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application ar- eas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology).