A Brief Introduction to Bayesian Statistics Bayes’ Theorem
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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. -
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. -
Bayesian Versus Frequentist Statistics for Uncertainty Analysis
- 1 - Disentangling Classical and Bayesian Approaches to Uncertainty Analysis Robin Willink1 and Rod White2 1email: [email protected] 2Measurement Standards Laboratory PO Box 31310, Lower Hutt 5040 New Zealand email(corresponding author): [email protected] Abstract Since the 1980s, we have seen a gradual shift in the uncertainty analyses recommended in the metrological literature, principally Metrologia, and in the BIPM’s guidance documents; the Guide to the Expression of Uncertainty in Measurement (GUM) and its two supplements. The shift has seen the BIPM’s recommendations change from a purely classical or frequentist analysis to a purely Bayesian analysis. Despite this drift, most metrologists continue to use the predominantly frequentist approach of the GUM and wonder what the differences are, why there are such bitter disputes about the two approaches, and should I change? The primary purpose of this note is to inform metrologists of the differences between the frequentist and Bayesian approaches and the consequences of those differences. It is often claimed that a Bayesian approach is philosophically consistent and is able to tackle problems beyond the reach of classical statistics. However, while the philosophical consistency of the of Bayesian analyses may be more aesthetically pleasing, the value to science of any statistical analysis is in the long-term success rates and on this point, classical methods perform well and Bayesian analyses can perform poorly. Thus an important secondary purpose of this note is to highlight some of the weaknesses of the Bayesian approach. We argue that moving away from well-established, easily- taught frequentist methods that perform well, to computationally expensive and numerically inferior Bayesian analyses recommended by the GUM supplements is ill-advised. -
Practical Statistics for Particle Physics Lecture 1 AEPS2018, Quy Nhon, Vietnam
Practical Statistics for Particle Physics Lecture 1 AEPS2018, Quy Nhon, Vietnam Roger Barlow The University of Huddersfield August 2018 Roger Barlow ( Huddersfield) Statistics for Particle Physics August 2018 1 / 34 Lecture 1: The Basics 1 Probability What is it? Frequentist Probability Conditional Probability and Bayes' Theorem Bayesian Probability 2 Probability distributions and their properties Expectation Values Binomial, Poisson and Gaussian 3 Hypothesis testing Roger Barlow ( Huddersfield) Statistics for Particle Physics August 2018 2 / 34 Question: What is Probability? Typical exam question Q1 Explain what is meant by the Probability PA of an event A [1] Roger Barlow ( Huddersfield) Statistics for Particle Physics August 2018 3 / 34 Four possible answers PA is number obeying certain mathematical rules. PA is a property of A that determines how often A happens For N trials in which A occurs NA times, PA is the limit of NA=N for large N PA is my belief that A will happen, measurable by seeing what odds I will accept in a bet. Roger Barlow ( Huddersfield) Statistics for Particle Physics August 2018 4 / 34 Mathematical Kolmogorov Axioms: For all A ⊂ S PA ≥ 0 PS = 1 P(A[B) = PA + PB if A \ B = ϕ and A; B ⊂ S From these simple axioms a complete and complicated structure can be − ≤ erected. E.g. show PA = 1 PA, and show PA 1.... But!!! This says nothing about what PA actually means. Kolmogorov had frequentist probability in mind, but these axioms apply to any definition. Roger Barlow ( Huddersfield) Statistics for Particle Physics August 2018 5 / 34 Classical or Real probability Evolved during the 18th-19th century Developed (Pascal, Laplace and others) to serve the gambling industry. -
This History of Modern Mathematical Statistics Retraces Their Development
BOOK REVIEWS GORROOCHURN Prakash, 2016, Classic Topics on the History of Modern Mathematical Statistics: From Laplace to More Recent Times, Hoboken, NJ, John Wiley & Sons, Inc., 754 p. This history of modern mathematical statistics retraces their development from the “Laplacean revolution,” as the author so rightly calls it (though the beginnings are to be found in Bayes’ 1763 essay(1)), through the mid-twentieth century and Fisher’s major contribution. Up to the nineteenth century the book covers the same ground as Stigler’s history of statistics(2), though with notable differences (see infra). It then discusses developments through the first half of the twentieth century: Fisher’s synthesis but also the renewal of Bayesian methods, which implied a return to Laplace. Part I offers an in-depth, chronological account of Laplace’s approach to probability, with all the mathematical detail and deductions he drew from it. It begins with his first innovative articles and concludes with his philosophical synthesis showing that all fields of human knowledge are connected to the theory of probabilities. Here Gorrouchurn raises a problem that Stigler does not, that of induction (pp. 102-113), a notion that gives us a better understanding of probability according to Laplace. The term induction has two meanings, the first put forward by Bacon(3) in 1620, the second by Hume(4) in 1748. Gorroochurn discusses only the second. For Bacon, induction meant discovering the principles of a system by studying its properties through observation and experimentation. For Hume, induction was mere enumeration and could not lead to certainty. Laplace followed Bacon: “The surest method which can guide us in the search for truth, consists in rising by induction from phenomena to laws and from laws to forces”(5). -
The Likelihood Principle
1 01/28/99 ãMarc Nerlove 1999 Chapter 1: The Likelihood Principle "What has now appeared is that the mathematical concept of probability is ... inadequate to express our mental confidence or diffidence in making ... inferences, and that the mathematical quantity which usually appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term 'Likelihood' to designate this quantity; since both the words 'likelihood' and 'probability' are loosely used in common speech to cover both kinds of relationship." R. A. Fisher, Statistical Methods for Research Workers, 1925. "What we can find from a sample is the likelihood of any particular value of r [a parameter], if we define the likelihood as a quantity proportional to the probability that, from a particular population having that particular value of r, a sample having the observed value r [a statistic] should be obtained. So defined, probability and likelihood are quantities of an entirely different nature." R. A. Fisher, "On the 'Probable Error' of a Coefficient of Correlation Deduced from a Small Sample," Metron, 1:3-32, 1921. Introduction The likelihood principle as stated by Edwards (1972, p. 30) is that Within the framework of a statistical model, all the information which the data provide concerning the relative merits of two hypotheses is contained in the likelihood ratio of those hypotheses on the data. ...For a continuum of hypotheses, this principle -
People's Intuitions About Randomness and Probability
20 PEOPLE’S INTUITIONS ABOUT RANDOMNESS AND PROBABILITY: AN EMPIRICAL STUDY4 MARIE-PAULE LECOUTRE ERIS, Université de Rouen [email protected] KATIA ROVIRA Laboratoire Psy.Co, Université de Rouen [email protected] BRUNO LECOUTRE ERIS, C.N.R.S. et Université de Rouen [email protected] JACQUES POITEVINEAU ERIS, Université de Paris 6 et Ministère de la Culture [email protected] ABSTRACT What people mean by randomness should be taken into account when teaching statistical inference. This experiment explored subjective beliefs about randomness and probability through two successive tasks. Subjects were asked to categorize 16 familiar items: 8 real items from everyday life experiences, and 8 stochastic items involving a repeatable process. Three groups of subjects differing according to their background knowledge of probability theory were compared. An important finding is that the arguments used to judge if an event is random and those to judge if it is not random appear to be of different natures. While the concept of probability has been introduced to formalize randomness, a majority of individuals appeared to consider probability as a primary concept. Keywords: Statistics education research; Probability; Randomness; Bayesian Inference 1. INTRODUCTION In recent years Bayesian statistical practice has considerably evolved. Nowadays, the frequentist approach is increasingly challenged among scientists by the Bayesian proponents (see e.g., D’Agostini, 2000; Lecoutre, Lecoutre & Poitevineau, 2001; Jaynes, 2003). In applied statistics, “objective Bayesian techniques” (Berger, 2004) are now a promising alternative to the traditional frequentist statistical inference procedures (significance tests and confidence intervals). Subjective Bayesian analysis also has a role to play in scientific investigations (see e.g., Kadane, 1996). -
The Interplay of Bayesian and Frequentist Analysis M.J.Bayarriandj.O.Berger
Statistical Science 2004, Vol. 19, No. 1, 58–80 DOI 10.1214/088342304000000116 © Institute of Mathematical Statistics, 2004 The Interplay of Bayesian and Frequentist Analysis M.J.BayarriandJ.O.Berger Abstract. Statistics has struggled for nearly a century over the issue of whether the Bayesian or frequentist paradigm is superior. This debate is far from over and, indeed, should continue, since there are fundamental philosophical and pedagogical issues at stake. At the methodological level, however, the debate has become considerably muted, with the recognition that each approach has a great deal to contribute to statistical practice and each is actually essential for full development of the other approach. In this article, we embark upon a rather idiosyncratic walk through some of these issues. Key words and phrases: Admissibility, Bayesian model checking, condi- tional frequentist, confidence intervals, consistency, coverage, design, hierar- chical models, nonparametric Bayes, objective Bayesian methods, p-values, reference priors, testing. CONTENTS 5. Areas of current disagreement 6. Conclusions 1. Introduction Acknowledgments 2. Inherently joint Bayesian–frequentist situations References 2.1. Design or preposterior analysis 2.2. The meaning of frequentism 2.3. Empirical Bayes, gamma minimax, restricted risk 1. INTRODUCTION Bayes Statisticians should readily use both Bayesian and 3. Estimation and confidence intervals frequentist ideas. In Section 2 we discuss situations 3.1. Computation with hierarchical, multilevel or mixed in which simultaneous frequentist and Bayesian think- model analysis ing is essentially required. For the most part, how- 3.2. Assessment of accuracy of estimation ever, the situations we discuss are situations in which 3.3. Foundations, minimaxity and exchangeability 3.4. -
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. -
Frequentism-As-Model
Frequentism-as-model Christian Hennig, Dipartimento di Scienze Statistiche, Universita die Bologna, Via delle Belle Arti, 41, 40126 Bologna [email protected] July 14, 2020 Abstract: Most statisticians are aware that probability models interpreted in a frequentist manner are not really true in objective reality, but only idealisations. I argue that this is often ignored when actually applying frequentist methods and interpreting the results, and that keeping up the awareness for the essential difference between reality and models can lead to a more appropriate use and in- terpretation of frequentist models and methods, called frequentism-as-model. This is elaborated showing connections to existing work, appreciating the special role of i.i.d. models and subject matter knowledge, giving an account of how and under what conditions models that are not true can be useful, giving detailed interpreta- tions of tests and confidence intervals, confronting their implicit compatibility logic with the inverse probability logic of Bayesian inference, re-interpreting the role of model assumptions, appreciating robustness, and the role of “interpretative equiv- alence” of models. Epistemic (often referred to as Bayesian) probability shares the issue that its models are only idealisations and not really true for modelling reasoning about uncertainty, meaning that it does not have an essential advantage over frequentism, as is often claimed. Bayesian statistics can be combined with frequentism-as-model, leading to what Gelman and Hennig (2017) call “falsifica- arXiv:2007.05748v1 [stat.OT] 11 Jul 2020 tionist Bayes”. Key Words: foundations of statistics, Bayesian statistics, interpretational equiv- alence, compatibility logic, inverse probability logic, misspecification testing, sta- bility, robustness 1 Introduction The frequentist interpretation of probability and frequentist inference such as hy- pothesis tests and confidence intervals have been strongly criticised recently (e.g., Hajek (2009); Diaconis and Skyrms (2018); Wasserstein et al. -
The P–T Probability Framework for Semantic Communication, Falsification, Confirmation, and Bayesian Reasoning
philosophies Article The P–T Probability Framework for Semantic Communication, Falsification, Confirmation, and Bayesian Reasoning Chenguang Lu School of Computer Engineering and Applied Mathematics, Changsha University, Changsha 410003, China; [email protected] Received: 8 July 2020; Accepted: 16 September 2020; Published: 2 October 2020 Abstract: Many researchers want to unify probability and logic by defining logical probability or probabilistic logic reasonably. This paper tries to unify statistics and logic so that we can use both statistical probability and logical probability at the same time. For this purpose, this paper proposes the P–T probability framework, which is assembled with Shannon’s statistical probability framework for communication, Kolmogorov’s probability axioms for logical probability, and Zadeh’s membership functions used as truth functions. Two kinds of probabilities are connected by an extended Bayes’ theorem, with which we can convert a likelihood function and a truth function from one to another. Hence, we can train truth functions (in logic) by sampling distributions (in statistics). This probability framework was developed in the author’s long-term studies on semantic information, statistical learning, and color vision. This paper first proposes the P–T probability framework and explains different probabilities in it by its applications to semantic information theory. Then, this framework and the semantic information methods are applied to statistical learning, statistical mechanics, hypothesis evaluation (including falsification), confirmation, and Bayesian reasoning. Theoretical applications illustrate the reasonability and practicability of this framework. This framework is helpful for interpretable AI. To interpret neural networks, we need further study. Keywords: statistical probability; logical probability; semantic information; rate-distortion; Boltzmann distribution; falsification; verisimilitude; confirmation measure; Raven Paradox; Bayesian reasoning 1.