Evidence-Based Crime Investigation. a Bayesian Approach
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The Bayesian Approach to Statistics
THE BAYESIAN APPROACH TO STATISTICS ANTHONY O’HAGAN INTRODUCTION the true nature of scientific reasoning. The fi- nal section addresses various features of modern By far the most widely taught and used statisti- Bayesian methods that provide some explanation for the rapid increase in their adoption since the cal methods in practice are those of the frequen- 1980s. tist school. The ideas of frequentist inference, as set out in Chapter 5 of this book, rest on the frequency definition of probability (Chapter 2), BAYESIAN INFERENCE and were developed in the first half of the 20th century. This chapter concerns a radically differ- We first present the basic procedures of Bayesian ent approach to statistics, the Bayesian approach, inference. which depends instead on the subjective defini- tion of probability (Chapter 3). In some respects, Bayesian methods are older than frequentist ones, Bayes’s Theorem and the Nature of Learning having been the basis of very early statistical rea- Bayesian inference is a process of learning soning as far back as the 18th century. Bayesian from data. To give substance to this statement, statistics as it is now understood, however, dates we need to identify who is doing the learning and back to the 1950s, with subsequent development what they are learning about. in the second half of the 20th century. Over that time, the Bayesian approach has steadily gained Terms and Notation ground, and is now recognized as a legitimate al- ternative to the frequentist approach. The person doing the learning is an individual This chapter is organized into three sections. -
Scalable and Robust Bayesian Inference Via the Median Posterior
Scalable and Robust Bayesian Inference via the Median Posterior CS 584: Big Data Analytics Material adapted from David Dunson’s talk (http://bayesian.org/sites/default/files/Dunson.pdf) & Lizhen Lin’s ICML talk (http://techtalks.tv/talks/scalable-and-robust-bayesian-inference-via-the-median-posterior/61140/) Big Data Analytics • Large (big N) and complex (big P with interactions) data are collected routinely • Both speed & generality of data analysis methods are important • Bayesian approaches offer an attractive general approach for modeling the complexity of big data • Computational intractability of posterior sampling is a major impediment to application of flexible Bayesian methods CS 584 [Spring 2016] - Ho Existing Frequentist Approaches: The Positives • Optimization-based approaches, such as ADMM or glmnet, are currently most popular for analyzing big data • General and computationally efficient • Used orders of magnitude more than Bayes methods • Can exploit distributed & cloud computing platforms • Can borrow some advantages of Bayes methods through penalties and regularization CS 584 [Spring 2016] - Ho Existing Frequentist Approaches: The Drawbacks • Such optimization-based methods do not provide measure of uncertainty • Uncertainty quantification is crucial for most applications • Scalable penalization methods focus primarily on convex optimization — greatly limits scope and puts ceiling on performance • For non-convex problems and data with complex structure, existing optimization algorithms can fail badly CS 584 [Spring 2016] - -
Using Epidemiological Evidence in Tort Law: a Practical Guide Aleksandra Kobyasheva
Using epidemiological evidence in tort law: a practical guide Aleksandra Kobyasheva Introduction Epidemiology is the study of disease patterns in populations which seeks to identify and understand causes of disease. By using this data to predict how and when diseases are likely to arise, it aims to prevent the disease and its consequences through public health regulation or the development of medication. But can this data also be used to tell us something about an event that has already occurred? To illustrate, consider a case where a claimant is exposed to a substance due to the defendant’s negligence and subsequently contracts a disease. There is not enough evidence available to satisfy the traditional ‘but-for’ test of causation.1 However, an epidemiological study shows that there is a potential causal link between the substance and the disease. What is, or what should be, the significance of this study? In Sienkiewicz v Greif members of the Supreme Court were sceptical that epidemiological evidence can have a useful role to play in determining questions of causation.2 The issue in that case was a narrow one that did not present the full range of concerns raised by epidemiological evidence, but the court’s general attitude was not unusual. Few English courts have thought seriously about the ways in which epidemiological evidence should be approached or interpreted; in particular, the factors which should be taken into account when applying the ‘balance of probability’ test, or the weight which should be attached to such factors. As a result, this article aims to explore this question from a practical perspective. -
1 Estimation and Beyond in the Bayes Universe
ISyE8843A, Brani Vidakovic Handout 7 1 Estimation and Beyond in the Bayes Universe. 1.1 Estimation No Bayes estimate can be unbiased but Bayesians are not upset! No Bayes estimate with respect to the squared error loss can be unbiased, except in a trivial case when its Bayes’ risk is 0. Suppose that for a proper prior ¼ the Bayes estimator ±¼(X) is unbiased, Xjθ (8θ)E ±¼(X) = θ: This implies that the Bayes risk is 0. The Bayes risk of ±¼(X) can be calculated as repeated expectation in two ways, θ Xjθ 2 X θjX 2 r(¼; ±¼) = E E (θ ¡ ±¼(X)) = E E (θ ¡ ±¼(X)) : Thus, conveniently choosing either EθEXjθ or EX EθjX and using the properties of conditional expectation we have, θ Xjθ 2 θ Xjθ X θjX X θjX 2 r(¼; ±¼) = E E θ ¡ E E θ±¼(X) ¡ E E θ±¼(X) + E E ±¼(X) θ Xjθ 2 θ Xjθ X θjX X θjX 2 = E E θ ¡ E θ[E ±¼(X)] ¡ E ±¼(X)E θ + E E ±¼(X) θ Xjθ 2 θ X X θjX 2 = E E θ ¡ E θ ¢ θ ¡ E ±¼(X)±¼(X) + E E ±¼(X) = 0: Bayesians are not upset. To check for its unbiasedness, the Bayes estimator is averaged with respect to the model measure (Xjθ), and one of the Bayesian commandments is: Thou shall not average with respect to sample space, unless you have Bayesian design in mind. Even frequentist agree that insisting on unbiasedness can lead to bad estimators, and that in their quest to minimize the risk by trading off between variance and bias-squared a small dosage of bias can help. -
Analysis of Scientific Research on Eyewitness Identification
ANALYSIS OF SCIENTIFIC RESEARCH ON EYEWITNESS IDENTIFICATION January 2018 Patricia A. Riley U.S. Attorney’s Office, DC Table of Contents RESEARCH DOES NOT PROVIDE A FIRM FOUNDATION FOR JURY INSTRUCTIONS OF THE TYPE ADOPTED BY SOME COURTS OR, IN SOME INSTANCES, FOR EXPERT TESTIMONY ...................................................... 6 RESEARCH DOES NOT PROVIDE A FIRM FOUNDATION FOR JURY INSTRUCTIONS OF THE TYPE ADOPTED BY SOME COURTS AND, IN SOME INSTANCES, FOR EXPERT TESTIMONY .................................................... 8 Introduction ............................................................................................................................................. 8 Courts should not comment on the evidence or take judicial notice of contested facts ................... 11 Jury Instructions based on flawed or outdated research should not be given .................................. 12 AN OVERVIEW OF THE RESEARCH ON EYEWITNESS IDENTIFICATION OF STRANGERS, NEW AND OLD .... 16 Important Points .................................................................................................................................... 16 System Variables .................................................................................................................................... 16 Simultaneous versus sequential presentation ................................................................................... 16 Double blind, blind, blinded administration (unless impracticable ................................................... -
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. -
Statistical Inference: Paradigms and Controversies in Historic Perspective
Jostein Lillestøl, NHH 2014 Statistical inference: Paradigms and controversies in historic perspective 1. Five paradigms We will cover the following five lines of thought: 1. Early Bayesian inference and its revival Inverse probability – Non-informative priors – “Objective” Bayes (1763), Laplace (1774), Jeffreys (1931), Bernardo (1975) 2. Fisherian inference Evidence oriented – Likelihood – Fisher information - Necessity Fisher (1921 and later) 3. Neyman- Pearson inference Action oriented – Frequentist/Sample space – Objective Neyman (1933, 1937), Pearson (1933), Wald (1939), Lehmann (1950 and later) 4. Neo - Bayesian inference Coherent decisions - Subjective/personal De Finetti (1937), Savage (1951), Lindley (1953) 5. Likelihood inference Evidence based – likelihood profiles – likelihood ratios Barnard (1949), Birnbaum (1962), Edwards (1972) Classical inference as it has been practiced since the 1950’s is really none of these in its pure form. It is more like a pragmatic mix of 2 and 3, in particular with respect to testing of significance, pretending to be both action and evidence oriented, which is hard to fulfill in a consistent manner. To keep our minds on track we do not single out this as a separate paradigm, but will discuss this at the end. A main concern through the history of statistical inference has been to establish a sound scientific framework for the analysis of sampled data. Concepts were initially often vague and disputed, but even after their clarification, various schools of thought have at times been in strong opposition to each other. When we try to describe the approaches here, we will use the notions of today. All five paradigms of statistical inference are based on modeling the observed data x given some parameter or “state of the world” , which essentially corresponds to stating the conditional distribution f(x|(or making some assumptions about it). -
Probability, Explanation, and Inference: a Reply
Alabama Law Scholarly Commons Working Papers Faculty Scholarship 3-2-2017 Probability, Explanation, and Inference: A Reply Michael S. Pardo University of Alabama - School of Law, [email protected] Ronald J. Allen Northwestern University - Pritzker School of Law, [email protected] Follow this and additional works at: https://scholarship.law.ua.edu/fac_working_papers Recommended Citation Michael S. Pardo & Ronald J. Allen, Probability, Explanation, and Inference: A Reply, (2017). Available at: https://scholarship.law.ua.edu/fac_working_papers/4 This Working Paper is brought to you for free and open access by the Faculty Scholarship at Alabama Law Scholarly Commons. It has been accepted for inclusion in Working Papers by an authorized administrator of Alabama Law Scholarly Commons. Probability, Explanation, and Inference: A Reply Ronald J. Allen Michael S. Pardo 11 INTERNATIONAL JOURNAL OF EVIDENCE AND PROOF 307 (2007) This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection: http://ssrn.com/abstract=2925866 Electronic copy available at: https://ssrn.com/abstract=2925866 PROBABILITY, EXPLANATION AND INFERENCE: A REPLY Probability, explanation and inference: a reply By Ronald J. Allen* and Wigmore Professor of Law, Northwestern University; Fellow, Procedural Law Research Center, China Political Science and Law University Michael S. Pardo† Assistant Professor, University of Alabama School of Law he inferences drawn from legal evidence may be understood in both probabilistic and explanatory terms. Consider evidence that a criminal T defendant confessed while in police custody. To evaluate the strength of this evidence in supporting the conclusion that the defendant is guilty, one could try to assess the probability that guilty and innocent persons confess while in police custody. -
Bayesian Inference for Median of the Lognormal Distribution K
Journal of Modern Applied Statistical Methods Volume 15 | Issue 2 Article 32 11-1-2016 Bayesian Inference for Median of the Lognormal Distribution K. Aruna Rao SDM Degree College, Ujire, India, [email protected] Juliet Gratia D'Cunha Mangalore University, Mangalagangothri, India, [email protected] Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Rao, K. Aruna and D'Cunha, Juliet Gratia (2016) "Bayesian Inference for Median of the Lognormal Distribution," Journal of Modern Applied Statistical Methods: Vol. 15 : Iss. 2 , Article 32. DOI: 10.22237/jmasm/1478003400 Available at: http://digitalcommons.wayne.edu/jmasm/vol15/iss2/32 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Bayesian Inference for Median of the Lognormal Distribution Cover Page Footnote Acknowledgements The es cond author would like to thank Government of India, Ministry of Science and Technology, Department of Science and Technology, New Delhi, for sponsoring her with an INSPIRE fellowship, which enables her to carry out the research program which she has undertaken. She is much honored to be the recipient of this award. This regular article is available in Journal of Modern Applied Statistical Methods: http://digitalcommons.wayne.edu/jmasm/vol15/ iss2/32 Journal of Modern Applied Statistical Methods Copyright © 2016 JMASM, Inc. November 2016, Vol. 15, No. 2, 526-535. -
9 Bayesian Inference
9 Bayesian inference 1702 - 1761 9.1 Subjective probability This is probability regarded as degree of belief. A subjective probability of an event A is assessed as p if you are prepared to stake £pM to win £M and equally prepared to accept a stake of £pM to win £M. In other words ... ... the bet is fair and you are assumed to behave rationally. 9.1.1 Kolmogorov’s axioms How does subjective probability fit in with the fundamental axioms? Let A be the set of all subsets of a countable sample space Ω. Then (i) P(A) ≥ 0 for every A ∈A; (ii) P(Ω)=1; 83 (iii) If {Aλ : λ ∈ Λ} is a countable set of mutually exclusive events belonging to A,then P Aλ = P (Aλ) . λ∈Λ λ∈Λ Obviously the subjective interpretation has no difficulty in conforming with (i) and (ii). (iii) is slightly less obvious. Suppose we have 2 events A and B such that A ∩ B = ∅. Consider a stake of £pAM to win £M if A occurs and a stake £pB M to win £M if B occurs. The total stake for bets on A or B occurring is £pAM+ £pBM to win £M if A or B occurs. Thus we have £(pA + pB)M to win £M and so P (A ∪ B)=P(A)+P(B) 9.1.2 Conditional probability Define pB , pAB , pA|B such that £pBM is the fair stake for £M if B occurs; £pABM is the fair stake for £M if A and B occur; £pA|BM is the fair stake for £M if A occurs given B has occurred − other- wise the bet is off. -
Patents & Legal Expenditures
Patents & Legal Expenditures Christopher J. Ryan, Jr. & Brian L. Frye* I. INTRODUCTION ................................................................................................ 577 A. A Brief History of University Patents ................................................. 578 B. The Origin of University Patents ........................................................ 578 C. University Patenting as a Function of Patent Policy Incentives ........ 580 D. The Business of University Patenting and Technology Transfer ....... 581 E. Trends in Patent Litigation ................................................................. 584 II. DATA AND ANALYSIS .................................................................................... 588 III. CONCLUSION ................................................................................................. 591 I. INTRODUCTION Universities are engines of innovation. To encourage further innovation, the federal government and charitable foundations give universities grants in order to enable university researchers to produce the inventions and discoveries that will continue to fuel our knowledge economy. Among other things, the Bayh-Dole Act of 1980 was supposed to encourage additional innovation by enabling universities to patent inventions and discoveries produced using federal funds and to license those patents to private companies, rather than turning their patent rights over to the government. The Bayh-Dole Act unquestionably encouraged universities to patent inventions and license their patents. -
Bayesian Inference
Bayesian Inference Thomas Nichols With thanks Lee Harrison Bayesian segmentation Spatial priors Posterior probability Dynamic Causal and normalisation on activation extent maps (PPMs) Modelling Attention to Motion Paradigm Results SPC V3A V5+ Attention – No attention Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain - fixation only - observe static dots + photic V1 - observe moving dots + motion V5 - task on moving dots + attention V5 + parietal cortex Attention to Motion Paradigm Dynamic Causal Models Model 1 (forward): Model 2 (backward): attentional modulation attentional modulation of V1→V5: forward of SPC→V5: backward Photic SPC Attention Photic SPC V1 V1 - fixation only V5 - observe static dots V5 - observe moving dots Motion Motion - task on moving dots Attention Bayesian model selection: Which model is optimal? Responses to Uncertainty Long term memory Short term memory Responses to Uncertainty Paradigm Stimuli sequence of randomly sampled discrete events Model simple computational model of an observers response to uncertainty based on the number of past events (extent of memory) 1 2 3 4 Question which regions are best explained by short / long term memory model? … 1 2 40 trials ? ? Overview • Introductory remarks • Some probability densities/distributions • Probabilistic (generative) models • Bayesian inference • A simple example – Bayesian linear regression • SPM applications – Segmentation – Dynamic causal modeling – Spatial models of fMRI time series Probability distributions and densities k=2 Probability distributions