Bayesian Analysis
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Statistical Methods for Data Science, Lecture 5 Interval Estimates; Comparing Systems
Statistical methods for Data Science, Lecture 5 Interval estimates; comparing systems Richard Johansson November 18, 2018 statistical inference: overview I estimate the value of some parameter (last lecture): I what is the error rate of my drug test? I determine some interval that is very likely to contain the true value of the parameter (today): I interval estimate for the error rate I test some hypothesis about the parameter (today): I is the error rate significantly different from 0.03? I are users significantly more satisfied with web page A than with web page B? -20pt “recipes” I in this lecture, we’ll look at a few “recipes” that you’ll use in the assignment I interval estimate for a proportion (“heads probability”) I comparing a proportion to a specified value I comparing two proportions I additionally, we’ll see the standard method to compute an interval estimate for the mean of a normal I I will also post some pointers to additional tests I remember to check that the preconditions are satisfied: what kind of experiment? what assumptions about the data? -20pt overview interval estimates significance testing for the accuracy comparing two classifiers p-value fishing -20pt interval estimates I if we get some estimate by ML, can we say something about how reliable that estimate is? I informally, an interval estimate for the parameter p is an interval I = [plow ; phigh] so that the true value of the parameter is “likely” to be contained in I I for instance: with 95% probability, the error rate of the spam filter is in the interval [0:05; 0:08] -20pt frequentists and Bayesians again. -
DSA 5403 Bayesian Statistics: an Advancing Introduction 16 Units – Each Unit a Week’S Work
DSA 5403 Bayesian Statistics: An Advancing Introduction 16 units – each unit a week’s work. The following are the contents of the course divided into chapters of the book Doing Bayesian Data Analysis . by John Kruschke. The course is structured around the above book but will be embellished with more theoretical content as needed. The book can be obtained from the library : http://www.sciencedirect.com/science/book/9780124058880 Topics: This is divided into three parts From the book: “Part I The basics: models, probability, Bayes' rule and r: Introduction: credibility, models, and parameters; The R programming language; What is this stuff called probability?; Bayes' rule Part II All the fundamentals applied to inferring a binomial probability: Inferring a binomial probability via exact mathematical analysis; Markov chain Monte C arlo; J AG S ; Hierarchical models; Model comparison and hierarchical modeling; Null hypothesis significance testing; Bayesian approaches to testing a point ("Null") hypothesis; G oals, power, and sample size; S tan Part III The generalized linear model: Overview of the generalized linear model; Metric-predicted variable on one or two groups; Metric predicted variable with one metric predictor; Metric predicted variable with multiple metric predictors; Metric predicted variable with one nominal predictor; Metric predicted variable with multiple nominal predictors; Dichotomous predicted variable; Nominal predicted variable; Ordinal predicted variable; C ount predicted variable; Tools in the trunk” Part 1: The Basics: Models, Probability, Bayes’ Rule and R Unit 1 Chapter 1, 2 Credibility, Models, and Parameters, • Derivation of Bayes’ rule • Re-allocation of probability • The steps of Bayesian data analysis • Classical use of Bayes’ rule o Testing – false positives etc o Definitions of prior, likelihood, evidence and posterior Unit 2 Chapter 3 The R language • Get the software • Variables types • Loading data • Functions • Plots Unit 3 Chapter 4 Probability distributions. -
Naïve Bayes Classifier
CS 60050 Machine Learning Naïve Bayes Classifier Some slides taken from course materials of Tan, Steinbach, Kumar Bayes Classifier ● A probabilistic framework for solving classification problems ● Approach for modeling probabilistic relationships between the attribute set and the class variable – May not be possible to certainly predict class label of a test record even if it has identical attributes to some training records – Reason: noisy data or presence of certain factors that are not included in the analysis Probability Basics ● P(A = a, C = c): joint probability that random variables A and C will take values a and c respectively ● P(A = a | C = c): conditional probability that A will take the value a, given that C has taken value c P(A,C) P(C | A) = P(A) P(A,C) P(A | C) = P(C) Bayes Theorem ● Bayes theorem: P(A | C)P(C) P(C | A) = P(A) ● P(C) known as the prior probability ● P(C | A) known as the posterior probability Example of Bayes Theorem ● Given: – A doctor knows that meningitis causes stiff neck 50% of the time – Prior probability of any patient having meningitis is 1/50,000 – Prior probability of any patient having stiff neck is 1/20 ● If a patient has stiff neck, what’s the probability he/she has meningitis? P(S | M )P(M ) 0.5×1/50000 P(M | S) = = = 0.0002 P(S) 1/ 20 Bayesian Classifiers ● Consider each attribute and class label as random variables ● Given a record with attributes (A1, A2,…,An) – Goal is to predict class C – Specifically, we want to find the value of C that maximizes P(C| A1, A2,…,An ) ● Can we estimate -
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. -
Posterior Propriety and Admissibility of Hyperpriors in Normal
The Annals of Statistics 2005, Vol. 33, No. 2, 606–646 DOI: 10.1214/009053605000000075 c Institute of Mathematical Statistics, 2005 POSTERIOR PROPRIETY AND ADMISSIBILITY OF HYPERPRIORS IN NORMAL HIERARCHICAL MODELS1 By James O. Berger, William Strawderman and Dejun Tang Duke University and SAMSI, Rutgers University and Novartis Pharmaceuticals Hierarchical modeling is wonderful and here to stay, but hyper- parameter priors are often chosen in a casual fashion. Unfortunately, as the number of hyperparameters grows, the effects of casual choices can multiply, leading to considerably inferior performance. As an ex- treme, but not uncommon, example use of the wrong hyperparameter priors can even lead to impropriety of the posterior. For exchangeable hierarchical multivariate normal models, we first determine when a standard class of hierarchical priors results in proper or improper posteriors. We next determine which elements of this class lead to admissible estimators of the mean under quadratic loss; such considerations provide one useful guideline for choice among hierarchical priors. Finally, computational issues with the resulting posterior distributions are addressed. 1. Introduction. 1.1. The model and the problems. Consider the block multivariate nor- mal situation (sometimes called the “matrix of means problem”) specified by the following hierarchical Bayesian model: (1) X ∼ Np(θ, I), θ ∼ Np(B, Σπ), where X1 θ1 X2 θ2 Xp×1 = . , θp×1 = . , arXiv:math/0505605v1 [math.ST] 27 May 2005 . X θ m m Received February 2004; revised July 2004. 1Supported by NSF Grants DMS-98-02261 and DMS-01-03265. AMS 2000 subject classifications. Primary 62C15; secondary 62F15. Key words and phrases. Covariance matrix, quadratic loss, frequentist risk, posterior impropriety, objective priors, Markov chain Monte Carlo. -
LINEAR REGRESSION MODELS Likelihood and Reference Bayesian
{ LINEAR REGRESSION MODELS { Likeliho o d and Reference Bayesian Analysis Mike West May 3, 1999 1 Straight Line Regression Mo dels We b egin with the simple straight line regression mo del y = + x + i i i where the design p oints x are xed in advance, and the measurement/sampling i 2 errors are indep endent and normally distributed, N 0; for each i = i i 1;::: ;n: In this context, wehavelooked at general mo delling questions, data and the tting of least squares estimates of and : Now we turn to more formal likeliho o d and Bayesian inference. 1.1 Likeliho o d and MLEs The formal parametric inference problem is a multi-parameter problem: we 2 require inferences on the three parameters ; ; : The likeliho o d function has a simple enough form, as wenow show. Throughout, we do not indicate the design p oints in conditioning statements, though they are implicitly conditioned up on. Write Y = fy ;::: ;y g and X = fx ;::: ;x g: Given X and the mo del 1 n 1 n parameters, each y is the corresp onding zero-mean normal random quantity i plus the term + x ; so that y is normal with this term as its mean and i i i 2 variance : Also, since the are indep endent, so are the y : Thus i i 2 2 y j ; ; N + x ; i i with conditional density function 2 2 2 2 1=2 py j ; ; = expfy x =2 g=2 i i i for each i: Also, by indep endence, the joint density function is n Y 2 2 : py j ; ; = pY j ; ; i i=1 1 Given the observed resp onse values Y this provides the likeliho o d function for the three parameters: the joint density ab ove evaluated at the observed and 2 hence now xed values of Y; now viewed as a function of ; ; as they vary across p ossible parameter values. -
The Deviance Information Criterion: 12 Years On
J. R. Statist. Soc. B (2014) The deviance information criterion: 12 years on David J. Spiegelhalter, University of Cambridge, UK Nicola G. Best, Imperial College School of Public Health, London, UK Bradley P. Carlin University of Minnesota, Minneapolis, USA and Angelika van der Linde Bremen, Germany [Presented to The Royal Statistical Society at its annual conference in a session organized by the Research Section on Tuesday, September 3rd, 2013, Professor G. A.Young in the Chair ] Summary.The essentials of our paper of 2002 are briefly summarized and compared with other criteria for model comparison. After some comments on the paper’s reception and influence, we consider criticisms and proposals for improvement made by us and others. Keywords: Bayesian; Model comparison; Prediction 1. Some background to model comparison Suppose that we have a given set of candidate models, and we would like a criterion to assess which is ‘better’ in a defined sense. Assume that a model for observed data y postulates a density p.y|θ/ (which may include covariates etc.), and call D.θ/ =−2log{p.y|θ/} the deviance, here considered as a function of θ. Classical model choice uses hypothesis testing for comparing nested models, e.g. the deviance (likelihood ratio) test in generalized linear models. For non- nested models, alternatives include the Akaike information criterion AIC =−2log{p.y|θˆ/} + 2k where θˆ is the maximum likelihood estimate and k is the number of parameters in the model (dimension of Θ). AIC is built with the aim of favouring models that are likely to make good predictions. -
Points for Discussion
Bayesian Analysis in Medicine EPIB-677 Points for Discussion 1. Precisely what information does a p-value provide? 2. What is the correct (and incorrect) way to interpret a confi- dence interval? 3. What is Bayes Theorem? How does it operate? 4. Summary of above three points: What exactly does one learn about a parameter of interest after having done a frequentist analysis? Bayesian analysis? 5. Example 8 from the book chapter by Jim Berger. 6. Example 13 from the book chapter by Jim Berger. 7. Example 17 from the book chapter by Jim Berger. 8. Examples where there is a choice between a binomial or a negative binomial likelihood, found in the paper by Berger and Berry. 9. Problems with specifying prior distributions. 10. In what types of epidemiology data analysis situations are Bayesian methods particularly useful? 11. What does decision analysis add to a Bayesian analysis? 1. Precisely what information does a p-value provide? Recall the definition of a p-value: The probability of observing a test statistic as extreme as or more extreme than the observed value, as- suming that the null hypothesis is true. 2. What is the correct (and incorrect) way to interpret a confi- dence interval? Does a 95% confidence interval provide a 95% probability region for the true parameter value? If not, what it the correct interpretation? In practice, it is usually helpful to consider the following graphic: Figure 1: How to interpret confidence intervals and/or credible regions. Depending on where the confidence/credible interval lies in relation to a region of clinical equivalence, different conclusions can be drawn. -
A Widely Applicable Bayesian Information Criterion
JournalofMachineLearningResearch14(2013)867-897 Submitted 8/12; Revised 2/13; Published 3/13 A Widely Applicable Bayesian Information Criterion Sumio Watanabe [email protected] Department of Computational Intelligence and Systems Science Tokyo Institute of Technology Mailbox G5-19, 4259 Nagatsuta, Midori-ku Yokohama, Japan 226-8502 Editor: Manfred Opper Abstract A statistical model or a learning machine is called regular if the map taking a parameter to a prob- ability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature 1/logn, where n is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for or unrealizable by a statistical model. -
The Bayesian New Statistics: Hypothesis Testing, Estimation, Meta-Analysis, and Power Analysis from a Bayesian Perspective
Published online: 2017-Feb-08 Corrected proofs submitted: 2017-Jan-16 Accepted: 2016-Dec-16 Published version at http://dx.doi.org/10.3758/s13423-016-1221-4 Revision 2 submitted: 2016-Nov-15 View only at http://rdcu.be/o6hd Editor action 2: 2016-Oct-12 In Press, Psychonomic Bulletin & Review. Revision 1 submitted: 2016-Apr-16 Version of November 15, 2016. Editor action 1: 2015-Aug-23 Initial submission: 2015-May-13 The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective John K. Kruschke and Torrin M. Liddell Indiana University, Bloomington, USA In the practice of data analysis, there is a conceptual distinction between hypothesis testing, on the one hand, and estimation with quantified uncertainty, on the other hand. Among frequentists in psychology a shift of emphasis from hypothesis testing to estimation has been dubbed “the New Statistics” (Cumming, 2014). A second conceptual distinction is between frequentist methods and Bayesian methods. Our main goal in this article is to explain how Bayesian methods achieve the goals of the New Statistics better than frequentist methods. The article reviews frequentist and Bayesian approaches to hypothesis testing and to estimation with confidence or credible intervals. The article also describes Bayesian approaches to meta-analysis, randomized controlled trials, and power analysis. Keywords: Null hypothesis significance testing, Bayesian inference, Bayes factor, confidence interval, credible interval, highest density interval, region of practical equivalence, meta-analysis, power analysis, effect size, randomized controlled trial. he New Statistics emphasizes a shift of empha- phasis. Bayesian methods provide a coherent framework for sis away from null hypothesis significance testing hypothesis testing, so when null hypothesis testing is the crux T (NHST) to “estimation based on effect sizes, confi- of the research then Bayesian null hypothesis testing should dence intervals, and meta-analysis” (Cumming, 2014, p. -
Bayes Theorem and Its Recent Applications
BAYES THEOREM AND ITS RECENT APPLICATIONS Nicholas Stylianides Eleni Kontou March 2020 Abstract Individuals who have studied maths to a specific level have come across Bayes’ Theorem or Bayes Formula. Bayes’ Theorem has many applications in areas such as mathematics, medicine, finance, marketing, engineering and many other. This paper covers Bayes’ Theorem at a basic level and explores how the formula was derived. We also, look at some extended forms of the formula and give an explicit example. Lastly, we discuss recent applications of Bayes’ Theorem in valuating depression and predicting water quality conditions. The goal of this expository paper is to break down the interesting topic of Bayes’ Theorem to a basic level which can be easily understood by a wide audience. What is Bayes Theorem? Thomas Bayes was an 18th-century British mathematician who developed a mathematical formula to calculate conditional probability in order to provide a way to re-examine current expectations. This mathematical formula is well known as Bayes Theorem or Bayes’ rule or Bayes’ Law. It is also the basis of a whole field of statistics known as Bayesian Statistics. For example, in finance, Bayes’ Theorem can be utilized to rate the risk of loaning cash to possible borrowers. The formula for Bayes’ Theorem is: [1] P(A) =The probability of A occurring P(B) = The probability of B occurring P(A|B) = The probability of A given B P(B|A) = The probability of B given A P(A∩B) = The probability of both A and B occurring MA3517 Mathematics Research Journal @Nicholas Stylianides, Eleni Kontou Bayes’ Theorem: , a $% &((!) > 0 %,- .// Proof: From Bayes’ Formula: (1) From the Total Probability (2) Theorem: Substituting equation (2) into equation (1) we obtain . -
More on Bayesian Methods: Part II
More on Bayesian Methods: Part II Rebecca C. Steorts Bayesian Methods and Modern Statistics: STA 360/601 Lecture 5 1 Today's menu I Confidence intervals I Credible Intervals I Example 2 Confidence intervals vs credible intervals A confidence interval for an unknown (fixed) parameter θ is an interval of numbers that we believe is likely to contain the true value of θ: Intervals are important because they provide us with an idea of how well we can estimate θ: 3 Confidence intervals vs credible intervals I A confidence interval is constructed to contain θ a percentage of the time, say 95%. I Suppose our confidence level is 95% and our interval is (L; U). Then we are 95% confident that the true value of θ is contained in (L; U) in the long run. I In the long run means that this would occur nearly 95% of the time if we repeated our study millions and millions of times. 4 Common Misconceptions in Statistical Inference I A confidence interval is a statement about θ (a population parameter). I It is not a statement about the sample. I It is also not a statement about individual subjects in the population. 5 Common Misconceptions in Statistical Inference I Let a 95% confidence interval for the average amount of television watched by Americans be (2.69, 6.04) hours. I This doesn't mean we can say that 95% of all Americans watch between 2.69 and 6.04 hours of television. I We also cannot say that 95% of Americans in the sample watch between 2.69 and 6.04 hours of television.