{"title":"The Shark Attack Problem: The Gamma-Poisson Conjugate","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0011","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0011","url":null,"abstract":"This chapter introduces the gamma-Poisson conjugate. Many Bayesian analyses consider alternative parameter values as hypotheses. The prior distribution for an unknown parameter can be represented by a continuous probability density function when the number of hypotheses is infinite. There are special cases where a Bayesian prior probability distribution for an unknown parameter of interest can be quickly updated to a posterior distribution of the same form as the prior. In the “Shark Attack Problem,” a gamma distribution is used as the prior distribution of λ, the mean number of shark attacks in a given year. Poisson data are then collected to determine the number of attacks in a given year. The prior distribution is updated to the posterior distribution in light of this new information. In short, a gamma prior distribution + Poisson data → gamma posterior distribution. The gamma distribution is said to be “conjugate to” the Poisson distribution.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123795112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Joint, Marginal, and Conditional Probability","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0002","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0002","url":null,"abstract":"This chapter introduces additional terms and concepts used in the study of probability, including Venn diagrams, independent events, and dependent events. The chapter focuses on two characteristics observed at the same time. In the example given in the chapter, the characteristics are eye dominance (i.e., left eye dominance or right eye dominance) and the presence or absence of “Morton’s toe” (Morton’s toe is a large second metatarsal which is longer than that the first metatarsal, or big toe; less than 20% of the human population has this condition). The chapter then analyses the distribution of these characteristics, both separately and simultaneously. In doing so, the chapter introduce the important concepts of joint probability, marginal probability, and conditional probability.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130670480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Once-ler Problem: Introduction to Decision Trees","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0020","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0020","url":null,"abstract":"In the “Once-ler Problem,” the decision tree is introduced as a very useful technique that can be used to answer a variety of questions and assist in making decisions. This chapter builds on the “Lorax Problem” introduced in Chapter 19, where Bayesian networks were introduced. A decision tree is a graphical representation of the alternatives in a decision. It is closely related to Bayesian networks except that the decision problem takes the shape of a tree instead. The tree itself consists of decision nodes, chance nodes, and end nodes, which provide an outcome. In a decision tree, probabilities associated with chance nodes are conditional probabilities, which Bayes’ Theorem can be used to estimate or update. The calculation of expected values (or expected utility) of competing alternative decisions is provided on a step-by-step basis with an example from The Lorax.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130784580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Author Problem: Bayesian Inference with Two Hypotheses","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0005","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0005","url":null,"abstract":"The “Author Problem” provides a concrete example of Bayesian inference. This chapter draws on work by Frederick Mosteller and David Wallace, who used Bayesian inference to assign authorship for unsigned Federalist Papers. The Federalist Papers were a collection of papers known to be written during the American Revolution. However, some papers were unsigned by the author, resulting in disputed authorship. The chapter provides a very basic Bayesian analysis of the unsigned “Paper 54,” which was written by Alexander Hamilton or James Madison. The example illustrates the principles of Bayesian inference for two competing hypotheses, including the concepts of alternative hypothesis, prior probability distribution, posterior probability distribution, prior probability of a hypothesis, likelihood of the observed data, and posterior probability of a hypothesis.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133153877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Lorax Problem: Introduction to Bayesian Networks","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0019","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0019","url":null,"abstract":"The “Lorax Problem” introduces Bayesian networks, another set of methods that makes use of Bayes’ Theorem. The ideas are first explained in terms of a small, standard example that explores two alternative hypotheses for why the grass is wet: the sprinkler is on versus it is raining. The chapter describes how to depict causal models graphically with the use of influence diagrams and directed acyclic graphs. Bayes’ Theorem is used to compute conditional probabilities and to update probabilities once new information is obtained or assumed. The software program Netica is introduced. Finally, the chapter provides a second example of Bayesian networks based on The Lorax by Dr. Seuss. The reader will gain a firm understanding of parent nodes (also known as root nodes), child nodes, conditional probability tables (CPTs), and the chain rule for joint probability.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124092991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Survivor Problem Continued: Introduction to Bayesian Model Selection","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0018","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0018","url":null,"abstract":"This chapter provides a very brief introduction to Bayesian model selection. The “Survivor Problem” is expanded in this chapter, where the focus is now on comparing two models that predict how long a contestant will last in a game of Survivor: one model uses years of formal education as a predictor, and a second model uses grit as a predictor. Gibbs sampling is used for parameter estimation. Deviance Information Criterion (commonly abbreviated as DIC) is used as a guide for model selection. Details of how this measure is computed are described. The chapter also discusses model assessment (model fit) and Occam’s razor.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125431076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bayes’ Theorem","authors":"Therese M. Donovan, R. Mickey","doi":"10.1093/oso/9780198841296.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198841296.003.0003","url":null,"abstract":"This chapter focuses on Bayes’ Theorem. The chapter first gives a brief introduction to Thomas Bayes, who first formulated the theorem. It then builds on the content presented in Chapters 1 and 2 to derive Bayes’ Theorem and describes two ways to think about it. First, Bayes’ Theorem describes the relationship between two inverse conditional probabilities, P(A | B) and P(B | A). Second, Bayes’ Theorem can be used to express how a degree of belief for a given hypothesis can be updated in light of new evidence. This chapter focuses on the first interpretation. The chapter also discusses the concepts of joint probability and marginal probability.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"167 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122069934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Probability Density Functions","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0009","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0009","url":null,"abstract":"This chapter builds on probability distributions. Its focus is on general concepts associated with probability density functions (pdf’s), which are distributions associated with continuous random variables. The continuous uniform and normal distributions are highlighted as examples of pdf’s. These and other pdf’s can be used to specify prior distributions, likelihoods, and/or posterior distributions in Bayesian inference. Although this chapter specifically focuses on the continuous uniform and normal distributions, the concepts discussed in this chapter will apply to other continuous probability distributions. By the end of the chapter, the reader should be able to define and use the following terms for a continuous random variable: random variable, probability distribution, parameter, probability density, likelihood, and likelihood profile.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125446467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The White House Problem Revisited: MCMC with the Metropolis–Hastings Algorithm","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0015","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0015","url":null,"abstract":"The “White House Problem” of Chapter 10 is revisited in this chapter. Markov Chain Monte Carlo (MCMC) is used to build the posterior distribution of the unknown parameter p, the probability that a famous person could gain access to the White House without invitation. The chapter highlights the Metropolis–Hastings algorithm in MCMC analysis, describing the process step by step. The posterior distribution generated in Chapter 10 using the beta-binomial conjugate is compared with the MCMC posterior distribution to show how successful the MCMC method can be. By the end of this chapter, the reader will have a firm understanding of the following concepts: Monte Carlo, Markov chain, Metropolis–Hastings algorithm, Metropolis–Hastings random walk, and Metropolis–Hastings independence sampler.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127381347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Maple Syrup Problem Revisited: MCMC with Gibbs Sampling","authors":"T. Donovan, R. Mickey","doi":"10.1093/OSO/9780198841296.003.0016","DOIUrl":"https://doi.org/10.1093/OSO/9780198841296.003.0016","url":null,"abstract":"This chapter introduces Markov Chain Monte Carlo (MCMC) with Gibbs sampling, revisiting the “Maple Syrup Problem” of Chapter 12, where the goal was to estimate the two parameters of a normal distribution, μ and σ. Chapter 12 used the normal-normal conjugate to derive the posterior distribution for the unknown parameter μ; the parameter σ was assumed to be known. This chapter uses MCMC with Gibbs sampling to estimate the joint posterior distribution of both μ and σ. Gibbs sampling is a special case of the Metropolis–Hastings algorithm. The chapter describes MCMC with Gibbs sampling step by step, which requires (1) computing the posterior distribution of a given parameter, conditional on the value of the other parameter, and (2) drawing a sample from the posterior distribution. In this chapter, Gibbs sampling makes use of the conjugate solutions to decompose the joint posterior distribution into full conditional distributions for each parameter.","PeriodicalId":285230,"journal":{"name":"Bayesian Statistics for Beginners","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126023557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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