The paper studies the relationship between p-values and Bayesian measures of evidence for testing point null hypotheses. It finds that p-values can be highly misleading about the evidence provided by data against the null hypothesis. Through examples, it shows that a small p-value does not necessarily mean high posterior probability for rejecting the null or a large Bayes factor against the null. The paper derives lower bounds on these Bayesian measures of evidence for several classes of distributions to demonstrate the conflicts between p-values and Bayesian analysis.
The EM algorithm is used to find maximum likelihood estimates for problems with latent variables. It works by alternating between an E-step (computing expected values of the latent variables) and an M-step (maximizing the likelihood with respect to the parameters). For mixture of Gaussians, the E-step computes the posterior probabilities that each data point belongs to each component. The M-step then updates the mixture weights, means, and covariances by taking weighted averages/sums of the data using these posteriors.
A Geometric Note on a Type of Multiple Testing-07-24-2015Junfeng Liu
This document summarizes new perspectives on false discovery rate (FDR) control procedures for multiple testing. It examines FDR control using linear and quadratic rejection cut-off routes applied to ordered p-values. Key findings include: 1) the FDR is controlled at π0q regardless of where the Ha p-value profile crosses the no-rejection boundary, 2) specificity approaches limits as the Ha mean increases, 3) quadratic cuts control FDR better when Ha means are close to zero. Numerical simulations explore the impact of factors like population size, variation levels, and mean profiles on discovery rates and FDR.
This document discusses treating the truth predicate (Tr) as a logical connective in truth theories like Friedman-Sheared's theory (FS). It analyzes FS from the perspective of proof theoretic semantics, where Tr's introduction and elimination rules are like those of a connective. However, FS violates the "harmony" requirement for connectives, as it is not conservatively extending and proves the consistency of PA. The document then discusses interpreting paradoxical sentences like McGee's using coinduction and how guarded corecursion relates to the failure of Tr's formal commutability in FS.
The document discusses statistical models and exponential families. It states that for most of the course, data is assumed to be a random sample from a distribution F. Repetition of observations via the law of large numbers and central limit theorem increases information about F. Exponential families are a class of parametric distributions with convenient analytic properties, where the density can be written as a function of natural parameters in an exponential form. Examples of exponential families include the binomial and normal distributions.
This document discusses Bayesian hypothesis testing and some of the challenges associated with it. It makes three key points:
1) There is tension between using posterior probabilities from a loss function approach versus Bayes factors, which eliminate prior dependence but have no direct connection to the posterior.
2) Bayesian hypothesis testing relies on choosing prior probabilities for hypotheses and prior distributions for parameters, which can strongly impact results and are often arbitrary.
3) Common Bayesian testing procedures like using Bayes factors can produce paradoxical results in some cases, like Lindley's paradox where the Bayes factor favors the null hypothesis as sample size increases despite evidence against it.
"reflections on the probability space induced by moment conditions with impli...Christian Robert
This document discusses using moment conditions to perform Bayesian inference when the likelihood function is intractable or unknown. It outlines some approaches that have been proposed, including approximating the likelihood using empirical likelihood or pseudo-likelihoods. However, these approaches do not guarantee the same consistency as a true likelihood. Alternative approximative Bayesian methods are also discussed, such as Approximate Bayesian Computation, Integrated Nested Laplace Approximation, and variational Bayes. The empirical likelihood method constructs a likelihood from generalized moment conditions, but its use in Bayesian inference requires further analysis of consistency in each application.
The EM algorithm is used to find maximum likelihood estimates for problems with latent variables. It works by alternating between an E-step (computing expected values of the latent variables) and an M-step (maximizing the likelihood with respect to the parameters). For mixture of Gaussians, the E-step computes the posterior probabilities that each data point belongs to each component. The M-step then updates the mixture weights, means, and covariances by taking weighted averages/sums of the data using these posteriors.
A Geometric Note on a Type of Multiple Testing-07-24-2015Junfeng Liu
This document summarizes new perspectives on false discovery rate (FDR) control procedures for multiple testing. It examines FDR control using linear and quadratic rejection cut-off routes applied to ordered p-values. Key findings include: 1) the FDR is controlled at π0q regardless of where the Ha p-value profile crosses the no-rejection boundary, 2) specificity approaches limits as the Ha mean increases, 3) quadratic cuts control FDR better when Ha means are close to zero. Numerical simulations explore the impact of factors like population size, variation levels, and mean profiles on discovery rates and FDR.
This document discusses treating the truth predicate (Tr) as a logical connective in truth theories like Friedman-Sheared's theory (FS). It analyzes FS from the perspective of proof theoretic semantics, where Tr's introduction and elimination rules are like those of a connective. However, FS violates the "harmony" requirement for connectives, as it is not conservatively extending and proves the consistency of PA. The document then discusses interpreting paradoxical sentences like McGee's using coinduction and how guarded corecursion relates to the failure of Tr's formal commutability in FS.
The document discusses statistical models and exponential families. It states that for most of the course, data is assumed to be a random sample from a distribution F. Repetition of observations via the law of large numbers and central limit theorem increases information about F. Exponential families are a class of parametric distributions with convenient analytic properties, where the density can be written as a function of natural parameters in an exponential form. Examples of exponential families include the binomial and normal distributions.
This document discusses Bayesian hypothesis testing and some of the challenges associated with it. It makes three key points:
1) There is tension between using posterior probabilities from a loss function approach versus Bayes factors, which eliminate prior dependence but have no direct connection to the posterior.
2) Bayesian hypothesis testing relies on choosing prior probabilities for hypotheses and prior distributions for parameters, which can strongly impact results and are often arbitrary.
3) Common Bayesian testing procedures like using Bayes factors can produce paradoxical results in some cases, like Lindley's paradox where the Bayes factor favors the null hypothesis as sample size increases despite evidence against it.
"reflections on the probability space induced by moment conditions with impli...Christian Robert
This document discusses using moment conditions to perform Bayesian inference when the likelihood function is intractable or unknown. It outlines some approaches that have been proposed, including approximating the likelihood using empirical likelihood or pseudo-likelihoods. However, these approaches do not guarantee the same consistency as a true likelihood. Alternative approximative Bayesian methods are also discussed, such as Approximate Bayesian Computation, Integrated Nested Laplace Approximation, and variational Bayes. The empirical likelihood method constructs a likelihood from generalized moment conditions, but its use in Bayesian inference requires further analysis of consistency in each application.
Presentation of Birnbaum's Likelihood Principle foundational paper at the Reading Statistical Classics seminar, Jan. 20, 2013, Université Paris-Dauphine
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
The document outlines the theory of the most efficient tests of statistical hypotheses as proposed by Neyman Pearson. It discusses the two types of errors in hypothesis testing, the likelihood principle, and provides solutions for finding the most efficient critical regions for simple and composite hypotheses. Examples are given to illustrate the theory and key points include that the most efficient critical region maximizes the probability of rejecting the null for a given probability of type 1 error.
Discussion of Persi Diaconis' lecture at ISBA 2016Christian Robert
This document discusses Monte Carlo methods for numerical integration and estimating normalizing constants. It summarizes several approaches: estimating normalizing constants using samples; reverse logistic regression for estimating constants in mixtures; Xiao-Li's maximum likelihood formulation for Monte Carlo integration; and Persi's probabilistic numerics which provide uncertainties for numerical calculations. The document advocates first approximating the distribution of an integrand before estimating its expectation to incorporate non-parametric information and account for multiple estimators.
Statistics (1): estimation Chapter 3: likelihood function and likelihood esti...Christian Robert
The document discusses likelihood functions and inference. It begins by defining the likelihood function as the function that gives the probability of observing a sample given a parameter value. The likelihood varies with the parameter, while the density function varies with the data. Maximum likelihood estimation chooses parameters that maximize the likelihood function. The score function is the gradient of the log-likelihood and has an expected value of zero at the true parameter value. The Fisher information matrix measures the curvature of the likelihood surface and provides information about the precision of parameter estimates. It relates to the concentration of likelihood functions around the true parameter value as sample size increases.
This document summarizes a presentation on testing hypotheses as mixture estimation and the challenges of Bayesian testing. The key points are:
1) Bayesian hypothesis testing faces challenges including the dependence on prior distributions, difficulties interpreting Bayes factors, and the inability to use improper priors in most situations.
2) Testing via mixtures is proposed as a paradigm shift that frames hypothesis testing as a model selection problem involving mixture models rather than distinct hypotheses.
3) Traditional Bayesian testing using Bayes factors and posterior probabilities depends strongly on prior distributions and choices that are difficult to justify, while not providing measures of uncertainty around decisions. Alternative approaches are needed to address these issues.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
Optimization Approach to Nash Euilibria with Applications to InterchangeabilityYosuke YASUDA
This document presents an optimization approach to characterizing Nash equilibria in games. It shows that the set of Nash equilibria is identical to the set of solutions that minimize an objective function defined over strategy profiles. This allows the equilibrium problem to be framed as an optimization problem. The approach provides a unified way to derive existing results on interchangeability of equilibria in zero-sum and supermodular games, by relating the properties of the objective function to the structure of the optimal solution set.
Statistics (1): estimation, Chapter 2: Empirical distribution and bootstrapChristian Robert
The document discusses the bootstrap method and its applications in statistical inference. It introduces the bootstrap as a technique for estimating properties of estimators like variance and distribution when the true sampling distribution is unknown. This is done by treating the observed sample as if it were the population and resampling with replacement to create new simulated samples. The bootstrap then approximates characteristics of the sampling distribution, allowing inferences like confidence intervals to be constructed.
Lecture slides on Decision Theory. The contents in large part come from the following excellent textbook.
Rubinstein, A. (2012). Lecture notes in microeconomic theory: the
economic agent, 2nd.
http://www.amazon.co.jp/dp/B0073X0J7Q/
On the vexing dilemma of hypothesis testing and the predicted demise of the B...Christian Robert
The document discusses hypothesis testing from both frequentist and Bayesian perspectives. It introduces the concept of statistical tests as functions that output accept or reject decisions for hypotheses. P-values are presented as a way to quantify uncertainty in these decisions. Bayes' original 1763 paper on Bayesian statistics is summarized, introducing the concept of the posterior distribution. Bayesian hypothesis testing is then discussed, including the optimal Bayes test and the use of Bayes factors to compare hypotheses without requiring prior probabilities on the hypotheses.
The document describes Approximate Bayesian Computation (ABC), a technique for performing Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC works by simulating data under different parameter values, and accepting simulations that are close to the observed data according to a distance measure and tolerance level. ABC provides an approximation to the posterior distribution that improves as the tolerance level decreases and more informative summary statistics are used. The document discusses the ABC algorithm, properties of the exact ABC posterior distribution, and challenges in selecting appropriate summary statistics.
11.[29 35]a unique common fixed point theorem under psi varphi contractive co...Alexander Decker
This document presents a unique common fixed point theorem for two self maps satisfying a generalized contraction condition in partial metric spaces using rational expressions. It begins by introducing basic definitions and lemmas related to partial metric spaces. It then presents the main theorem, which states that if two self maps T and f satisfy certain contractive and completeness conditions, including being weakly compatible, then they have a unique common fixed point. The proof considers two cases - when the sequences constructed from the maps are eventually equal, and when they are not eventually equal but form a Cauchy sequence. It is shown in both cases that the maps must have a unique common fixed point.
This document proposes representing hypothesis testing problems as estimating mixture models. Specifically, two competing models are embedded within an encompassing mixture model with a weight parameter between 0 and 1. Inference is then drawn on the mixture representation, treating each observation as coming from the mixture model. This avoids difficulties with traditional Bayesian testing approaches like computing marginal likelihoods. It also allows for a more intuitive interpretation of the weight parameter compared to posterior model probabilities. The weight parameter can be estimated using standard mixture estimation algorithms like Gibbs sampling or Metropolis-Hastings. Several illustrations of the approach are provided, including comparisons of Poisson and geometric distributions.
This document discusses various importance sampling methods for approximating Bayes factors, which are used for Bayesian model selection. It compares regular importance sampling, bridge sampling, harmonic means, mixtures to bridge sampling, and Chib's solution. An example application to probit modeling of diabetes in Pima Indian women is presented to illustrate regular importance sampling. Markov chain Monte Carlo methods like the Metropolis-Hastings algorithm and Gibbs sampling can be used to sample from the probit models.
The document summarizes Approximate Bayesian Computation (ABC). It discusses how ABC provides a way to approximate Bayesian inference when the likelihood function is intractable or too computationally expensive to evaluate directly. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure and tolerance level. Key points discussed include:
- ABC provides an approximation to the posterior distribution by sampling from simulations that fall within a tolerance of the observed data.
- Summary statistics are often used to reduce the dimension of the data and improve the signal-to-noise ratio when applying the tolerance criterion.
- Random forests can help select informative summary statistics and provide semi-automated ABC
This document discusses various methods for estimating normalizing constants that arise when evaluating integrals numerically. It begins by noting there are many computational methods for approximating normalizing constants across different communities. It then lists the topics that will be covered in the upcoming workshop, including discussions on estimating constants using Monte Carlo methods and Bayesian versus frequentist approaches. The document provides examples of estimating normalizing constants using Monte Carlo integration, reverse logistic regression, and Xiao-Li Meng's maximum likelihood estimation approach. It concludes by discussing some of the challenges in bringing a statistical framework to constant estimation problems.
This document discusses using the Wasserstein distance for inference in generative models. It begins with an overview of approximate Bayesian computation (ABC) and how distances between samples are used. It then introduces the Wasserstein distance as an alternative distance that can have lower variance than the Euclidean distance. Computational aspects and asymptotics of using the Wasserstein distance are discussed. The document also covers how transport distances can handle time series data.
Reading the Lindley-Smith 1973 paper on linear Bayes estimatorsChristian Robert
The document outlines a seminar on Bayes estimates for the linear model. It introduces the linear model and Bayesian methods. It then discusses exchangeability, providing an example of an exchangeable distribution. It also discusses the general Bayesian linear model, including the posterior distribution of the parameters using a three stage model.
Asymptotics for discrete random measuresJulyan Arbel
This document provides an introduction to asymptotics for discrete random measures, specifically the Dirichlet process and two-parameter Poisson-Dirichlet process. It discusses several key aspects in 3 sentences or less:
1) It outlines the stick-breaking construction of the two-parameter Poisson-Dirichlet process and defines related notation. 2) It introduces the truncation error Rn and discusses how its asymptotic behavior differs between the Dirichlet and two-parameter Poisson-Dirichlet cases. 3) It briefly describes some applications of these processes in mixture modeling and summarizes different sampling approaches like blocked Gibbs and slice sampling that rely on truncation of the infinite-dimensional distributions.
Presentation of Birnbaum's Likelihood Principle foundational paper at the Reading Statistical Classics seminar, Jan. 20, 2013, Université Paris-Dauphine
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
The document outlines the theory of the most efficient tests of statistical hypotheses as proposed by Neyman Pearson. It discusses the two types of errors in hypothesis testing, the likelihood principle, and provides solutions for finding the most efficient critical regions for simple and composite hypotheses. Examples are given to illustrate the theory and key points include that the most efficient critical region maximizes the probability of rejecting the null for a given probability of type 1 error.
Discussion of Persi Diaconis' lecture at ISBA 2016Christian Robert
This document discusses Monte Carlo methods for numerical integration and estimating normalizing constants. It summarizes several approaches: estimating normalizing constants using samples; reverse logistic regression for estimating constants in mixtures; Xiao-Li's maximum likelihood formulation for Monte Carlo integration; and Persi's probabilistic numerics which provide uncertainties for numerical calculations. The document advocates first approximating the distribution of an integrand before estimating its expectation to incorporate non-parametric information and account for multiple estimators.
Statistics (1): estimation Chapter 3: likelihood function and likelihood esti...Christian Robert
The document discusses likelihood functions and inference. It begins by defining the likelihood function as the function that gives the probability of observing a sample given a parameter value. The likelihood varies with the parameter, while the density function varies with the data. Maximum likelihood estimation chooses parameters that maximize the likelihood function. The score function is the gradient of the log-likelihood and has an expected value of zero at the true parameter value. The Fisher information matrix measures the curvature of the likelihood surface and provides information about the precision of parameter estimates. It relates to the concentration of likelihood functions around the true parameter value as sample size increases.
This document summarizes a presentation on testing hypotheses as mixture estimation and the challenges of Bayesian testing. The key points are:
1) Bayesian hypothesis testing faces challenges including the dependence on prior distributions, difficulties interpreting Bayes factors, and the inability to use improper priors in most situations.
2) Testing via mixtures is proposed as a paradigm shift that frames hypothesis testing as a model selection problem involving mixture models rather than distinct hypotheses.
3) Traditional Bayesian testing using Bayes factors and posterior probabilities depends strongly on prior distributions and choices that are difficult to justify, while not providing measures of uncertainty around decisions. Alternative approaches are needed to address these issues.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
Optimization Approach to Nash Euilibria with Applications to InterchangeabilityYosuke YASUDA
This document presents an optimization approach to characterizing Nash equilibria in games. It shows that the set of Nash equilibria is identical to the set of solutions that minimize an objective function defined over strategy profiles. This allows the equilibrium problem to be framed as an optimization problem. The approach provides a unified way to derive existing results on interchangeability of equilibria in zero-sum and supermodular games, by relating the properties of the objective function to the structure of the optimal solution set.
Statistics (1): estimation, Chapter 2: Empirical distribution and bootstrapChristian Robert
The document discusses the bootstrap method and its applications in statistical inference. It introduces the bootstrap as a technique for estimating properties of estimators like variance and distribution when the true sampling distribution is unknown. This is done by treating the observed sample as if it were the population and resampling with replacement to create new simulated samples. The bootstrap then approximates characteristics of the sampling distribution, allowing inferences like confidence intervals to be constructed.
Lecture slides on Decision Theory. The contents in large part come from the following excellent textbook.
Rubinstein, A. (2012). Lecture notes in microeconomic theory: the
economic agent, 2nd.
http://www.amazon.co.jp/dp/B0073X0J7Q/
On the vexing dilemma of hypothesis testing and the predicted demise of the B...Christian Robert
The document discusses hypothesis testing from both frequentist and Bayesian perspectives. It introduces the concept of statistical tests as functions that output accept or reject decisions for hypotheses. P-values are presented as a way to quantify uncertainty in these decisions. Bayes' original 1763 paper on Bayesian statistics is summarized, introducing the concept of the posterior distribution. Bayesian hypothesis testing is then discussed, including the optimal Bayes test and the use of Bayes factors to compare hypotheses without requiring prior probabilities on the hypotheses.
The document describes Approximate Bayesian Computation (ABC), a technique for performing Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC works by simulating data under different parameter values, and accepting simulations that are close to the observed data according to a distance measure and tolerance level. ABC provides an approximation to the posterior distribution that improves as the tolerance level decreases and more informative summary statistics are used. The document discusses the ABC algorithm, properties of the exact ABC posterior distribution, and challenges in selecting appropriate summary statistics.
11.[29 35]a unique common fixed point theorem under psi varphi contractive co...Alexander Decker
This document presents a unique common fixed point theorem for two self maps satisfying a generalized contraction condition in partial metric spaces using rational expressions. It begins by introducing basic definitions and lemmas related to partial metric spaces. It then presents the main theorem, which states that if two self maps T and f satisfy certain contractive and completeness conditions, including being weakly compatible, then they have a unique common fixed point. The proof considers two cases - when the sequences constructed from the maps are eventually equal, and when they are not eventually equal but form a Cauchy sequence. It is shown in both cases that the maps must have a unique common fixed point.
This document proposes representing hypothesis testing problems as estimating mixture models. Specifically, two competing models are embedded within an encompassing mixture model with a weight parameter between 0 and 1. Inference is then drawn on the mixture representation, treating each observation as coming from the mixture model. This avoids difficulties with traditional Bayesian testing approaches like computing marginal likelihoods. It also allows for a more intuitive interpretation of the weight parameter compared to posterior model probabilities. The weight parameter can be estimated using standard mixture estimation algorithms like Gibbs sampling or Metropolis-Hastings. Several illustrations of the approach are provided, including comparisons of Poisson and geometric distributions.
This document discusses various importance sampling methods for approximating Bayes factors, which are used for Bayesian model selection. It compares regular importance sampling, bridge sampling, harmonic means, mixtures to bridge sampling, and Chib's solution. An example application to probit modeling of diabetes in Pima Indian women is presented to illustrate regular importance sampling. Markov chain Monte Carlo methods like the Metropolis-Hastings algorithm and Gibbs sampling can be used to sample from the probit models.
The document summarizes Approximate Bayesian Computation (ABC). It discusses how ABC provides a way to approximate Bayesian inference when the likelihood function is intractable or too computationally expensive to evaluate directly. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure and tolerance level. Key points discussed include:
- ABC provides an approximation to the posterior distribution by sampling from simulations that fall within a tolerance of the observed data.
- Summary statistics are often used to reduce the dimension of the data and improve the signal-to-noise ratio when applying the tolerance criterion.
- Random forests can help select informative summary statistics and provide semi-automated ABC
This document discusses various methods for estimating normalizing constants that arise when evaluating integrals numerically. It begins by noting there are many computational methods for approximating normalizing constants across different communities. It then lists the topics that will be covered in the upcoming workshop, including discussions on estimating constants using Monte Carlo methods and Bayesian versus frequentist approaches. The document provides examples of estimating normalizing constants using Monte Carlo integration, reverse logistic regression, and Xiao-Li Meng's maximum likelihood estimation approach. It concludes by discussing some of the challenges in bringing a statistical framework to constant estimation problems.
This document discusses using the Wasserstein distance for inference in generative models. It begins with an overview of approximate Bayesian computation (ABC) and how distances between samples are used. It then introduces the Wasserstein distance as an alternative distance that can have lower variance than the Euclidean distance. Computational aspects and asymptotics of using the Wasserstein distance are discussed. The document also covers how transport distances can handle time series data.
Reading the Lindley-Smith 1973 paper on linear Bayes estimatorsChristian Robert
The document outlines a seminar on Bayes estimates for the linear model. It introduces the linear model and Bayesian methods. It then discusses exchangeability, providing an example of an exchangeable distribution. It also discusses the general Bayesian linear model, including the posterior distribution of the parameters using a three stage model.
Asymptotics for discrete random measuresJulyan Arbel
This document provides an introduction to asymptotics for discrete random measures, specifically the Dirichlet process and two-parameter Poisson-Dirichlet process. It discusses several key aspects in 3 sentences or less:
1) It outlines the stick-breaking construction of the two-parameter Poisson-Dirichlet process and defines related notation. 2) It introduces the truncation error Rn and discusses how its asymptotic behavior differs between the Dirichlet and two-parameter Poisson-Dirichlet cases. 3) It briefly describes some applications of these processes in mixture modeling and summarizes different sampling approaches like blocked Gibbs and slice sampling that rely on truncation of the infinite-dimensional distributions.
Species sampling models in Bayesian NonparametricsJulyan Arbel
This document discusses species sampling models and discovery probabilities. It introduces the problem of estimating the probability of observing a new species given a sample. Good and Turing proposed an estimator for this during World War II. Bayesian nonparametric models provide an alternative approach by placing a prior on unknown species proportions. The document outlines BNP estimators for discovery probabilities and how credible intervals can be derived. It applies these methods to genomic datasets of expressed sequence tags to estimate discovery probabilities for observing new genes.
A Gentle Introduction to Bayesian NonparametricsJulyan Arbel
The document provides an introduction to Bayesian nonparametrics and the Dirichlet process. It explains that Bayesian nonparametrics aims to fit models that can adapt their complexity based on the data, without strictly imposing a fixed structure. The Dirichlet process is described as a prior distribution on the space of all probability distributions, allowing the model to utilize an infinite number of parameters. Nonparametric mixture models using the Dirichlet process provide a flexible approach to density estimation and clustering.
Bayesian Nonparametrics, Applications to biology, ecology, and marketingJulyan Arbel
This document discusses applications of Bayesian nonparametric methods to various domains including toxicology, ecology, marketing, human fertility, and more. It provides examples of using rounded Gaussian mixtures and Dirichlet process mixtures to model count data from developmental toxicity studies and animal abundance data. Applications to modeling multivariate mobile phone usage data and basal body temperature curves are also described. The document emphasizes that Bayesian nonparametric approaches allow inclusion of prior information and flexible modeling of complex data structures.
This document discusses nested sampling, a technique for Bayesian computation and evidence evaluation. It begins by introducing Bayesian inference and the evidence integral. It then shows that nested sampling transforms the multidimensional evidence integral into a one-dimensional integral over the prior mass constrained to have likelihood above a given value. The document outlines the nested sampling algorithm and shows that it provides samples from the posterior distribution. It also discusses termination criteria and choices of sample size for the algorithm. Finally, it provides a numerical example of nested sampling applied to a Gaussian model.
Presentation of Bassoum Abou on Stein's 1981 AoS paperChristian Robert
The document discusses estimation of the mean of a multivariate normal distribution. It covers basic formulas, Bayes estimates, choosing a scalar factor, applications including symmetric moving averages, and the case of unknown variance. The main results are theorems on the risk and unbiased risk estimates for different Bayes estimates of the mean.
Dependent processes in Bayesian NonparametricsJulyan Arbel
This document summarizes dependent processes in Bayesian nonparametrics. It motivates the need for dependent random probability measures to accommodate temporal dependence structures beyond the exchangeability assumption. It describes modeling collections of random probability measures indexed by time as either discrete-time or continuous-time processes. The diffusive Dirichlet process is introduced as a dependent Dirichlet process with Dirichlet marginal distributions at each time point and continuous sample paths. Simulation and estimation methods are discussed for this model.
This document provides a list of 33 papers related to Bayesian statistics for students to choose from for a presentation. It includes brief descriptions of several theoretical and general audience journals. The papers cover a range of topics in Bayesian statistics published between 1763 and 2013. Students will be evaluated on their understanding and presentation of the chosen paper.
This document discusses sampling-based approaches for calculating marginal densities from conditional distributions. It introduces substitution algorithms, substitution sampling, Gibbs sampling, and importance sampling. Substitution algorithms iteratively estimate marginal densities by substituting conditional distributions. Substitution sampling generates samples by iteratively drawing from conditional distributions. Gibbs sampling repeatedly draws values from conditional distributions to estimate joint and marginal distributions.
The document describes several bootstrap methods for estimating parameters from sample data when the underlying distribution is unknown. It outlines the bootstrap procedure, which involves resampling the original data with replacement to create bootstrap samples and estimating the parameter from each resample. Three methods for calculating the bootstrap distribution are described: direct theoretical calculation, simulation-based resampling, and Bayesian approaches. The document also provides an example of using the bootstrap to estimate the median from a sample.
Reading the Lasso 1996 paper by Robert TibshiraniChristian Robert
The document outlines a presentation on regression analysis using the LASSO (Least Absolute Shrinkage and Selection Operator) method. It includes an introduction to the topic, definitions of key terms like OLS (ordinary least squares) estimates, and descriptions of standard techniques like subset selection and ridge regression. The bulk of the presentation covers LASSO specifically - its definition, motivation, behavior in certain cases, examples of its use, and algorithms for finding LASSO solutions. It concludes with a discussion of simulations. The presenter's goal is to explain the LASSO method for regression shrinkage and variable selection.
The document describes the k-means clustering algorithm. It introduces clustering and its aim to divide data points into k clusters to minimize within-cluster sums of squares. The algorithm involves initializing cluster centers, then iteratively performing optimal transfers of points between clusters and quick transfers until convergence is reached. Optimal transfers minimize an objective function to determine the best cluster for a point, while quick transfers perform simpler transfers without minimizing the objective.
This document discusses hypothesis testing and p-values. It begins by defining a hypothesis as a proposition or prediction about the outcome of an experiment. Hypotheses are formulated and tested through science to evaluate their credibility. There are two main types of hypotheses: the null hypothesis, which corresponds to a default or general position, and the alternative hypothesis, which asserts a rival relationship. Hypothesis testing uses sample data to evaluate whether differences observed could be due to chance (the null hypothesis) or are real effects (the alternative hypothesis). Key concepts discussed include type 1 and type 2 errors, significance levels, one-sided and two-sided tests, and the relationship between p-values, confidence intervals, and the strength of evidence against
This document discusses p-values and their significance in statistical hypothesis testing. It defines a p-value as the probability of obtaining a result equal to or more extreme than what was observed assuming the null hypothesis is true. Lower p-values indicate stronger evidence against the null hypothesis. The document outlines the steps in hypothesis testing which include stating hypotheses, determining acceptable type I and type II error rates, selecting a statistical test to calculate a test statistic, determining the p-value, making inferences, and forming conclusions. It emphasizes that statistical significance does not necessarily imply real-world significance.
Excursion 4 Tour II: Rejection Fallacies: Whose Exaggerating What?jemille6
This document discusses criticisms of p-values and proposes reforms based on Bayesian statistics. It summarizes debates between Fisher and Bayesians regarding p-values exaggerating evidence against the null hypothesis when using certain priors. When a lump prior of 0.5 is given to the null and the remaining 0.5 spread over the alternative, as the sample size increases, a statistically significant result can correspond to a posterior probability for the null that exceeds the prior of 0.5. Reforms are proposed based on likelihood ratios and Bayes factors to define statistical significance in a way more consistent with Bayesian evidence standards.
This document provides lecture notes on hypothesis testing. It begins with an introduction to hypothesis testing and how it differs from estimation in its hypothetical reasoning approach. It then discusses Fisher's significance testing approach, including defining a test statistic, its sampling distribution under the null hypothesis, and calculating a p-value. It provides examples of applying this approach. Finally, it discusses some weaknesses of Fisher's approach identified by Neyman and Pearson and how their approach improved upon it by introducing the concept of alternative hypotheses and pre-data error probabilities.
Mayo Slides: Part I Meeting #2 (Phil 6334/Econ 6614)jemille6
Slides Meeting #2 (Phil 6334/Econ 6614: Current Debates on Statistical Inference and Modeling (D. Mayo and A. Spanos)
Part I: Bernoulli trials: Plane Jane Version
Likelihoodist vs. significance tester w bernoulli trialsjemille6
This document provides background on the difference between the likelihoodist and significance testing approaches to analyzing Bernoulli trials. It explains that likelihoodists compare the likelihood of different parameter values, like comparing the likelihood of θ = 0.2 versus θ = 0.8 given observed data. Significance tests compare the likelihood of the null hypothesis versus an alternative, without considering specific alternatives, and reject the null if the data is deemed unlikely under the null. The document uses an example to illustrate how the two approaches can reach different conclusions about the evidence provided by data.
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
The document summarizes key concepts in hypothesis testing including:
- The null and alternative hypotheses are formulated, with the null hypothesis stating the parameter equals a specific value and the alternative allowing other values.
- There are two types of errors - type I rejects the null when true, type II accepts when false. Tests aim to minimize both.
- The power of a test is the probability it correctly rejects the null when an alternative is true.
- One-tailed tests have critical regions in one tail, two-tailed in both. P-values are used to determine if results are significant.
- Steps of hypothesis testing are outlined along with examples of tests for single and two means/proportions.
Equational axioms for probability calculus and modelling of Likelihood ratio ...Advanced-Concepts-Team
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Reading Testing a point-null hypothesis, by Jiahuan Li, Feb. 25, 2013
1. Testing a Point Null Hypothesisi: The
Irreconcilability of P Values and Evidence
JAMES O.BERGER and THOMAS SELLKE
25.02.2013
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
2. Content
1 INTRODUCTION
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
3. Content
1 INTRODUCTION
2 POSTERIOR PROBABILITIES AND ODDS
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
4. Content
1 INTRODUCTION
2 POSTERIOR PROBABILITIES AND ODDS
3 LOWER BOUNDS ON POSTERIOR PROBABILI-
TIES
Introduction
Lower Bounds for GA ={All Distributions}
Lower Bounds for GS ={Symmetric Distributions}
Lower Bounds for GUS ={Unimodal,Symmetric Distributions}
Lower Bounds for GNOR ={Normal Distributions}
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
5. Content
1 INTRODUCTION
2 POSTERIOR PROBABILITIES AND ODDS
3 LOWER BOUNDS ON POSTERIOR PROBABILI-
TIES
Introduction
Lower Bounds for GA ={All Distributions}
Lower Bounds for GS ={Symmetric Distributions}
Lower Bounds for GUS ={Unimodal,Symmetric Distributions}
Lower Bounds for GNOR ={Normal Distributions}
4 MORE GENERAL HYPOTHESES AND CONDITIONAL
CALCULATOINS
General Formulation
More General Hypotheses
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
6. Content
1 INTRODUCTION
2 POSTERIOR PROBABILITIES AND ODDS
3 LOWER BOUNDS ON POSTERIOR PROBABILI-
TIES
Introduction
Lower Bounds for GA ={All Distributions}
Lower Bounds for GS ={Symmetric Distributions}
Lower Bounds for GUS ={Unimodal,Symmetric Distributions}
Lower Bounds for GNOR ={Normal Distributions}
4 MORE GENERAL HYPOTHESES AND CONDITIONAL
CALCULATOINS
General Formulation
More General Hypotheses
5 CONCLUSIONS
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
7. 1. Introduction
The paper studies the problem of testing a point null hypothesis,
of interest is the relationship between the P value and conditional
and Bayesian measures of evidence against the null hypothesis
∗ The overall conclusion is that P value can be highly
misleading measures of the evidence provided by le data
against the null hypothesis
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
8. 1. Introduction
Consider the simple situation of observing a random quantity
X having density f (x | θ) , θ ⊂ R 1 , it is desired to test the
null hypothesis H0 : θ = θ0 versus the alternative hypothesis
H1 : θ = θ0 .
p = Prθ=θ0 (T (X ) ≥ T (x))
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
9. 1. lntroduction
Example
Suppose that X = (X1 , ......., Xn ) where the Xi are iid N(θ, σ 2 )
Then the usual test statistic is
√
¯
T (X ) = n | X − θ0 | /σ
¯
where X is the sample mean, and
p = 2(1 − Φ(t))
where Φ is the standard normal cdf and
√
t = T (x) = n | x − θ0 | /σ
¯
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
10. 1. Introduction
We presume that the classical approcach is the report of p,
rather than the report of a Neyman-Perason error probability.
This is because
Most statistician prefer use of P values, feeling it to be impor-
tant to indicate how strong the evidence against H0 is .
The alternative measures of evidence we consider are based on
knowledge of x.
There are several well-known criticisms of testing a point null
hypothesis.
One is the issue of ’statistical’ versus ’practical’ significance,
that one can get a very small p even when | θ − θ0 | is so small
as to make θ equivalent to θ0 for practical purposes.
Another well known is ’Jeffrey’s paradox’
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
11. 1. Introduction
Example
Consider a Bayesian who chooses the prior distribution on θ, which
gives probability 0,5 to H0 and H1 and spreads mass out on H1
according to an N(θ, σ 2 ) density. It will be seen in Section 2 that
the posterior probability, Pr(H0 | x), of H0 is given by
Pr (H0 | x) = (1 + (1 + n)−1/2 exp{t 2 /[2[(1 + 1/n)]})−1
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
12. 1. Introduction
Table 1 : Pr(H0 | x) for Jeffreys-Type Prior)
n
p t 1 5 10 20 50 100 1,000
.10 1.645 .42 .44 .47 .56 .65 .72 .89
.05 .1.960 .35 .33 .37 .42 .52 .60 .82
.01 .2.576 .21 .13 .14 .16 .22 .27 .53
.001 3.291 .086 .026 .024 .026 .034 .045 .124
The conflict between p and Pr(H0 | x) is apparent.
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
13. 1. Introduction
Example
Again consider a Bayesian who gives each hypothesis prior probabil-
ity 0.5, but now suppose that he decides to spread out the mass on
H1 in the symmetric fashion that is as favorable to H1 as possible.
The corresponding values of Pr (H0 | x) are determined in Section 3
and are given in Table 2 for certain values of t.
Table 2 : Pr(H0 | x) for a Prior Biased Towar H1
P value(p) t Pr (H0 | x)
.10 1.645 .340
.05 .1.960 .227
.01 .2.576 .068
.001 3.291 .0088
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
14. 1. Introduction
Example (A Likelihood Analysis)
It is common to perceive the comparative evidence provided by x for
two pssible parameter values, θ1 andθ2 , as being measured by the
likelihood ratio
lx (θ1 : θ2 ) = f (x | θ1 )/f (x | θ2 )
A lower bound on the comparative evidence would be
f (x | θ0 )
lx = inf lx (θ0 : θ) = = exp{−t 2 /2}
θ supθ f (x | θ)
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
15. 1. Introduction
Values of lx for various t are given in Table 3
Table 3 : Bounds on the Comparative Likelihood
Likelihood ratio
P value(p) t lower bound (lx )
.10 1.645 .340
.05 .1.960 .227
.01 .2.576 .068
.001 3.291 .0088
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
16. 2. Posterior probabilities and odds
let 0< π0 < 1 denote the prior probability of H0 , and let π1 = 1−π0
denote the prior probability of H1 , suppose that the mass on H1 is
spread out according to the density g (θ).
Realistic hypothesis: H0 :| θ − θ0 |≤ b
Prior probability π0 would be assigned to {θ :| θ − θ0 |≤ b}
(To a Bayesian, a point null test is typically reasonable only when
the prior distribution is of this form)
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
17. 2. Posterior probabilities and odds
Noting that the marginal density of X is
m(x) = f (x | θ0 )π0 + (1 − π0 )mg (x)
Where
mg (x) = f (x | θ)g (θ)dθ
The posterior probability of H0 is given by
Pr (H0 | x) = f (x | θ0 ) × π0 /m(x)
(1 − π0 ) mg (x) −1
= [1 + × ]
π0 f (x | θ0 )
Also of interest is the posterior odds ratio of H0 to H1 which is
Pr (H0 | x) π0 f (x | θ0 )
= ×
1 − Pr (H0 | x) (1 − π0 ) mg (x)
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
18. 2. Posterior probabilities and odds
The posterior odds ratio of H0 to H1
Pr (H0 | x) π0 f (x | θ0 )
= ×
1 − Pr (H0 | x) (1 − π0 ) mg (x)
Post odds Prior odds Bayes factor Bg (x)
Interest in the Bayes factor centers around the fact that it does
not involve the prior probabilities of the hypotheses and hence is
sometimes interpreted as the actual odds of the hypotheses implied
by the data alone.
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
19. 2. Posterior probabilities and odds
Example (Jeffreys-Lindley paradox)
Suppose that π0 is arbitrary and g is again N(θ0 , σ 2 ).Since a suffi-
cent statistic for θ is X ¯ N(θ0 , σ 2 /n) ,we have that mg (¯) is an
x
N(θ0 , σ 2 (1 + n−1 )) distribution. Thus
Bg (x) = f (x | θ0 )/mg (¯)
x
[2πσ 2 /n]−2 exp{− n (¯ − θ0 )2 /σ 2 }
2 x
= 2 (1 + n−1 )]−1/2 exp{− 1 (¯ − θ 2 )/[σ 2 (1 + n−1 ]}
[2πσ 2 x 0
1
= (1 + n)1/2 exp{− t 2 /(1 + n−1 )}
2
and
Pr (H0 | x) = [1 + (1 − π0 )/(π0 Bg )]−1
(1 − π0 ) 1
= [1 + (1 + n)−1/2 × exp{ t 2 /(1 + n−1 )}]−1
π0 2
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
20. 3. Lower bounds on posterior probabilites
3.1 Introduction
This section will examine some lower bounds on Pr (H0 | x)
when g (θ), the distribution of θ given that H1 is true is
allowed to vary within some class of distribitions G
GA ={all distributions}
GS ={all distributions symmetric about θ0 }
GUS ={all unimodal distribution symmetric about θ0 }
GNOR ={all N(θ0 , τ 2 )distributions, 0≤ τ 2 < ∞}
Even though these G’s are supposed to consist only of distribution
on {θ | θ = θ0 }, il will be convenient to allow them to include
distributions with mass at θ0 , so the lower bounds we compute are
always attained.
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
21. 3. Lower bounds on posterior probabilites
3.1 Introduction
Letting
Pr (H0 | x, G ) = inf Pr (H0 | x)
g ∈G
and
B(xmG ) = inf Bg (x)
g ∈G
we see immediately form formulas before that
B(x, G ) = f (x | θ0 )/ sup mg (x)
g ∈G
and
(1 − π0 ) 1
Pr (H0 | xmG ) = [1 + × ]−1
π0 B(x, G )
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
22. 3. Lower bounds on posterior probabilites
3.2 Lower bounds for GA ={All distributions}
Theorem
Suppose that a maximum likelihood estimate of θ0 , exists for the
observed x. Then
ˆ
B(x, GA ) = f (x | θ0 )/f (x | θ(x))
and
ˆ
(1 − π0 ) f (x | θ(x)) −1
Pr (H0 | x, GA ) = [1 + × ]
π0 f (x | θ0 )
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
23. 3. Lower bounds on posterior probabilites
3.2 Lower bounds for GA ={All distributions}
Example
In this situation,we have
2
B(x, GA ) = e −t /2
¯
and
(1 − π0 ) t 2 /2 −1
Pr (H0 | x, GA ) = [1 + e ]
π0
For servral choices of t, Table 4 gives the corresponding two-sided
P values,p, and the values of Pr (H0 | x, GA ),with π0 = 0.5.
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
24. 3. Lower bounds on posterior probabilites
3.2 Lower bounds for GA ={All distributions}
For servral choices of t, Table 4 gives the corresponding
two-sided P values,p, and the values of Pr (H0 | x, GA ),with
π0 = 0.5.
Table 4 : Comparison of P values and Pr (H0 | x, GA ) when π0 = 0.5
P value(p) t Pr (H0 | x, GA ) Pr (H0 | x, GA )/(pt)
.10 1.645 .205 1.25
.05 .1.960 .128 1.30
.01 .2.576 .035 1.36
.001 3.291 .0044 1.35
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
25. 3. Lower bounds on posterior probabilites
3.2 Lower bounds for GA ={All distributions}
Theorem
For t > 1.68 and π0 = 0.5 in Example 1,
Pr (H0 | x, GA )/pt > π/2 1.253
Furthermore
lim Pr (H0 | x, GA )/pt = π/2
t→∞
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
26. 3. Lower bounds on posterior probabilites
3.3 Lower bounds for GS ={Symmetric distributions}
There is a large gap between Pr (H0 | x, GA )andPr (H0 | x) for
the Jeffreys-type single prior analysis.This reinforces the
suspicion that using GA unduly biases the conclusion against
H0 and suggests use of more reasonable classes of priors.
Theorem
sup mg (x) = sup mg (x),
g ∈G2ps g ∈GS
so
B(x, G2PS ) = B(x, GS )
and
Pr (H0 | x, G2ps ) = Pr (H0 | x, GS )
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
27. 3. Lower bounds on posterior probabilites
3.3 Lower bounds for GS ={Symmetric distributions}
Example
If t ≤ 1, a calculus argument show that the symmetric two point
distribution that strictly maximizes mg (x) is the degenerate ”two-
point”distribution putting all mass at θ0 . Thus B(x, GS ) = 1 and
Pr (H0 | x, GS ) = π0 for t ≤ 1.
If t ≥ 1 , then mg (x) is maximized by a nondegenerate element
of G2ps . For moderately large t, the maximum value of mg (x) for
g∈ G2ps is very well approximated by taking g to be the two-point
ˆ ˆ
distribution putting equal mass at θ(x) and at 2θ − θ(x). so
ϕ(t)
B(x, GS ) 2 exp {−0.5t 2 }
0.5ϕ(0) + 0.5ϕ(2t)
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
28. 3. Lower bounds on posterior probabilites
3.3 Lower bounds for GS ={Symmetric distributions}
Example
For t ≤ 1.645, the first approximation is accurate to within 1 in the
fourth significant digit and the second approximation to within 2 in
the third significant digit.
Table 5 gives the value of Pr (H0 | x, Gs ) of several choices of t.
Table 5 : Comparison of P values and Pr (H0 | x, GS ) when π0 = 0.5
P value(p) t Pr (H0 | x, GS ) Pr (H0 | x, GS )/(pt)
.10 1.645 .340 2.07
.05 .1.960 .227 2.31
.01 .2.576 .068 2.62
.001 3.291 .0088 2.68
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
29. 3. Lower bounds on posterior probabilites
3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions}
Minimizing Pr (H0 | x) over all symmetric priors still involves
considerable bias against H0 . A further ’objective’ restriction,
which would seem reasonable to many, is to require the prior
to be unimodal, or non-increasing in | θ − θ0 | .
Theorem
sup mg (x) = sup mg (x),
g ∈Gus g ∈US
with US ={all symmetric uniform distributions}
so B(x, GUS ) = B(x, US ) and Pr (H0 | x, GUS ) = Pr (H0 | x, US )
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
30. 3. Lower bounds on posterior probabilites
3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions}
Theorem
If t ≤ 1 in example 1, then B(x, GUS ) = 1 and Pr (H0 | x.GUS ) = π0 .
Ift > 1 then
2ϕ(t)
B(x.GUS ) =
ϕ(K + t) + ϕ(K − t)
and
(1 − π0 ) ϕ(K + t) + ϕ(K = t) −1
Pr (H0 | x, GUS ) = [1 + × ]
π0 2ϕ(t)
where K > 0
Figures 1 and 2 give values of K and B for various val-
ues of t in this problem
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
31. 3. Lower bounds on posterior probabilites
3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions}
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
32. 3. Lower bounds on posterior probabilites
3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions}
Table 6 gives Pr (H0 | x, GUS ) for some specific important
values of t and π0 = 0.5
Table 6 : Comparison of P values and Pr (H0 | x, GUS ) when π0 = 0.5
P value(p) t Pr (H0 | x, GUS ) Pr (H0 | x, GUS )/(pt)
.10 1.645 .390 1.44
.05 .1.960 .290 1.51
.01 .2.576 .109 1.64
.001 3.291 .018 1.66
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
33. 3. Lower bounds on posterior probabilites
3.5 Lower bounds for GNOR ={Normal distributions}
We have seene that minimizing Pr (H | x)overg ∈ GUS is the same
as minimizing over g ∈ US . Althought using US is much more
reasonable than using GA ,there is still some residual bias against
H0 involved in using US .
Theorem
If t ≤ 1in Example 1, then B(x, GNOR ) = 1 and Pr (H0 | x, GNOR ) =
π0 . If t > 1, then
√ 2
B(x, GNOR ) = ete −t /2
and
(1 − π0 ) exp{t 2 /2} −1
Pr (H0 | x, GNOR ) = [1 + × √ ]
π0 et
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
34. 3. Lower bounds on posterior probabilites
3.5 Lower bounds for GNOR ={Normal distributions}
Table 7 gives Pr (H0 | x, GNOR ) for servral values of t
Table 7 : Comparison of P values and Pr (H0 | x, GNOR ) when π0 = 0.5
P value(p) t Pr (H0 | x, GNOR ) Pr (H0 | x, GNOR )/(pt)
.10 1.645 .412 1.52
.05 .1.960 .321 1.67
.01 .2.576 .133 2.01
.001 3.291 .0235 2.18
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
35. 4. More general hypotheses and conditional calculations
4.1 General formulation
Consider the Bayesian calculation of Pr (H0 | A) , where H0 is
of the form H0 : θ ∈ Θ0 and A is the set in which x is known
to reside. Then letting π0 and π1 denote the prior
probabilities of H0 and H1 and g1 and g2 as the densities on
Θ0 and Θ1 . Then we have :
1 − π0 mg 1 (A) −1
Pr (H0 | A) = [1 + × ]
π+0 mg 0 (A)
where
mg 1 (A) = Prθ (A)gi (θ)dθ
Θ0
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
36. 4. More general hypotheses and conditional calculations
4.1 General formulation
For the general formulation, one can determine lower bounds
on Pr (H0 | A) by choosing sets G0 and G1 of g0 and g1 ,
respectively, calculating
B(A, G0 , G1 ) = inf mg0 (A)/ sup mg 1 (A)
g0 ∈G0 g1 ∈G1
(1−π0 ) 1 −1
and defining Pr (H0 | A, G0 , G1 ) = [1 + π0 × B(A,G0 ,G1 ) ]
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
37. 4. More general hypotheses and conditional calculations
4.2 More general hypotheses
Assume in this section that A= {x}. The lower bounds can
be applied to a variety of generalizations of point null
hypotheses. If Θ0 is a small set about θ0 , the negeral lower
bounds turn out to be essentially equivalent to the point null
lower bounds.
In example 1, suppose that the hypotheses were H0 : θ ∈ (θ0 −
√
b, θ0 + b) and H1 : θ ∈ (θ0 − b, θ0 + b) If | t − nb/σ |≥ 1
/
and G0 = G1 = GS , then B(x, G0 , G1 ) and Pr (H0 | x, G0 , G1 ) are
exactly the same as B and P for testing the point null.
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
38. 5. Conclusion
B(x, GUS ) and the P value calculated at (t − 1)+ in instead of t,
this last will be called the P value of (t − 1)+ Figure shows that
this comparative likelihood is close to the P value that would be
obtained if we replaced t by (t − 1)+ . The implication is that :
t = 1 means only mild evidence against H0 , t = 2 means
significant evidence against H0 , and t = 3 means highly evidence
against H0 , and t = 4 means over whelming evidence against H0 ,
should at least replaced by the rule of thumb, that is :
t = 1 means no evidence against H0 , t = 2 means only mild
evidence against H0 , and t = 3 means significant evidence against
H0 , and t = 4 means highly significant evidence against H0 .
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
39. 5. Conclusion
t = 1 means no evidence against H0 , t = 2 means only mild
evidence against H0 , and t = 3 means significant evidence against
H0 , and t = 4 means highly significant evidence against H0 .
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
40. References
EDWARDS, W, LINDMAN, H, and SAVAGE, L.J (1963).
Bayesian statistical inference for psychological research. Psycho-
logical Review 70, 193-242
Jayanta Ghosha, Sumitra Purkayastha and Tapas Samanta
Role of P-values and other Measures of Evidence in Bayesian Anal-
ysis. Handbook of Statistics, Vol. 25
James O. Berger Statistical Decision Theory and Bayesian Analy-
sis. Springer Series in Statistics
JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val