Naive Bayes Classifier Assumes That the Presence (Or Absence) of a Particular Feature of a Class Is Unrelated to the Presence (Or Absence) of Any Other Feature
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Small-Sample Properties of Tests for Heteroscedasticity in the Conditional Logit Model
HEDG Working Paper 06/04 Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole May 2006 ISSN 1751-1976 york.ac.uk/res/herc/hedgwp Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole National Primary Care Research and Development Centre Centre for Health Economics University of York May 16, 2006 Abstract This paper compares the small-sample properties of several asymp- totically equivalent tests for heteroscedasticity in the conditional logit model. While no test outperforms the others in all of the experiments conducted, the likelihood ratio test and a particular variety of the Wald test are found to have good properties in moderate samples as well as being relatively powerful. Keywords: conditional logit, heteroscedasticity JEL classi…cation: C25 National Primary Care Research and Development Centre, Centre for Health Eco- nomics, Alcuin ’A’Block, University of York, York YO10 5DD, UK. Tel.: +44 1904 321404; fax: +44 1904 321402. E-mail address: [email protected]. NPCRDC receives funding from the Department of Health. The views expressed are not necessarily those of the funders. 1 1 Introduction In most applications of the conditional logit model the error term is assumed to be homoscedastic. Recently, however, there has been a growing interest in testing the homoscedasticity assumption in applied work (Hensher et al., 1999; DeShazo and Fermo, 2002). This is partly due to the well-known result that heteroscedasticity causes the coe¢ cient estimates in discrete choice models to be inconsistent (Yatchew and Griliches, 1985), but also re‡ects a behavioural interest in factors in‡uencing the variance of the latent variables in the model (Louviere, 2001; Louviere et al., 2002). -
Statistical Modelling
Statistical Modelling Dave Woods and Antony Overstall (Chapters 1–2 closely based on original notes by Anthony Davison and Jon Forster) c 2016 StatisticalModelling ............................... ......................... 0 1. Model Selection 1 Overview............................................ .................. 2 Basic Ideas 3 Whymodel?.......................................... .................. 4 Criteriaformodelselection............................ ...................... 5 Motivation......................................... .................... 6 Setting ............................................ ................... 9 Logisticregression................................... .................... 10 Nodalinvolvement................................... .................... 11 Loglikelihood...................................... .................... 14 Wrongmodel.......................................... ................ 15 Out-of-sampleprediction ............................. ..................... 17 Informationcriteria................................... ................... 18 Nodalinvolvement................................... .................... 20 Theoreticalaspects.................................. .................... 21 PropertiesofAIC,NIC,BIC............................... ................. 22 Linear Model 23 Variableselection ................................... .................... 24 Stepwisemethods.................................... ................... 25 Nuclearpowerstationdata............................ .................... -
Multinomial Logistic Regression
26-Osborne (Best)-45409.qxd 10/9/2007 5:52 PM Page 390 26 MULTINOMIAL LOGISTIC REGRESSION CAROLYN J. ANDERSON LESLIE RUTKOWSKI hapter 24 presented logistic regression Many of the concepts used in binary logistic models for dichotomous response vari- regression, such as the interpretation of parame- C ables; however, many discrete response ters in terms of odds ratios and modeling prob- variables have three or more categories (e.g., abilities, carry over to multicategory logistic political view, candidate voted for in an elec- regression models; however, two major modifica- tion, preferred mode of transportation, or tions are needed to deal with multiple categories response options on survey items). Multicate- of the response variable. One difference is that gory response variables are found in a wide with three or more levels of the response variable, range of experiments and studies in a variety of there are multiple ways to dichotomize the different fields. A detailed example presented in response variable. If J equals the number of cate- this chapter uses data from 600 students from gories of the response variable, then J(J – 1)/2 dif- the High School and Beyond study (Tatsuoka & ferent ways exist to dichotomize the categories. Lohnes, 1988) to look at the differences among In the High School and Beyond study, the three high school students who attended academic, program types can be dichotomized into pairs general, or vocational programs. The students’ of programs (i.e., academic and general, voca- socioeconomic status (ordinal), achievement tional and general, and academic and vocational). test scores (numerical), and type of school How the response variable is dichotomized (nominal) are all examined as possible explana- depends, in part, on the nature of the variable. -
Logit and Ordered Logit Regression (Ver
Getting Started in Logit and Ordered Logit Regression (ver. 3.1 beta) Oscar Torres-Reyna Data Consultant [email protected] http://dss.princeton.edu/training/ PU/DSS/OTR Logit model • Use logit models whenever your dependent variable is binary (also called dummy) which takes values 0 or 1. • Logit regression is a nonlinear regression model that forces the output (predicted values) to be either 0 or 1. • Logit models estimate the probability of your dependent variable to be 1 (Y=1). This is the probability that some event happens. PU/DSS/OTR Logit odelm From Stock & Watson, key concept 9.3. The logit model is: Pr(YXXXFXX 1 | 1= , 2 ,...=k β ) +0 β ( 1 +2 β 1 +βKKX 2 + ... ) 1 Pr(YXXX 1= | 1 , 2k = ,... ) 1−+(eβ0 + βXX 1 1 + β 2 2 + ...βKKX + ) 1 Pr(YXXX 1= | 1 , 2= ,... ) k ⎛ 1 ⎞ 1+ ⎜ ⎟ (⎝ eβ+0 βXX 1 1 + β 2 2 + ...βKK +X ⎠ ) Logit nd probita models are basically the same, the difference is in the distribution: • Logit – Cumulative standard logistic distribution (F) • Probit – Cumulative standard normal distribution (Φ) Both models provide similar results. PU/DSS/OTR It tests whether the combined effect, of all the variables in the model, is different from zero. If, for example, < 0.05 then the model have some relevant explanatory power, which does not mean it is well specified or at all correct. Logit: predicted probabilities After running the model: logit y_bin x1 x2 x3 x4 x5 x6 x7 Type predict y_bin_hat /*These are the predicted probabilities of Y=1 */ Here are the estimations for the first five cases, type: 1 x2 x3 x4 x5 x6 x7 y_bin_hatbrowse y_bin x Predicted probabilities To estimate the probability of Y=1 for the first row, replace the values of X into the logit regression equation. -
On Discriminative Bayesian Network Classifiers and Logistic Regression
On Discriminative Bayesian Network Classifiers and Logistic Regression Teemu Roos and Hannes Wettig Complex Systems Computation Group, Helsinki Institute for Information Technology, P.O. Box 9800, FI-02015 HUT, Finland Peter Gr¨unwald Centrum voor Wiskunde en Informatica, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlands Petri Myllym¨aki and Henry Tirri Complex Systems Computation Group, Helsinki Institute for Information Technology, P.O. Box 9800, FI-02015 HUT, Finland Abstract. Discriminative learning of the parameters in the naive Bayes model is known to be equivalent to a logistic regression problem. Here we show that the same fact holds for much more general Bayesian network models, as long as the corresponding network structure satisfies a certain graph-theoretic property. The property holds for naive Bayes but also for more complex structures such as tree-augmented naive Bayes (TAN) as well as for mixed diagnostic-discriminative structures. Our results imply that for networks satisfying our property, the condi- tional likelihood cannot have local maxima so that the global maximum can be found by simple local optimization methods. We also show that if this property does not hold, then in general the conditional likelihood can have local, non-global maxima. We illustrate our theoretical results by empirical experiments with local optimization in a conditional naive Bayes model. Furthermore, we provide a heuristic strategy for pruning the number of parameters and relevant features in such models. For many data sets, we obtain good results with heavily pruned submodels containing many fewer parameters than the original naive Bayes model. Keywords: Bayesian classifiers, Bayesian networks, discriminative learning, logistic regression 1. -
Introduction to Bayesian Estimation
Introduction to Bayesian Estimation March 2, 2016 The Plan 1. What is Bayesian statistics? How is it different from frequentist methods? 2. 4 Bayesian Principles 3. Overview of main concepts in Bayesian analysis Main reading: Ch.1 in Gary Koop's Bayesian Econometrics Additional material: Christian P. Robert's The Bayesian Choice: From decision theoretic foundations to Computational Implementation Gelman, Carlin, Stern, Dunson, Wehtari and Rubin's Bayesian Data Analysis Probability and statistics; What's the difference? Probability is a branch of mathematics I There is little disagreement about whether the theorems follow from the axioms Statistics is an inversion problem: What is a good probabilistic description of the world, given the observed outcomes? I There is some disagreement about how we interpret data/observations and how we make inference about unobservable parameters Why probabilistic models? Is the world characterized by randomness? I Is the weather random? I Is a coin flip random? I ECB interest rates? It is difficult to say with certainty whether something is \truly" random. Two schools of statistics What is the meaning of probability, randomness and uncertainty? Two main schools of thought: I The classical (or frequentist) view is that probability corresponds to the frequency of occurrence in repeated experiments I The Bayesian view is that probabilities are statements about our state of knowledge, i.e. a subjective view. The difference has implications for how we interpret estimated statistical models and there is no general -
Lecture 5: Naive Bayes Classifier 1 Introduction
CSE517A Machine Learning Spring 2020 Lecture 5: Naive Bayes Classifier Instructor: Marion Neumann Scribe: Jingyu Xin Reading: fcml 5.2.1 (Bayes Classifier, Naive Bayes, Classifying Text, and Smoothing) Application Sentiment analysis aims at discovering people's opinions, emo- tions, feelings about a subject matter, product, or service from text. Take some time to answer the following warm-up questions: (1) Is this a regression or classification problem? (2) What are the features and how would you represent them? (3) What is the prediction task? (4) How well do you think a linear model will perform? 1 Introduction Thought: Can we model p(y j x) without model assumptions, e.g. Gaussian/Bernoulli distribution etc.? Idea: Estimate p(y j x) from the data directly, then use the Bayes classifier. Let y be discrete, Pn i=1 I(xi = x \ yi = y) p(y j x) = Pn (1) i=1 I(xi = x) where the numerator counts all examples with input x that have label y and the denominator counts all examples with input x. Figure 1: Illustration of estimatingp ^(y j x) From Figure ??, it is clear that, using the MLE method, we can estimatep ^(y j x) as jCj p^(y j x) = (2) jBj But there is a big problem with this method: The MLE estimate is only good if there are many training vectors with the same identical features as x. 1 2 This never happens for high-dimensional or continuous feature spaces. Solution: Bayes Rule. p(x j y) p(y) p(y j x) = (3) p(x) Let's estimate p(x j y) and p(y) instead! 2 Naive Bayes We have a discrete label space C that can either be binary f+1; −1g or multi-class f1; :::; Kg. -
Hybrid of Naive Bayes and Gaussian Naive Bayes for Classification: a Map Reduce Approach
International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-6S3, April 2019 Hybrid of Naive Bayes and Gaussian Naive Bayes for Classification: A Map Reduce Approach Shikha Agarwal, Balmukumd Jha, Tisu Kumar, Manish Kumar, Prabhat Ranjan Abstract: Naive Bayes classifier is well known machine model could be used without accepting Bayesian probability learning algorithm which has shown virtues in many fields. In or using any Bayesian methods. Despite their naive design this work big data analysis platforms like Hadoop distributed and simple assumptions, Naive Bayes classifiers have computing and map reduce programming is used with Naive worked quite well in many complex situations. An analysis Bayes and Gaussian Naive Bayes for classification. Naive Bayes is manily popular for classification of discrete data sets while of the Bayesian classification problem showed that there are Gaussian is used to classify data that has continuous attributes. sound theoretical reasons behind the apparently implausible Experimental results show that Hybrid of Naive Bayes and efficacy of types of classifiers[5]. A comprehensive Gaussian Naive Bayes MapReduce model shows the better comparison with other classification algorithms in 2006 performance in terms of classification accuracy on adult data set showed that Bayes classification is outperformed by other which has many continuous attributes. approaches, such as boosted trees or random forests [6]. An Index Terms: Naive Bayes, Gaussian Naive Bayes, Map Reduce, Classification. advantage of Naive Bayes is that it only requires a small number of training data to estimate the parameters necessary I. INTRODUCTION for classification. The fundamental property of Naive Bayes is that it works on discrete value. -
Adjusted Probability Naive Bayesian Induction
Adjusted Probability Naive Bayesian Induction 1 2 Geo rey I. Webb and Michael J. Pazzani 1 Scho ol of Computing and Mathematics Deakin University Geelong, Vic, 3217, Australia. 2 Department of Information and Computer Science University of California, Irvine Irvine, Ca, 92717, USA. Abstract. NaiveBayesian classi ers utilise a simple mathematical mo del for induction. While it is known that the assumptions on which this mo del is based are frequently violated, the predictive accuracy obtained in discriminate classi cation tasks is surprisingly comp etitive in compar- ison to more complex induction techniques. Adjusted probability naive Bayesian induction adds a simple extension to the naiveBayesian clas- si er. A numeric weight is inferred for each class. During discriminate classi cation, the naiveBayesian probability of a class is multiplied by its weight to obtain an adjusted value. The use of this adjusted value in place of the naiveBayesian probabilityisshown to signi cantly improve predictive accuracy. 1 Intro duction The naiveBayesian classi er Duda & Hart, 1973 provides a simple approachto discriminate classi cation learning that has demonstrated comp etitive predictive accuracy on a range of learning tasks Clark & Niblett, 1989; Langley,P., Iba, W., & Thompson, 1992. The naiveBayesian classi er is also attractive as it has an explicit and sound theoretical basis which guarantees optimal induction given a set of explicit assumptions. There is a drawback, however, in that it is known that some of these assumptions will b e violated in many induction scenarios. In particular, one key assumption that is frequently violated is that the attributes are indep endent with resp ect to the class variable. -
Locally Differentially Private Naive Bayes Classification
Locally Differentially Private Naive Bayes Classification EMRE YILMAZ, Case Western Reserve University MOHAMMAD AL-RUBAIE, University of South Florida J. MORRIS CHANG, University of South Florida In machine learning, classification models need to be trained in order to predict class labels. When the training data contains personal information about individuals, collecting training data becomes difficult due to privacy concerns. Local differential privacy isa definition to measure the individual privacy when there is no trusted data curator. Individuals interact with an untrusteddata aggregator who obtains statistical information about the population without learning personal data. In order to train a Naive Bayes classifier in an untrusted setting, we propose to use methods satisfying local differential privacy. Individuals send theirperturbed inputs that keep the relationship between the feature values and class labels. The data aggregator estimates all probabilities needed by the Naive Bayes classifier. Then, new instances can be classified based on the estimated probabilities. We propose solutions forboth discrete and continuous data. In order to eliminate high amount of noise and decrease communication cost in multi-dimensional data, we propose utilizing dimensionality reduction techniques which can be applied by individuals before perturbing their inputs. Our experimental results show that the accuracy of the Naive Bayes classifier is maintained even when the individual privacy is guaranteed under local differential privacy, and that using dimensionality reduction enhances the accuracy. CCS Concepts: • Security and privacy → Privacy-preserving protocols; • Computing methodologies → Supervised learning by classification; Dimensionality reduction and manifold learning. Additional Key Words and Phrases: Local Differential Privacy, Naive Bayes, Classification, Dimensionality Reduction 1 INTRODUCTION Predictive analytics is the process of making prediction about future events by analyzing the current data using statistical techniques. -
Improving Feature Selection Algorithms Using Normalised Feature Histograms
Page 1 of 10 Improving feature selection algorithms using normalised feature histograms A. P. James and A. K. Maan The proposed feature selection method builds a histogram of the most stable features from random subsets of training set and ranks the features based on a classifier based cross validation. This approach reduces the instability of features obtained by conventional feature selection methods that occur with variation in training data and selection criteria. Classification results on 4 microarray and 3 image datasets using 3 major feature selection criteria and a naive bayes classifier show considerable improvement over benchmark results. Introduction : Relevance of features selected by conventional feature selection method is tightly coupled with the notion of inter-feature redundancy within the patterns [1-3]. This essentially results in design of data-driven methods that are designed for meeting criteria of redundancy using various measures of inter-feature correlations [4-5]. These measures behave differently to different types of data, and to different levels of natural variability. This introduces instability in selection of features with change in number of samples in the training set and quality of features from one database to another. The use of large number of statistically distinct training samples can increase the quality of the selected features, although it can be practically unrealistic due to increase in computational complexity and unavailability of sufficient training samples. As an example, in realistic high dimensional databases such as gene expression databases of a rare cancer, there is a practical limitation on availability of large number of training samples. In such cases, instability that occurs with selecting features from insufficient training data would lead to a wrong conclusion that genes responsible for a cancer be strongly subject dependent and not general for a cancer. -
Lecture 9: Logit/Probit Prof
Lecture 9: Logit/Probit Prof. Sharyn O’Halloran Sustainable Development U9611 Econometrics II Review of Linear Estimation So far, we know how to handle linear estimation models of the type: Y = β0 + β1*X1 + β2*X2 + … + ε ≡ Xβ+ ε Sometimes we had to transform or add variables to get the equation to be linear: Taking logs of Y and/or the X’s Adding squared terms Adding interactions Then we can run our estimation, do model checking, visualize results, etc. Nonlinear Estimation In all these models Y, the dependent variable, was continuous. Independent variables could be dichotomous (dummy variables), but not the dependent var. This week we’ll start our exploration of non- linear estimation with dichotomous Y vars. These arise in many social science problems Legislator Votes: Aye/Nay Regime Type: Autocratic/Democratic Involved in an Armed Conflict: Yes/No Link Functions Before plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric model We have one example of this already: logs Start with Y = Xβ+ ε Then change to log(Y) ≡ Y′ = Xβ+ ε Run this like a regular OLS equation Then you have to “back out” the results Link Functions Before plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric model We have one example of this already: logs Start with Y = Xβ+ ε Different β’s here Then change to log(Y) ≡ Y′ = Xβ + ε Run this like a regular OLS equation Then you have to “back out” the results Link Functions If the coefficient on some particular X is β, then a 1 unit ∆X Æ β⋅∆(Y′) = β⋅∆[log(Y))] = eβ ⋅∆(Y) Since for small values of β, eβ ≈ 1+β , this is almost the same as saying a β% increase in Y (This is why you should use natural log transformations rather than base-10 logs) In general, a link function is some F(⋅) s.t.