Evaluating Classifiers. Lecture 2 Instructor: Max Welling
|
|
- Samuel Ball
- 6 years ago
- Views:
Transcription
1 Evaluating Classifiers Lecture 2 Instructor: Max Welling
2 Evaluation of Results How do you report classification error? How certain are you about the error you claim? How do you compare two algorithms? How certain are you if you state one algorithm performs better than another?
3 Evaluation Given: Hypothesis h(x): X C, in hypothesis space H, mapping attributes x to classes c=[1,2,3,...c] A data-sample S(n) of size n. Questions: What is the error of h on unseen data? If we have two competing hypotheses, which one is better on unseen data? How do we compare two learning algorithms in the face of limited data? How certain are we about our answers?
4 Sample and True Error We can define two errors: 1) Error(h S) is the error on the sample S: n 1 error( h S ) [ h( xi) yi] n i 1 2) Error(h P) is the true error on the unseen data sampled from the distribution P(x): error ( h P ) dx P ( x ) [ h( x ) f ( x )] where f(x) is the true hypothesis.
5 Binomial Distributions Assume you toss a coin n times. And it has probability p of coming heads (which we will call success) What is the probability distribution governing the number of heads in n trials? Answer: the Binomial distribution. n! r p(# heads r p, n) p (1 p) r!( n r )! n r
6 Distribution over Errors Consider some hypothesis h(x) Draw n samples Xk~P(X). Do this k times. Compute e1=n*error(h X1), e2=n*error(h X2),...,ek=n*error(h Xk). {e1,...,ek} are samples from a Binomial distribution! Why? imagine a magic coin, where God secretly determines the probability of heads by the following procedure. First He takes some random hypothesis h. Then, He draws x~p(x) and observes if h(x) correctly predicts the label correctly. If it does, he makes sure the coin lands heads up... You have a single sample S, for which you observe e(s) errors. What would be a reasonable estimate for Error(h P) you think?
7 Binomial Moments mean ( r ) E [ r n, p] np var r E r E r np p 2 ( ) [( [ ]) ] (1 ) var mean If we match the mean, np, with the observed value n*error(h S) we find: E [ error ( h P )] E [ r / n] p error ( h S ) If we match the variance we can obtain an estimate of the width: var[ error ( h P )] var[ r / n] error( h S ) (1 error( h S )) n
8 Confidence Intervals We would like to state: With N% confidence we believe that error(h P) is contained in the interval: error ( h P ) error ( h S ) zn error ( h S ) (1 error ( h S )) n 80% z Normal(0,1) 1.28 In principle z N is hard to compute exactly, but for np(1-p)>5 or n>30 it is safe to approximate a Binomial by a Gaussian for which we can easily compute z-values.
9 Bias-Variance The estimator is unbiased if E [ error ( h X )] p Imagine again you have infinitely many sample sets X1,X2,.. of size n. Use these to compute estimates E1,E2,... of p where Ei=error(h Xi) If the average of E1,E2,.. converges to p, then error(h X) is an unbiased estimator. Two unbiased estimators can still differ in their variance (efficiency). Which one do you prefer? p Eav
10 Flow of Thought Determine the property you want to know about the future data (e.g. error(h P)) Find an unbiased estimator E for this quantity based on observing data X (e.g. error(h X)) Determine the distribution P(E) of E under the assumption you have infinitely many sample sets X1,X2,...of some size n. (e.g. p(e)=binomial(p,n), p=error(h P)) Estimate the parameters of P(E) from an actual data sample S (e.g. p=error(h S)) Compute mean and variance of P(E) and pray P(E) it is close to a Normal distribution. (sums of random variables converge to normal distributions central limit theorem) State you confidence interval as: with confidence N% error(h P) is contained in the interval Y mean z N var
11 Assumptions We only consider discrete valued hypotheses (i.e. classes) Training data and test data are drawn IID from the same distribution P(x). (IID: independently & identically distributed) The hypothesis must be chosen independently from the data sample S! When you obtain a hypothesis from a learning algorithm, split the data into a training set and a testing set. Find the hypothesis using the training set and estimate error on the testing set.
12 Comparing Hypotheses Assume we like to compare 2 hypothesis h1 and h2, which we have tested on two independent samples S1 and S2 of size n1 and n2. I.e. we are interested in the quantity: d error ( h1 P ) error ( h2 P )? Define estimator for d: dˆ error ( h1 X 1) error ( h2 X 2) with X1,X2 sample sets of size n1,n2. Since error(h1 S1) and error(h2 S2) are both approximately Normal their difference is approximately Normal with: mean d error ( h1 S1) error ( h2 S 2) var error ( h1 S1)(1 error ( h1 S1)) error ( h2 S 2)(1 error ( h2 S 2)) n1 n2 Hence, with N% confidence we believe that d is contained in the interval: d mean z N var
13 Paired Tests Consider the following data: error(h1 s1)=0.1 error(h2 s1)=0.11 error(h1 s2)=0.2 error(h2 s2)=0.21 error(h1 s3)=0.66 error(h2 s3)=0.67 error(h1 s4)=0.45 error(h2 s4)=0.46 and so on. We have var(error(h1)) = large, var(error(h2)) = large. The total variance of error(h1)-error(h2) is their sum. However, h1 is consistently better than h2. We ignored the fact that we compare on the same data. We want a different estimator that compares data one by one. You can use a paired t-test (e.g. in matlab) to see if the two errors are significantly different, or if one error is significantly larger than the other.
14 Paired t-test Chunk the data up in subsets T1,...,Tk with Ti >30 On each subset compute the error and compute: i error ( h1 Ti ) error ( h2 Ti ) Now compute: 1 k k i i 1 k 1 s ( ) ( i ) kk ( 1) i 1 2 State: With N% confidence the difference in error between h1 and h2 is: tnk, 1 s( ) t is the t-statistic which is related to the student-t distribution (table 5.6).
15 Comparing Learning Algorithms In general it is a really bad idea to estimate error rates on the same data on which a learning algorithm is trained. WHY? So just as in x-validation, we split the data into k subsets: S {T1,T2,...Tk}. Train both learning algorithm 1 (L1) and learning algorithm 2 (L2) on the complement of each subset: {S-T1,S-T2,...) to produce hypotheses {L1(S-Ti), L2(S-Ti)} for all i. Compute for all i : error ( L ( S T ) T ) error ( L ( S T ) T ) i 1 i i 2 i i Note: we train on S-Ti, but test on Ti. As in the last slide perform a paired t-test on these differences to compute an estimate and a confidence interval for the relative error of the hypothesis produced by L1 and L2.
16 Evaluation: ROC curves moving threshold class 0 (negatives) class 1 (positives) TP = true positive rate = # positives classified as positive divided by # positives FP = false positive rate = # negatives classified as positives divided by # negatives TN = true negative rate = # negatives classified as negatives divided by # negatives Identify a threshold in your classifier that you can shift. Plot ROC curve while you shift that parameter. FN = false negatives = # positives classified as negative divided by # positives
17 Conclusion Never (ever) draw error-curves without confidence intervals (The second most important sentence of this course)
Evaluating Hypotheses
Evaluating Hypotheses IEEE Expert, October 1996 1 Evaluating Hypotheses Sample error, true error Confidence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution,
More informationEstimating the accuracy of a hypothesis Setting. Assume a binary classification setting
Estimating the accuracy of a hypothesis Setting Assume a binary classification setting Assume input/output pairs (x, y) are sampled from an unknown probability distribution D = p(x, y) Train a binary classifier
More information[Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4]
Evaluating Hypotheses [Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4] Sample error, true error Condence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution,
More informationStephen Scott.
1 / 35 (Adapted from Ethem Alpaydin and Tom Mitchell) sscott@cse.unl.edu In Homework 1, you are (supposedly) 1 Choosing a data set 2 Extracting a test set of size > 30 3 Building a tree on the training
More informationHypothesis Evaluation
Hypothesis Evaluation Machine Learning Hamid Beigy Sharif University of Technology Fall 1395 Hamid Beigy (Sharif University of Technology) Hypothesis Evaluation Fall 1395 1 / 31 Table of contents 1 Introduction
More informationPerformance Evaluation and Hypothesis Testing
Performance Evaluation and Hypothesis Testing 1 Motivation Evaluating the performance of learning systems is important because: Learning systems are usually designed to predict the class of future unlabeled
More informationPerformance Evaluation and Comparison
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Cross Validation and Resampling 3 Interval Estimation
More informationRegularization. CSCE 970 Lecture 3: Regularization. Stephen Scott and Vinod Variyam. Introduction. Outline
Other Measures 1 / 52 sscott@cse.unl.edu learning can generally be distilled to an optimization problem Choose a classifier (function, hypothesis) from a set of functions that minimizes an objective function
More informationEvaluation. Andrea Passerini Machine Learning. Evaluation
Andrea Passerini passerini@disi.unitn.it Machine Learning Basic concepts requires to define performance measures to be optimized Performance of learning algorithms cannot be evaluated on entire domain
More informationHow do we compare the relative performance among competing models?
How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model
More informationComputational Learning Theory
CS 446 Machine Learning Fall 2016 OCT 11, 2016 Computational Learning Theory Professor: Dan Roth Scribe: Ben Zhou, C. Cervantes 1 PAC Learning We want to develop a theory to relate the probability of successful
More informationLecture Slides for INTRODUCTION TO. Machine Learning. ETHEM ALPAYDIN The MIT Press,
Lecture Slides for INTRODUCTION TO Machine Learning ETHEM ALPAYDIN The MIT Press, 2004 alpaydin@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/i2ml CHAPTER 14: Assessing and Comparing Classification Algorithms
More informationEmpirical Evaluation (Ch 5)
Empirical Evaluation (Ch 5) how accurate is a hypothesis/model/dec.tree? given 2 hypotheses, which is better? accuracy on training set is biased error: error train (h) = #misclassifications/ S train error
More informationEvaluation requires to define performance measures to be optimized
Evaluation Basic concepts Evaluation requires to define performance measures to be optimized Performance of learning algorithms cannot be evaluated on entire domain (generalization error) approximation
More information6.867 Machine Learning
6.867 Machine Learning Problem set 1 Due Thursday, September 19, in class What and how to turn in? Turn in short written answers to the questions explicitly stated, and when requested to explain or prove.
More informationSmart Home Health Analytics Information Systems University of Maryland Baltimore County
Smart Home Health Analytics Information Systems University of Maryland Baltimore County 1 IEEE Expert, October 1996 2 Given sample S from all possible examples D Learner L learns hypothesis h based on
More information6.867 Machine Learning
6.867 Machine Learning Problem set 1 Solutions Thursday, September 19 What and how to turn in? Turn in short written answers to the questions explicitly stated, and when requested to explain or prove.
More informationCptS 570 Machine Learning School of EECS Washington State University. CptS Machine Learning 1
CptS 570 Machine Learning School of EECS Washington State University CptS 570 - Machine Learning 1 IEEE Expert, October 1996 CptS 570 - Machine Learning 2 Given sample S from all possible examples D Learner
More informationhttp://imgs.xkcd.com/comics/electoral_precedent.png Statistical Learning Theory CS4780/5780 Machine Learning Fall 2012 Thorsten Joachims Cornell University Reading: Mitchell Chapter 7 (not 7.4.4 and 7.5)
More informationCHAPTER EVALUATING HYPOTHESES 5.1 MOTIVATION
CHAPTER EVALUATING HYPOTHESES Empirically evaluating the accuracy of hypotheses is fundamental to machine learning. This chapter presents an introduction to statistical methods for estimating hypothesis
More informationCS 543 Page 1 John E. Boon, Jr.
CS 543 Machine Learning Spring 2010 Lecture 05 Evaluating Hypotheses I. Overview A. Given observed accuracy of a hypothesis over a limited sample of data, how well does this estimate its accuracy over
More informationLecture Notes 5 Convergence and Limit Theorems. Convergence with Probability 1. Convergence in Mean Square. Convergence in Probability, WLLN
Lecture Notes 5 Convergence and Limit Theorems Motivation Convergence with Probability Convergence in Mean Square Convergence in Probability, WLLN Convergence in Distribution, CLT EE 278: Convergence and
More informationCS 446 Machine Learning Fall 2016 Nov 01, Bayesian Learning
CS 446 Machine Learning Fall 206 Nov 0, 206 Bayesian Learning Professor: Dan Roth Scribe: Ben Zhou, C. Cervantes Overview Bayesian Learning Naive Bayes Logistic Regression Bayesian Learning So far, we
More informationSupervised Machine Learning (Spring 2014) Homework 2, sample solutions
58669 Supervised Machine Learning (Spring 014) Homework, sample solutions Credit for the solutions goes to mainly to Panu Luosto and Joonas Paalasmaa, with some additional contributions by Jyrki Kivinen
More informationEmpirical Risk Minimization, Model Selection, and Model Assessment
Empirical Risk Minimization, Model Selection, and Model Assessment CS6780 Advanced Machine Learning Spring 2015 Thorsten Joachims Cornell University Reading: Murphy 5.7-5.7.2.4, 6.5-6.5.3.1 Dietterich,
More informationIntroduction to Bayesian Learning. Machine Learning Fall 2018
Introduction to Bayesian Learning Machine Learning Fall 2018 1 What we have seen so far What does it mean to learn? Mistake-driven learning Learning by counting (and bounding) number of mistakes PAC learnability
More informationCSE 103 Homework 8: Solutions November 30, var(x) = np(1 p) = P r( X ) 0.95 P r( X ) 0.
() () a. X is a binomial distribution with n = 000, p = /6 b. The expected value, variance, and standard deviation of X is: E(X) = np = 000 = 000 6 var(x) = np( p) = 000 5 6 666 stdev(x) = np( p) = 000
More informationSTAT 285: Fall Semester Final Examination Solutions
Name: Student Number: STAT 285: Fall Semester 2014 Final Examination Solutions 5 December 2014 Instructor: Richard Lockhart Instructions: This is an open book test. As such you may use formulas such as
More informationBayesian Learning (II)
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning (II) Niels Landwehr Overview Probabilities, expected values, variance Basic concepts of Bayesian learning MAP
More informationDiagnostics. Gad Kimmel
Diagnostics Gad Kimmel Outline Introduction. Bootstrap method. Cross validation. ROC plot. Introduction Motivation Estimating properties of an estimator. Given data samples say the average. x 1, x 2,...,
More informationWhat is a random variable
OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE MATH 256 Probability and Random Processes 04 Random Variables Fall 20 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr
More informationDecision Trees. Tirgul 5
Decision Trees Tirgul 5 Using Decision Trees It could be difficult to decide which pet is right for you. We ll find a nice algorithm to help us decide what to choose without having to think about it. 2
More informationCSC314 / CSC763 Introduction to Machine Learning
CSC314 / CSC763 Introduction to Machine Learning COMSATS Institute of Information Technology Dr. Adeel Nawab More on Evaluating Hypotheses/Learning Algorithms Lecture Outline: Review of Confidence Intervals
More informationLecture 3. STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher
Lecture 3 STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher Previous lectures What is machine learning? Objectives of machine learning Supervised and
More informationQuantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing
Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationModel Accuracy Measures
Model Accuracy Measures Master in Bioinformatics UPF 2017-2018 Eduardo Eyras Computational Genomics Pompeu Fabra University - ICREA Barcelona, Spain Variables What we can measure (attributes) Hypotheses
More informationIntroduction to Machine Learning. Lecture 2
Introduction to Machine Learning Lecturer: Eran Halperin Lecture 2 Fall Semester Scribe: Yishay Mansour Some of the material was not presented in class (and is marked with a side line) and is given for
More informationEnsemble Methods. NLP ML Web! Fall 2013! Andrew Rosenberg! TA/Grader: David Guy Brizan
Ensemble Methods NLP ML Web! Fall 2013! Andrew Rosenberg! TA/Grader: David Guy Brizan How do you make a decision? What do you want for lunch today?! What did you have last night?! What are your favorite
More informationEvaluation & Credibility Issues
Evaluation & Credibility Issues What measure should we use? accuracy might not be enough. How reliable are the predicted results? How much should we believe in what was learned? Error on the training data
More information(It's not always good, but we can always make it.) (4) Convert the normal distribution N to the standard normal distribution Z. Specically.
. Introduction The quick summary, going forwards: Start with random variable X. 2 Compute the mean EX and variance 2 = varx. 3 Approximate X by the normal distribution N with mean µ = EX and standard deviation.
More informationHypothesis tests
6.1 6.4 Hypothesis tests Prof. Tesler Math 186 February 26, 2014 Prof. Tesler 6.1 6.4 Hypothesis tests Math 186 / February 26, 2014 1 / 41 6.1 6.2 Intro to hypothesis tests and decision rules Hypothesis
More informationCS340 Machine learning Lecture 5 Learning theory cont'd. Some slides are borrowed from Stuart Russell and Thorsten Joachims
CS340 Machine learning Lecture 5 Learning theory cont'd Some slides are borrowed from Stuart Russell and Thorsten Joachims Inductive learning Simplest form: learn a function from examples f is the target
More informationLinear Regression. Machine Learning CSE546 Kevin Jamieson University of Washington. Oct 5, Kevin Jamieson 1
Linear Regression Machine Learning CSE546 Kevin Jamieson University of Washington Oct 5, 2017 1 The regression problem Given past sales data on zillow.com, predict: y = House sale price from x = {# sq.
More informationIntroduction to Supervised Learning. Performance Evaluation
Introduction to Supervised Learning Performance Evaluation Marcelo S. Lauretto Escola de Artes, Ciências e Humanidades, Universidade de São Paulo marcelolauretto@usp.br Lima - Peru Performance Evaluation
More information2.1 Lecture 5: Probability spaces, Interpretation of probabilities, Random variables
Chapter 2 Kinetic Theory 2.1 Lecture 5: Probability spaces, Interpretation of probabilities, Random variables In the previous lectures the theory of thermodynamics was formulated as a purely phenomenological
More informationExample continued. Math 425 Intro to Probability Lecture 37. Example continued. Example
continued : Coin tossing Math 425 Intro to Probability Lecture 37 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan April 8, 2009 Consider a Bernoulli trials process with
More informationBayesian Methods: Naïve Bayes
Bayesian Methods: aïve Bayes icholas Ruozzi University of Texas at Dallas based on the slides of Vibhav Gogate Last Time Parameter learning Learning the parameter of a simple coin flipping model Prior
More informationThe exam is closed book, closed notes except your one-page (two sides) or two-page (one side) crib sheet.
CS 189 Spring 013 Introduction to Machine Learning Final You have 3 hours for the exam. The exam is closed book, closed notes except your one-page (two sides) or two-page (one side) crib sheet. Please
More informationLeast Squares Classification
Least Squares Classification Stephen Boyd EE103 Stanford University November 4, 2017 Outline Classification Least squares classification Multi-class classifiers Classification 2 Classification data fitting
More informationCOMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017
COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University FEATURE EXPANSIONS FEATURE EXPANSIONS
More informationLecture 2 Sep 5, 2017
CS 388R: Randomized Algorithms Fall 2017 Lecture 2 Sep 5, 2017 Prof. Eric Price Scribe: V. Orestis Papadigenopoulos and Patrick Rall NOTE: THESE NOTES HAVE NOT BEEN EDITED OR CHECKED FOR CORRECTNESS 1
More informationLecture 4 Discriminant Analysis, k-nearest Neighbors
Lecture 4 Discriminant Analysis, k-nearest Neighbors Fredrik Lindsten Division of Systems and Control Department of Information Technology Uppsala University. Email: fredrik.lindsten@it.uu.se fredrik.lindsten@it.uu.se
More informationLinear Classifiers: Expressiveness
Linear Classifiers: Expressiveness Machine Learning Spring 2018 The slides are mainly from Vivek Srikumar 1 Lecture outline Linear classifiers: Introduction What functions do linear classifiers express?
More informationNaïve Bayes classification
Naïve Bayes classification 1 Probability theory Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. Examples: A person s height, the outcome of a coin toss
More informationStatistical and Computational Learning Theory
Statistical and Computational Learning Theory Fundamental Question: Predict Error Rates Given: Find: The space H of hypotheses The number and distribution of the training examples S The complexity of the
More informationStatistical Learning Theory: Generalization Error Bounds
Statistical Learning Theory: Generalization Error Bounds CS6780 Advanced Machine Learning Spring 2015 Thorsten Joachims Cornell University Reading: Murphy 6.5.4 Schoelkopf/Smola Chapter 5 (beginning, rest
More informationDECISION TREE LEARNING. [read Chapter 3] [recommended exercises 3.1, 3.4]
1 DECISION TREE LEARNING [read Chapter 3] [recommended exercises 3.1, 3.4] Decision tree representation ID3 learning algorithm Entropy, Information gain Overfitting Decision Tree 2 Representation: Tree-structured
More informationMethods and Criteria for Model Selection. CS57300 Data Mining Fall Instructor: Bruno Ribeiro
Methods and Criteria for Model Selection CS57300 Data Mining Fall 2016 Instructor: Bruno Ribeiro Goal } Introduce classifier evaluation criteria } Introduce Bias x Variance duality } Model Assessment }
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationPractice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
More informationCOMP9444: Neural Networks. Vapnik Chervonenkis Dimension, PAC Learning and Structural Risk Minimization
: Neural Networks Vapnik Chervonenkis Dimension, PAC Learning and Structural Risk Minimization 11s2 VC-dimension and PAC-learning 1 How good a classifier does a learner produce? Training error is the precentage
More informationLECTURE 5. Introduction to Econometrics. Hypothesis testing
LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will
More informationDetection theory. H 0 : x[n] = w[n]
Detection Theory Detection theory A the last topic of the course, we will briefly consider detection theory. The methods are based on estimation theory and attempt to answer questions such as Is a signal
More informationCOMP90051 Statistical Machine Learning
COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Trevor Cohn 2. Statistical Schools Adapted from slides by Ben Rubinstein Statistical Schools of Thought Remainder of lecture is to provide
More informationComplex Numbers. Essential Question What are the subsets of the set of complex numbers? Integers. Whole Numbers. Natural Numbers
3.4 Complex Numbers Essential Question What are the subsets of the set of complex numbers? In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically
More informationLinear Models: Comparing Variables. Stony Brook University CSE545, Fall 2017
Linear Models: Comparing Variables Stony Brook University CSE545, Fall 2017 Statistical Preliminaries Random Variables Random Variables X: A mapping from Ω to ℝ that describes the question we care about
More informationLearning Theory. Aar$ Singh and Barnabas Poczos. Machine Learning / Apr 17, Slides courtesy: Carlos Guestrin
Learning Theory Aar$ Singh and Barnabas Poczos Machine Learning 10-701/15-781 Apr 17, 2014 Slides courtesy: Carlos Guestrin Learning Theory We have explored many ways of learning from data But How good
More informationCOMPUTATIONAL LEARNING THEORY
COMPUTATIONAL LEARNING THEORY XIAOXU LI Abstract. This paper starts with basic concepts of computational learning theory and moves on with examples in monomials and learning rays to the final discussion
More informationNaïve Bayes classification. p ij 11/15/16. Probability theory. Probability theory. Probability theory. X P (X = x i )=1 i. Marginal Probability
Probability theory Naïve Bayes classification Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s height, the outcome of a coin toss Distinguish
More informationPAC Model and Generalization Bounds
PAC Model and Generalization Bounds Overview Probably Approximately Correct (PAC) model Basic generalization bounds finite hypothesis class infinite hypothesis class Simple case More next week 2 Motivating
More informationNull Hypothesis Significance Testing p-values, significance level, power, t-tests Spring 2017
Null Hypothesis Significance Testing p-values, significance level, power, t-tests 18.05 Spring 2017 Understand this figure f(x H 0 ) x reject H 0 don t reject H 0 reject H 0 x = test statistic f (x H 0
More informationProperties of Random Variables
Properties of Random Variables 1 Definitions A discrete random variable is defined by a probability distribution that lists each possible outcome and the probability of obtaining that outcome If the random
More informationThings to remember when learning probability distributions:
SPECIAL DISTRIBUTIONS Some distributions are special because they are useful They include: Poisson, exponential, Normal (Gaussian), Gamma, geometric, negative binomial, Binomial and hypergeometric distributions
More informationLecture Topic 4: Chapter 7 Sampling and Sampling Distributions
Lecture Topic 4: Chapter 7 Sampling and Sampling Distributions Statistical Inference: The aim is to obtain information about a population from information contained in a sample. A population is the set
More informationMachine Learning. Lecture Slides for. ETHEM ALPAYDIN The MIT Press, h1p://www.cmpe.boun.edu.
Lecture Slides for INTRODUCTION TO Machine Learning ETHEM ALPAYDIN The MIT Press, 2010 alpaydin@boun.edu.tr h1p://www.cmpe.boun.edu.tr/~ethem/i2ml2e CHAPTER 19: Design and Analysis of Machine Learning
More informationMachine Learning Linear Classification. Prof. Matteo Matteucci
Machine Learning Linear Classification Prof. Matteo Matteucci Recall from the first lecture 2 X R p Regression Y R Continuous Output X R p Y {Ω 0, Ω 1,, Ω K } Classification Discrete Output X R p Y (X)
More informationAnnouncements. Proposals graded
Announcements Proposals graded Kevin Jamieson 2018 1 Bayesian Methods Machine Learning CSE546 Kevin Jamieson University of Washington November 1, 2018 2018 Kevin Jamieson 2 MLE Recap - coin flips Data:
More informationLecture 5: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics. 1 Executive summary
ECE 830 Spring 207 Instructor: R. Willett Lecture 5: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics Executive summary In the last lecture we saw that the likelihood
More informationLecture Learning infinite hypothesis class via VC-dimension and Rademacher complexity;
CSCI699: Topics in Learning and Game Theory Lecture 2 Lecturer: Ilias Diakonikolas Scribes: Li Han Today we will cover the following 2 topics: 1. Learning infinite hypothesis class via VC-dimension and
More informationMACHINE LEARNING. Probably Approximately Correct (PAC) Learning. Alessandro Moschitti
MACHINE LEARNING Probably Approximately Correct (PAC) Learning Alessandro Moschitti Department of Information Engineering and Computer Science University of Trento Email: moschitti@disi.unitn.it Objectives:
More informationCMPT 882 Machine Learning
CMPT 882 Machine Learning Lecture Notes Instructor: Dr. Oliver Schulte Scribe: Qidan Cheng and Yan Long Mar. 9, 2004 and Mar. 11, 2004-1 - Basic Definitions and Facts from Statistics 1. The Binomial Distribution
More informationPointwise Exact Bootstrap Distributions of Cost Curves
Pointwise Exact Bootstrap Distributions of Cost Curves Charles Dugas and David Gadoury University of Montréal 25th ICML Helsinki July 2008 Dugas, Gadoury (U Montréal) Cost curves July 8, 2008 1 / 24 Outline
More informationMachine Learning CSE546 Carlos Guestrin University of Washington. September 30, 2013
Bayesian Methods Machine Learning CSE546 Carlos Guestrin University of Washington September 30, 2013 1 What about prior n Billionaire says: Wait, I know that the thumbtack is close to 50-50. What can you
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More informationNaive Bayes classification
Naive Bayes classification Christos Dimitrakakis December 4, 2015 1 Introduction One of the most important methods in machine learning and statistics is that of Bayesian inference. This is the most fundamental
More informationMachine Learning. Lecture 9: Learning Theory. Feng Li.
Machine Learning Lecture 9: Learning Theory Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2018 Why Learning Theory How can we tell
More information2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationPoisson approximations
Chapter 9 Poisson approximations 9.1 Overview The Binn, p) can be thought of as the distribution of a sum of independent indicator random variables X 1 + + X n, with {X i = 1} denoting a head on the ith
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 3: Probability, Bayes Theorem, and Bayes Classification Peter Belhumeur Computer Science Columbia University Probability Should you play this game? Game: A fair
More informationWeek 1 Quantitative Analysis of Financial Markets Distributions A
Week 1 Quantitative Analysis of Financial Markets Distributions A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More information6 The normal distribution, the central limit theorem and random samples
6 The normal distribution, the central limit theorem and random samples 6.1 The normal distribution We mentioned the normal (or Gaussian) distribution in Chapter 4. It has density f X (x) = 1 σ 1 2π e
More informationEfficient and Principled Online Classification Algorithms for Lifelon
Efficient and Principled Online Classification Algorithms for Lifelong Learning Toyota Technological Institute at Chicago Chicago, IL USA Talk @ Lifelong Learning for Mobile Robotics Applications Workshop,
More informationECE 661: Homework 10 Fall 2014
ECE 661: Homework 10 Fall 2014 This homework consists of the following two parts: (1) Face recognition with PCA and LDA for dimensionality reduction and the nearest-neighborhood rule for classification;
More informationLecture 25 of 42. PAC Learning, VC Dimension, and Mistake Bounds
Lecture 25 of 42 PAC Learning, VC Dimension, and Mistake Bounds Thursday, 15 March 2007 William H. Hsu, KSU http://www.kddresearch.org/courses/spring2007/cis732 Readings: Sections 7.4.17.4.3, 7.5.17.5.3,
More informationprobability of k samples out of J fall in R.
Nonparametric Techniques for Density Estimation (DHS Ch. 4) n Introduction n Estimation Procedure n Parzen Window Estimation n Parzen Window Example n K n -Nearest Neighbor Estimation Introduction Suppose
More informationClass 4: Classification. Quaid Morris February 11 th, 2011 ML4Bio
Class 4: Classification Quaid Morris February 11 th, 211 ML4Bio Overview Basic concepts in classification: overfitting, cross-validation, evaluation. Linear Discriminant Analysis and Quadratic Discriminant
More informationMathematical Statistics
Mathematical Statistics MAS 713 Chapter 8 Previous lecture: 1 Bayesian Inference 2 Decision theory 3 Bayesian Vs. Frequentist 4 Loss functions 5 Conjugate priors Any questions? Mathematical Statistics
More informationCS340 Machine learning Lecture 4 Learning theory. Some slides are borrowed from Sebastian Thrun and Stuart Russell
CS340 Machine learning Lecture 4 Learning theory Some slides are borrowed from Sebastian Thrun and Stuart Russell Announcement What: Workshop on applying for NSERC scholarships and for entry to graduate
More informationProbability and Statistics. Terms and concepts
Probability and Statistics Joyeeta Dutta Moscato June 30, 2014 Terms and concepts Sample vs population Central tendency: Mean, median, mode Variance, standard deviation Normal distribution Cumulative distribution
More information