Bayesian decision theory. Nuno Vasconcelos ECE Department, UCSD
|
|
- Mervin Bond
- 5 years ago
- Views:
Transcription
1 Bayesan decson theory Nuno Vasconcelos ECE Department, UCSD
2 Bayesan decson theory recall that we have state of the world observatons decson functon L[,y] loss of predctn y wth the epected value of the loss s called the rsk Rsk E whch can be wrtten as [ L, ], Rsk M, L[, ] d,
3 Bayesan decson theory from ths Rsk by chan rule where Rsk M, L[, ] d, M L[, ] d R R d E [ R ] M L[, ] 0 0 s the condtonal rsk, ven the observaton 3
4 Bayesan decson theory snce, by defnton, t follows that L [, ] 0,, y 0 R M L[, ] 0, hence Rsk E [ R ] s mnmum f we mnmze R at all,.e., f we use pck the decson functon ar mn M L [, ] 4
5 Bayesan decson theory ths s the Bayes decson rule M whch decson rule has the assocated rsk ], [ ar mn L M whch decson rule has smaller condtonal rsk? the assocated rsk d L R M ], [,, 0 or d L R M ] [ s the Bayes rsk and cannot be beaten d L R ], [ 5 s the Bayes rsk, and cannot be beaten
6 Eample let s consder a bnary classfcaton problem {0,} for whch the condtonal rsk s M R we have two optons L[, ] 0 0 L [,0] L [,] + 0 R 0 L [0,0] 0] + L [0,] 0 + L R 0 L[,0] L[,] + and should pck the one of smaller condtonal rsk 6
7 Eample.e. pck 0 f R 0 < R and otherwse ths can be wrtten as, pck 0 f 0 L[0,0] + < 0 L[,0] + L[0,] < L[,] or { L[0,0] [,0] } { L[,] L[0,] } 0 L < < 0 usually there s no loss assocated wth the correct decson L[,] L[0,0] 0 and ths s the same as 0 L[,0] > L[0,] 7
8 Eample or, pck 0 f > and applyn Bayes rule 0 0 L[0,] L [,0] 0 L whch s equvalent to pck 0 f 0 > T > L [0,] L[,0] L[0,] L[,0] 0 0.e. we pck 0, when the probablty of ven that 0 dvded by that ven s reater than a threshold the optmal threshold h T depends d on the costs of the two types of error and the probabltes of the two classes 8
9 Eample let s consder the 0- loss n ths case the optmal decson functon s y y y L 0,, ], [ n ths case the optmal decson functon s ], [ ar mn L M ar mn [ ] mn ar [ ] mn ar ma ar 9 ma ar
10 Eample for the 0- loss the optmal decson rule s the mamum a-posteror probablty rule ar ma what s the assocated rsk? M R L[, ] d M d y d, y, d 0
11 Eample but R y, d, s really just the probablty of error of the decson rule note that the same result would hold for any,.e. R would be the probablty of error of ths mples the follown for the 0- loss the Bayes decson rule s the MA rule ar ma the rsk s the probablty of error of ths rule Bayes error there s no other decson functon wth lower error
12 MA rule usually can be wrtten n a smple form ven a probablstc model for and consder the two-class problem,.e. 0 or the BDR s ar ma 0,, f f 0 0 < 0 pck 0 when and otherwse usn Bayes rule
13 MA rule notn that s a non-neatve quantty ths s the same as pck 0 when 0 0 by usn the same reasonn, ths can be easly eneralzed to note that: ar ma many class-condtonal dstrbutons are eponental e.. the Gaussan ths product can be trcky to compute e.. the tal probabltes are qute small we can take advantae of the fact that we only care about the order of the terms on the rht-hand sde 3
14 The lo trck ths s the lo trck whch s to take los note that the lo s a monotoncally ncreasn functon a > b lo a > lob lo a lo b from whch ar ma ar ma lo ar ma lo the order s preserved + lo b a 4
15 MA rule n summary for the zero/one loss, the follown three decson rules are optmal and equvalent ar ma [ ] ar ma 3 ar ma[ lo + lo ] s usually hard to use, 3 s frequently easer than 5
16 Eample the Bayes decson rule s usually hhly ntutve eample: communcatons a bt s transmtted by a source, corrupted by nose, and receved by a decoder channel Q: what should the optmal decoder do to recover? 6
17 Eample ntutvely, t appears that t should just threshold pck T decson rule 0,, f f < T > T what s the threshold value? let s solve the problem wth the BDR 7
18 Eample we need class probabltes: n the absence of any other nfo let s say 0 class-condtonal denstes: nose results from thermal processes, electrons movn around and bumpn each other a lot of ndependent events that add up by the central lmt theorem t appears reasonable to assume that the nose s Gaussan we denote a Gaussan random varable of mean and varance by ~ N, 8
19 Eample the Gaussan probablty densty functon s G,, e π snce nose s Gaussan, and assumn t s just added to the snal we have channel + ε, ε ~ N0, n both cases, corresponds to a constant plus zero-mean Gaussan nose ths smply adds to the mean of the Gaussan 9
20 Eample n summary 0 G,0, G,, 0 or, raphcally, 0 0
21 Eample to compute the BDR, we recall that ar ma lo and note that t [ + lo ] terms whch are constant as a functon of can be dropped snce we are just lookn for the that mamzes the functon snce ths s the case for the class-probabltes we have 0 ar ma lo
22 BDR ths s ntutve we pck the class that best eplans ves hher probablty the observaton n ths case, we can solve vsually 0 pck 0 pck but the mathematcal t soluton s equally smple
23 BDR let s consder the more eneral case 0 G G for whch,,,, 0 0 G G ar ma lo ar ma lo π e lo ar ma π 3 ar mn
24 BDR or ar mn ar mn + ar mn + the optmal decson s, therefore pck 0 f or, pck 0 f < 0 + < 0 0 < + 4
25 BDR for a problem wth Gaussan classes, equal varances and equal class probabltes optmal decson boundary s the threshold at the md-pont between the two means pck 0 0 pck 5
26 BDR back to our snal decodn problem n ths case T 0.5 decson rule 0,, f f < 0.5 > 0.5 ths s, once aan, ntutve we place the threshold mdway alon the nose sources 6
27 BDR what s the pont of on throuh all the math? now we know that the ntutve threshold s actually optmal, and n whch sense t s optmal mnmum probablty or error the Bayesan soluton keeps us honest. t forces us to make all our assumptons eplct assumptons we have made unform class probabltes 0 Gaussanty the varance s the same under the two states G,,, nose s addtve even for a trval problem, we have made lots of assumptons + ε 7
28 BDR what f the class probabltes are not the same? e codn scheme 7 0 e.. codn scheme 7 0 n ths case >> 0 how does ths chane the optmal decson rule? { } lo lo ar ma + + lo lo ar ma e π + lo lo ar ma π 8 lo mn ar
29 BDR or lo ar mn lo ar mn lo ar mn + + the optmal decson s, therefore k 0 f lo mn ar + pck 0 f 0 l lo 0 lo < + < + or, pck 0 f lo < lo <
30 BDR what s the role of the pror for class probabltes? + < lo 0 the pror moves the threshold up or down, n an ntutve way 0> : threshold ncreases snce 0 has hher probablty, we care more about errors on the 0 sde by usn a hher threshold we are makn t more lkely to pck 0 f 0, all we care about s 0, the threshold becomes nfnte we never say how relevant s the pror? t s wehed by 0 30
31 BDR how relevant s the pror? t s wehed by the nverse of the normalzed dstance between the means 0 dstance between the means n unts of varance f the classes are very far apart, the pror makes no dfference ths s the easy stuaton, the observatons are very clear, Bayes says foret the pror knowlede f the classes are eactly equal same mean the pror ets nfnte weht n ths case the observatons do not say anythn about the class, Bayes says foret about the data, just use the knowlede that you started wth even f that means always say 0 or always say 3
32 3
The Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informationBayesian decision theory. Nuno Vasconcelos ECE Department, UCSD
Bayesan decson theory Nuno Vasconcelos ECE Department UCSD Notaton the notaton n DHS s qute sloppy e.. show that error error z z dz really not clear what ths means we wll use the follown notaton subscrpts
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that e have state of the orld X observatons decson functon L[,y] loss of predctn y th Bayes decson rule s the rule
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More informationMIMA Group. Chapter 2 Bayesian Decision Theory. School of Computer Science and Technology, Shandong University. Xin-Shun SDU
Group M D L M Chapter Bayesan Decson heory Xn-Shun Xu @ SDU School of Computer Scence and echnology, Shandong Unversty Bayesan Decson heory Bayesan decson theory s a statstcal approach to data mnng/pattern
More informationMaximum Likelihood Estimation (MLE)
Maxmum Lkelhood Estmaton (MLE) Ken Kreutz-Delgado (Nuno Vasconcelos) ECE 175A Wnter 01 UCSD Statstcal Learnng Goal: Gven a relatonshp between a feature vector x and a vector y, and d data samples (x,y
More informationDepartment of Computer Science Artificial Intelligence Research Laboratory. Iowa State University MACHINE LEARNING
MACHINE LEANING Vasant Honavar Bonformatcs and Computatonal Bology rogram Center for Computatonal Intellgence, Learnng, & Dscovery Iowa State Unversty honavar@cs.astate.edu www.cs.astate.edu/~honavar/
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More informationwhere v means the change in velocity, and t is the
1 PHYS:100 LECTURE 4 MECHANICS (3) Ths lecture covers the eneral case of moton wth constant acceleraton and free fall (whch s one of the more mportant examples of moton wth constant acceleraton) n a more
More informationS Advanced Digital Communication (4 cr) Targets today
S.72-3320 Advanced Dtal Communcaton (4 cr) Convolutonal Codes Tarets today Why to apply convolutonal codn? Defnn convolutonal codes Practcal encodn crcuts Defnn qualty of convolutonal codes Decodn prncples
More informationWhy Bayesian? 3. Bayes and Normal Models. State of nature: class. Decision rule. Rev. Thomas Bayes ( ) Bayes Theorem (yes, the famous one)
Why Bayesan? 3. Bayes and Normal Models Alex M. Martnez alex@ece.osu.edu Handouts Handoutsfor forece ECE874 874Sp Sp007 If all our research (n PR was to dsappear and you could only save one theory, whch
More informationCommunication with AWGN Interference
Communcaton wth AWG Interference m {m } {p(m } Modulator s {s } r=s+n Recever ˆm AWG n m s a dscrete random varable(rv whch takes m wth probablty p(m. Modulator maps each m nto a waveform sgnal s m=m
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationLinear discriminants. Nuno Vasconcelos ECE Department, UCSD
Lnear dscrmnants Nuno Vasconcelos ECE Department UCSD Classfcaton a classfcaton problem as to tpes of varables e.g. X - vector of observatons features n te orld Y - state class of te orld X R 2 fever blood
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationError Probability for M Signals
Chapter 3 rror Probablty for M Sgnals In ths chapter we dscuss the error probablty n decdng whch of M sgnals was transmtted over an arbtrary channel. We assume the sgnals are represented by a set of orthonormal
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLecture 12: Classification
Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna
More informationLecture 3: Shannon s Theorem
CSE 533: Error-Correctng Codes (Autumn 006 Lecture 3: Shannon s Theorem October 9, 006 Lecturer: Venkatesan Guruswam Scrbe: Wdad Machmouch 1 Communcaton Model The communcaton model we are usng conssts
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition
EG 880/988 - Specal opcs n Computer Engneerng: Pattern Recognton Memoral Unversty of ewfoundland Pattern Recognton Lecture 7 May 3, 006 http://wwwengrmunca/~charlesr Offce Hours: uesdays hursdays 8:30-9:30
More informationChapter 7 Channel Capacity and Coding
Chapter 7 Channel Capacty and Codng Contents 7. Channel models and channel capacty 7.. Channel models Bnary symmetrc channel Dscrete memoryless channels Dscrete-nput, contnuous-output channel Waveform
More informationClustering & Unsupervised Learning
Clusterng & Unsupervsed Learnng Ken Kreutz-Delgado (Nuno Vasconcelos) ECE 175A Wnter 2012 UCSD Statstcal Learnng Goal: Gven a relatonshp between a feature vector x and a vector y, and d data samples (x,y
More informationProbability Theory. The nth coefficient of the Taylor series of f(k), expanded around k = 0, gives the nth moment of x as ( ik) n n!
8333: Statstcal Mechancs I Problem Set # 3 Solutons Fall 3 Characterstc Functons: Probablty Theory The characterstc functon s defned by fk ep k = ep kpd The nth coeffcent of the Taylor seres of fk epanded
More informationClassification as a Regression Problem
Target varable y C C, C,, ; Classfcaton as a Regresson Problem { }, 3 L C K To treat classfcaton as a regresson problem we should transform the target y nto numercal values; The choce of numercal class
More informationxp(x µ) = 0 p(x = 0 µ) + 1 p(x = 1 µ) = µ
CSE 455/555 Sprng 2013 Homework 7: Parametrc Technques Jason J. Corso Computer Scence and Engneerng SUY at Buffalo jcorso@buffalo.edu Solutons by Yngbo Zhou Ths assgnment does not need to be submtted and
More informationb ), which stands for uniform distribution on the interval a x< b. = 0 elsewhere
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Some mportant probablty dstrbutons: Unform Bnomal Posson Gaussan/ormal The Unform dstrbuton s often called U( a, b ), hch stands for unform
More informationClassification learning II
Lecture 8 Classfcaton learnng II Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Logstc regresson model Defnes a lnear decson boundar Dscrmnant functons: g g g g here g z / e z f, g g - s a logstc functon
More informationMLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012
MLE and Bayesan Estmaton Je Tang Department of Computer Scence & Technology Tsnghua Unversty 01 1 Lnear Regresson? As the frst step, we need to decde how we re gong to represent the functon f. One example:
More informationEngineering Risk Benefit Analysis
Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007
More informationClustering & (Ken Kreutz-Delgado) UCSD
Clusterng & Unsupervsed Learnng Nuno Vasconcelos (Ken Kreutz-Delgado) UCSD Statstcal Learnng Goal: Gven a relatonshp between a feature vector x and a vector y, and d data samples (x,y ), fnd an approxmatng
More information9.913 Pattern Recognition for Vision. Class IV Part I Bayesian Decision Theory Yuri Ivanov
9.93 Class IV Part I Bayesan Decson Theory Yur Ivanov TOC Roadmap to Machne Learnng Bayesan Decson Makng Mnmum Error Rate Decsons Mnmum Rsk Decsons Mnmax Crteron Operatng Characterstcs Notaton x - scalar
More informationChapter 7 Channel Capacity and Coding
Wreless Informaton Transmsson System Lab. Chapter 7 Channel Capacty and Codng Insttute of Communcatons Engneerng atonal Sun Yat-sen Unversty Contents 7. Channel models and channel capacty 7.. Channel models
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for
More informationβ0 + β1xi and want to estimate the unknown
SLR Models Estmaton Those OLS Estmates Estmators (e ante) v. estmates (e post) The Smple Lnear Regresson (SLR) Condtons -4 An Asde: The Populaton Regresson Functon B and B are Lnear Estmators (condtonal
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More informationFinite Mixture Models and Expectation Maximization. Most slides are from: Dr. Mario Figueiredo, Dr. Anil Jain and Dr. Rong Jin
Fnte Mxture Models and Expectaton Maxmzaton Most sldes are from: Dr. Maro Fgueredo, Dr. Anl Jan and Dr. Rong Jn Recall: The Supervsed Learnng Problem Gven a set of n samples X {(x, y )},,,n Chapter 3 of
More informationCHAPTER 3: BAYESIAN DECISION THEORY
HATER 3: BAYESIAN DEISION THEORY Decson mang under uncertanty 3 Data comes from a process that s completely not nown The lac of nowledge can be compensated by modelng t as a random process May be the underlyng
More informationThe big picture. Outline
The bg pcture Vncent Claveau IRISA - CNRS, sldes from E. Kjak INSA Rennes Notatons classes: C = {ω = 1,.., C} tranng set S of sze m, composed of m ponts (x, ω ) per class ω representaton space: R d (=
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationBayesian Learning. Smart Home Health Analytics Spring Nirmalya Roy Department of Information Systems University of Maryland Baltimore County
Smart Home Health Analytcs Sprng 2018 Bayesan Learnng Nrmalya Roy Department of Informaton Systems Unversty of Maryland Baltmore ounty www.umbc.edu Bayesan Learnng ombnes pror knowledge wth evdence to
More information8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore
8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø
More informationFlux-Uncertainty from Aperture Photometry. F. Masci, version 1.0, 10/14/2008
Flux-Uncertanty from Aperture Photometry F. Masc, verson 1.0, 10/14/008 1. Summary We derve a eneral formula for the nose varance n the flux of a source estmated from aperture photometry. The 1-σ uncertanty
More informationAssuming that the transmission delay is negligible, we have
Baseband Transmsson of Bnary Sgnals Let g(t), =,, be a sgnal transmtted over an AWG channel. Consder the followng recever g (t) + + Σ x(t) LTI flter h(t) y(t) t = nt y(nt) threshold comparator Decson ˆ
More informationOther NN Models. Reinforcement learning (RL) Probabilistic neural networks
Other NN Models Renforcement learnng (RL) Probablstc neural networks Support vector machne (SVM) Renforcement learnng g( (RL) Basc deas: Supervsed dlearnng: (delta rule, BP) Samples (x, f(x)) to learn
More informationLogistic Regression. CAP 5610: Machine Learning Instructor: Guo-Jun QI
Logstc Regresson CAP 561: achne Learnng Instructor: Guo-Jun QI Bayes Classfer: A Generatve model odel the posteror dstrbuton P(Y X) Estmate class-condtonal dstrbuton P(X Y) for each Y Estmate pror dstrbuton
More informationPattern Classification
attern Classfcaton All materals n these sldes were taken from attern Classfcaton nd ed by R. O. Duda,. E. Hart and D. G. Stork, John Wley & Sons, 000 wth the ermsson of the authors and the ublsher Chater
More informationMarginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It
More informationStatistical pattern recognition
Statstcal pattern recognton Bayes theorem Problem: decdng f a patent has a partcular condton based on a partcular test However, the test s mperfect Someone wth the condton may go undetected (false negatve
More informationArtificial Intelligence Bayesian Networks
Artfcal Intellgence Bayesan Networks Adapted from sldes by Tm Fnn and Mare desjardns. Some materal borrowed from Lse Getoor. 1 Outlne Bayesan networks Network structure Condtonal probablty tables Condtonal
More informationEvaluation for sets of classes
Evaluaton for Tet Categorzaton Classfcaton accuracy: usual n ML, the proporton of correct decsons, Not approprate f the populaton rate of the class s low Precson, Recall and F 1 Better measures 21 Evaluaton
More informationQuantifying Uncertainty
Partcle Flters Quantfyng Uncertanty Sa Ravela M. I. T Last Updated: Sprng 2013 1 Quantfyng Uncertanty Partcle Flters Partcle Flters Appled to Sequental flterng problems Can also be appled to smoothng problems
More informationA Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach
A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland
More informationINF 5860 Machine learning for image classification. Lecture 3 : Image classification and regression part II Anne Solberg January 31, 2018
INF 5860 Machne learnng for mage classfcaton Lecture 3 : Image classfcaton and regresson part II Anne Solberg January 3, 08 Today s topcs Multclass logstc regresson and softma Regularzaton Image classfcaton
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationDigital Modems. Lecture 2
Dgtal Modems Lecture Revew We have shown that both Bayes and eyman/pearson crtera are based on the Lkelhood Rato Test (LRT) Λ ( r ) < > η Λ r s called observaton transformaton or suffcent statstc The crtera
More informationA random variable is a function which associates a real number to each element of the sample space
Introducton to Random Varables Defnton of random varable Defnton of of random varable Dscrete and contnuous random varable Probablty blt functon Dstrbuton functon Densty functon Sometmes, t s not enough
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationGenerative classification models
CS 675 Intro to Machne Learnng Lecture Generatve classfcaton models Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Data: D { d, d,.., dn} d, Classfcaton represents a dscrete class value Goal: learn
More informationINTRODUCTION TO MACHINE LEARNING 3RD EDITION
ETHEM ALPAYDIN The MIT Press, 2014 Lecture Sldes for INTRODUCTION TO MACHINE LEARNING 3RD EDITION alpaydn@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/2ml3e CHAPTER 3: BAYESIAN DECISION THEORY Probablty
More informationDecision-making and rationality
Reslence Informatcs for Innovaton Classcal Decson Theory RRC/TMI Kazuo URUTA Decson-makng and ratonalty What s decson-makng? Methodology for makng a choce The qualty of decson-makng determnes success or
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationSupport Vector Machines
Separatng boundary, defned by w Support Vector Machnes CISC 5800 Professor Danel Leeds Separatng hyperplane splts class 0 and class 1 Plane s defned by lne w perpendcular to plan Is data pont x n class
More informationHomework Assignment 3 Due in class, Thursday October 15
Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationLecture 14 (03/27/18). Channels. Decoding. Preview of the Capacity Theorem.
Lecture 14 (03/27/18). Channels. Decodng. Prevew of the Capacty Theorem. A. Barg The concept of a communcaton channel n nformaton theory s an abstracton for transmttng dgtal (and analog) nformaton from
More informationSupport Vector Machines
/14/018 Separatng boundary, defned by w Support Vector Machnes CISC 5800 Professor Danel Leeds Separatng hyperplane splts class 0 and class 1 Plane s defned by lne w perpendcular to plan Is data pont x
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationThe conjugate prior to a Bernoulli is. A) Bernoulli B) Gaussian C) Beta D) none of the above
The conjugate pror to a Bernoull s A) Bernoull B) Gaussan C) Beta D) none of the above The conjugate pror to a Gaussan s A) Bernoull B) Gaussan C) Beta D) none of the above MAP estmates A) argmax θ p(θ
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationWhich Separator? Spring 1
Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal
More informationLearning from Data 1 Naive Bayes
Learnng from Data 1 Nave Bayes Davd Barber dbarber@anc.ed.ac.uk course page : http://anc.ed.ac.uk/ dbarber/lfd1/lfd1.html c Davd Barber 2001, 2002 1 Learnng from Data 1 : c Davd Barber 2001,2002 2 1 Why
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationAssignment 2. Tyler Shendruk February 19, 2010
Assgnment yler Shendruk February 9, 00 Kadar Ch. Problem 8 We have an N N symmetrc matrx, M. he symmetry means M M and we ll say the elements of the matrx are m j. he elements are pulled from a probablty
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate
More informationTAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES
TAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES SVANTE JANSON Abstract. We gve explct bounds for the tal probabltes for sums of ndependent geometrc or exponental varables, possbly wth dfferent
More informationPropagation of error for multivariable function
Propagaton o error or multvarable uncton ow consder a multvarable uncton (u, v, w, ). I measurements o u, v, w,. All have uncertant u, v, w,., how wll ths aect the uncertant o the uncton? L tet) o (Equaton
More informationConsider the following passband digital communication system model. c t. modulator. t r a n s m i t t e r. signal decoder.
PASSBAND DIGITAL MODULATION TECHNIQUES Consder the followng passband dgtal communcaton system model. cos( ω + φ ) c t message source m sgnal encoder s modulator s () t communcaton xt () channel t r a n
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationIntroduction to Random Variables
Introducton to Random Varables Defnton of random varable Defnton of random varable Dscrete and contnuous random varable Probablty functon Dstrbuton functon Densty functon Sometmes, t s not enough to descrbe
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More information15-381: Artificial Intelligence. Regression and cross validation
15-381: Artfcal Intellgence Regresson and cross valdaton Where e are Inputs Densty Estmator Probablty Inputs Classfer Predct category Inputs Regressor Predct real no. Today Lnear regresson Gven an nput
More informationLecture 3: Boltzmann distribution
Lecture 3: Boltzmann dstrbuton Statstcal mechancs: concepts Ams: Dervaton of Boltzmann dstrbuton: Basc postulate of statstcal mechancs. All states equally lkely. Equal a-pror probablty: Statstcal vew of
More informationSDMML HT MSc Problem Sheet 4
SDMML HT 06 - MSc Problem Sheet 4. The recever operatng characterstc ROC curve plots the senstvty aganst the specfcty of a bnary classfer as the threshold for dscrmnaton s vared. Let the data space be
More informationECE 534: Elements of Information Theory. Solutions to Midterm Exam (Spring 2006)
ECE 534: Elements of Informaton Theory Solutons to Mdterm Eam (Sprng 6) Problem [ pts.] A dscrete memoryless source has an alphabet of three letters,, =,, 3, wth probabltes.4,.4, and., respectvely. (a)
More informationCS 3710: Visual Recognition Classification and Detection. Adriana Kovashka Department of Computer Science January 13, 2015
CS 3710: Vsual Recognton Classfcaton and Detecton Adrana Kovashka Department of Computer Scence January 13, 2015 Plan for Today Vsual recognton bascs part 2: Classfcaton and detecton Adrana s research
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationU-Pb Geochronology Practical: Background
U-Pb Geochronology Practcal: Background Basc Concepts: accuracy: measure of the dfference between an expermental measurement and the true value precson: measure of the reproducblty of the expermental result
More informationVQ widely used in coding speech, image, and video
at Scalar quantzers are specal cases of vector quantzers (VQ): they are constraned to look at one sample at a tme (memoryless) VQ does not have such constrant better RD perfomance expected Source codng
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationMarkov Chain Monte Carlo (MCMC), Gibbs Sampling, Metropolis Algorithms, and Simulated Annealing Bioinformatics Course Supplement
Markov Chan Monte Carlo MCMC, Gbbs Samplng, Metropols Algorthms, and Smulated Annealng 2001 Bonformatcs Course Supplement SNU Bontellgence Lab http://bsnuackr/ Outlne! Markov Chan Monte Carlo MCMC! Metropols-Hastngs
More informationPattern Classification
Pattern Classfcaton All materals n these sldes ere taken from Pattern Classfcaton (nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wley & Sons, 000 th the permsson of the authors and the publsher
More informationNatural Language Processing and Information Retrieval
Natural Language Processng and Informaton Retreval Support Vector Machnes Alessandro Moschtt Department of nformaton and communcaton technology Unversty of Trento Emal: moschtt@ds.untn.t Summary Support
More informationExplaining the Stein Paradox
Explanng the Sten Paradox Kwong Hu Yung 1999/06/10 Abstract Ths report offers several ratonale for the Sten paradox. Sectons 1 and defnes the multvarate normal mean estmaton problem and ntroduces Sten
More information} Often, when learning, we deal with uncertainty:
Uncertanty and Learnng } Often, when learnng, we deal wth uncertanty: } Incomplete data sets, wth mssng nformaton } Nosy data sets, wth unrelable nformaton } Stochastcty: causes and effects related non-determnstcally
More informationMaximal Margin Classifier
CS81B/Stat41B: Advanced Topcs n Learnng & Decson Makng Mamal Margn Classfer Lecturer: Mchael Jordan Scrbes: Jana van Greunen Corrected verson - /1/004 1 References/Recommended Readng 1.1 Webstes www.kernel-machnes.org
More informationExpectation Maximization Mixture Models HMMs
-755 Machne Learnng for Sgnal Processng Mture Models HMMs Class 9. 2 Sep 200 Learnng Dstrbutons for Data Problem: Gven a collecton of eamples from some data, estmate ts dstrbuton Basc deas of Mamum Lelhood
More informationEffects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012
Effects of Ignorng Correlatons When Computng Sample Ch-Square John W. Fowler February 6, 0 It can happen that ch-square must be computed for a sample whose elements are correlated to an unknown extent.
More information