Nuno Vasconcelos UCSD
|
|
- Jean Lawson
- 7 years ago
- Views:
Transcription
1 Bayesan parameter estmaton Nuno Vasconcelos UCSD 1
2 Maxmum lkelhood parameter estmaton n three steps: 1 choose a parametrc model for probabltes to make ths clear we denote the vector of parameters by Θ P X ( x; Θ note that ths means that Θ s NOT a random varable 2 assemble D = {x 1,..., x n } of examples drawn ndependently 3 select the parameters that maxmze the probablty of the data Θ * = arg max Θ P X = arg max log P Θ ( D; Θ P X ( D; Θ P X (D;Θ s the lkelhood of parameter Θ wth respect to the data 2
3 Least squares there are nterestng connectons between ML estmaton and least squares methods e.g. n a regresson problem we have two random varables X and Y a dataset of examples D = {(x 1,y 1, (x n,y n } a parametrc model of the form y = f (x; Θ + ε where Θ s a parameter vector, and ε a random varable that accounts for nose e.g. ε ~ N(0,σ 2 3
4 Least squares assumng that the famly of models s known, e.g. K f ( x ; Θ = f = 0 x ths s really just a problem of parameter estmaton where the data s dstrbuted as P Z X ( 2 z, f ( x ; Θ X ( D x ; Θ = G f, σ note that X s always known, and the mean s a functon of x and Θ n the homework, you wll show that Θ * = [ T 1 T Γ Γ] Γ y 4
5 Least squares where Γ = 1 K 1 K x1 M K K x n concluson: least squares estmaton s really just ML estmaton under the assumpton of Gaussan nose ndependent d sample ε ~ N(0,σ 2 once agan, probablty blt makes the assumptons explct t 5
6 Least squares soluton due to the connecton to parameter estmaton we can also talk about the qualty of the least squares soluton n partcular, we know that t s unbased varance goes to zero as the number of ponts ncreases t s the BLUE estmator for f(x;θ under the statstcal formulaton we can also see how the optmal estmator changes wth assumptons ML estmaton can also lead to (homework weghted least squares mnmzaton of L p norms robust estmators 6
7 Bayesan parameter estmaton Bayesan parameter estmaton s an alternatve framework for parameter estmaton t turns out that the dvson between Bayesan and ML methods s qute fundamental t stems from a dfferent way of nterpretng probabltes frequentst vs Bayesan there s a long debate about whch s best ths debate goes to the core of what probabltes blt mean to understand t, we have to dstngush two components the defnton of probablty (ths does not change the assessment of probablty (ths changes let s start wth a bref revew of the part that does not change 7
8 Probablty probablty s a language to deal wth processes that are non-determnstc examples: f I flp a con 100 tmes, how many can I expect to see heads? what s the weather gong to be lke tomorrow? are my stocks gong to be up or down? am I n front of a classroom or s ths just a pcture of t? 8
9 Sample space the most mportant concept s that of a sample space our process defnes a set of events these are the outcomes or states of the process example: we roll a par of dce call the value on the up face at the n th toss x n note that possble events such as odd number on second throw two sxes x 1 = 2 and x 2 = 6 can all be expressed as combnatons x 2 6 of the sample space events x 1 9
10 Sample space s the lst of possble events that satsfes the followng propertes: fnest gran: all possble dstngushable events are lsted separately mutually exclusve: f one event happens the other does not (f x 1 = 5 t cannot be anythng else collectvely exhaustve: any possble outcome can be expressed as unons of sample space events x x 1 mutually exclusve property smplfes the calculaton of the probablty of complex events collectvely exhaustve means that there s no possble outcome to whch h we cannot assgn a probablty blt 10
11 Probablty measure probablty of an event: number expressng the chance that the event wll be the outcome of the process probablty measure: satsfes three axoms P(A 0 for any event A P(unversal event = 1 f A B =, then P(A+B = P(A + P(B all of ths has to do wth the defnton of probablty 1 s the same under Bayes and frequentst vews what changes s how probabltes are assessed x x 1 11
12 Frequentst vew under the frequentst vew probabltes are relatve frequences I throw my dce n tmes n m of those the sum s 5 I say that P ( sum = 5 = m n ths s ntmately connected wth the ML method t s the ML estmate for the probablty of a Bernoull process wth states ( 5, everythng else makes sense when we have a lot of observatons no bas; decreasng varance; converges to true probablty blt 12
13 Problems many nstances where we do not have a large number of observatons consder the problem of crossng a street ths s a decson problem wth two states Y = 0: I am gong to get hurt Y = 1: I wll make t safely optmal decson computable by Bayes decson rule collect some measurements that are nformatve e.g. (X = {sze, dstance, speed} of ncomng cars collect examples under both states and estmate all probabltes somehow ths does not sound lke a great dea! 13
14 Problems under the frequentst vew you need to repeat an experment a large number of tmes to estmate any probabltes yet, people are very good at estmatng probabltes for problems n whch t s mpossble to set up such experments for example: wll I de f I jon the army? wll Democrats or Republcans wn the next electon? s there a God? wll I graduate n two years? to the pont where they make lfe-changng decsons based on these probablty estmates (enlstng n the army, etc. 14
15 Subjectve probablty ths motvates an alternatve defnton of probabltes note that ths has to do more wth how probabltes are assessed than wth the probablty defnton tself we stll have a sample space, a probablty measure, etc however the probabltes are not equated to relatve counts ths s usually referred to as subjectve probablty probabltes are degrees of belef on the outcomes of the experment they are ndvdual (vary from person to person they are not ratos of expermental outcomes e.g. for very relgous person P(god exsts ~ 1 for casual churchgoer P(god exsts ~ 0.8 (e.g. accepts evoluton, etc. for non-relgous P(god exsts ~ 0 15
16 Problems n practce, why do we care about ths? under the noton of subjectve probablty, the entre ML framework makes lttle sense there s a magc number that s estmated from the world and determnes our belefs to evaluate my estmates I have to run experments over and over agan and measure quanttes lke bas and varance ths s not how people behave, when we make estmates we attach a degree of confdence to them, wthout further experments there s only one model (the ML model for the probablty of the data, no multple explanatons there s no way to specfy that some models are, a pror, better than others 16
17 Bayesan parameter estmaton the man dfference wth respect to ML s that n the Bayesan case Θ s a random varable basc concepts tranng set D = {x 1,..., x n } of examples drawn ndependently probablty densty for observatons gven parameter P X Θ ( x pror dstrbuton b t for parameter confguratons P Θ ( that encodes pror belefs about them goal: to compute the posteror dstrbuton PΘ X D ( D 17
18 Bayes vs ML there are a number of sgnfcant dfferences between Bayesan and ML estmates D 1 : ML produces a number, the best estmate to measure ts goodness we need to measure bas and varance ths can only be done wth repeated experments Bayes produces a complete characterzaton of the parameter from the sngle dataset n addton to the most probable estmate, we obtan a characterzaton of the uncertanty lower uncertanty hgher uncertanty 18
19 Bayes vs ML D 2 : optmal estmate under ML there s one best estmate under Bayes there s no best estmate only a random varable that takes dfferent values wth dfferent probabltes techncally speakng, t makes no sense to talk about the best estmate D 3 : predctons remember that we do not really care about the parameters themselves they are needed only n the sense that they allow us to buld models that can be used to make predctons (e.g. the BDR unlke ML, Bayes uses ALL nformaton n the tranng set to make predctons 19
20 Bayes vs ML let s consder the BDR under the 0-1 loss and an ndependent sample D = {x 1,..., x n } ML-BDR: pck f two steps: fnd * * * ( x = arg max P ( x ; * where plug nto the BDR X Y = arg max P X Y P ( Y ( D, all nformaton not captured by * s lost, not used at decson tme 20
21 Bayes vs ML note that we know that nformaton s lost e.g. we can t even know how good of an estmate * s unless we run multple experments and measure bas/varance Bayesan BDR under the Bayesan framework, everythng s condtoned on the tranng data denote T = {X 1,..., X n } the set of random varables from whch the tranng sample D = {x 1,..., x n n} s drawn B-BDR: pck f * ( x = arg max PX Y, ( x, D P ( the decson s condtoned d on the entre tranng set T Y 21
22 Bayesan BDR to compute the condtonal probabltes, we use the margnalzaton equaton P X Y, T ( x, D ( ( PX Θ, Y, T x,, D PΘ Y, T, D = d note 1: when the parameter value s known, x no longer depends on T, e.g. XΘ ~ N(,σ 2 we can, smplfy equaton above nto P ( x, D ( ( PX Θ, Y x, PΘ Y, T D = d X Y, T, note 2: once agan can be done n two steps (per class fnd P ΘT (D compute P XY,T (x, D and plug nto the BDR no tranng nformaton s lost 22
23 Bayesan BDR n summary pck f * note: ( x = arg max PX Y, where P T ( x, D P Y ( ( x, D P ( x, P ( D d X Y, T X Y, Θ Θ Y, T, = as before the bottom equaton s repeated for each class hence, we can drop the dependence on the class and consder the more general problem of estmatng P ( x D P ( x P ( D d X T X Θ Θ T = 23
24 The predctve dstrbuton the dstrbuton ( x D P ( x P ( D d P = X T X Θ Θ T s known as the predctve dstrbuton ths follows from the fact that t allows us to predct the value of x gven ALL the nformaton avalable n the tranng set note that t t can also be wrtten as P ( x D E P ( x [ T D] X T = Θ T X Θ = snce each parameter value defnes a model ths s an expectaton over all possble models each model s weghted by ts posteror probablty, gven tranng data 24
25 The predctve dstrbuton suppose that 2 P ( x ~ N(,1 and P ( D ~ N( µ σ X Θ Θ T, P T ( D π P X T 1 ( x D weght π 2 Θ weght π 1 weght π 2 π σ µ 2 µ µ 1 µ 2 µ µ 1 the predctve dstrbuton s an average of all these Gaussans P ( x D P ( x P ( D d X T X Θ Θ T = 1 1 x 25
26 The predctve dstrbuton Bayes vs ML ML: pck one model Bayes: average all models are Bayesan predctons very dfferent than those of ML? they can be, unless the pror s narrow P T ( D Θ P T ( D Θ max max Bayes ~ ML very dfferent 26
27 The predctve dstrbuton hence, ML can be seen as a specal case of Bayes when you are very confdent about the model pckng one s good enough n comng lectures we wll see that f the sample s qute large, the pror tends to be narrow ntutve: gven a lot of tranng data, there s lttle uncertanty about what the model s Bayes can make a dfference when there s lttle data we have already seen that ths s the mportant case snce the varance of ML tends to go down as the sample ncreases overall Bayes regularzes the ML estmate when ths s uncertan converges to ML when there s a lot of certanty 27
28 MAP approxmaton ths sounds good, why use ML at all? the man problem wth Bayes s that the ntegral P can be qute nasty ( x D P ( x P ( D d = X T X Θ Θ T n practce one s frequently forced to use approxmatons one possblty s to do somethng smlar to ML,.e. pck only one model ths can be made to account for the pror by pckng the model that has the largest posteror probablty gven the tranng data ( D MAP = arg max P Θ T 28
29 MAP approxmaton ths can usually be computed snce arg max P ( D MAP P Θ T = D T Θ ( D ( = arg max P P and corresponds to approxmatng the pror by a delta functon centered at ts maxmum Θ ( D PΘ T ( D P T Θ MAP MAP 29
30 MAP approxmaton n ths case P X T the BDR becomes pck f * ( x D = PX Θ ( x δ ( MAP d d = P ( x X Θ ( x = arg max PX Y MAP ( MAP x ; ( ( D, P ( MAP where = arg max PT Y, Θ Θ Y P Y when compared to the ML ths has the advantage of stll accountng for the pror (although only approxmately 30
31 MAP vs ML ML-BDR pck f * * ( x = arg max P ( x ; where Bayes MAP-BDR pck f * ( x where * = MAP X Y arg max = arg max P X Y P X Y = arg max P P ( Y ( D, ( MAP x ; P ( T Y, Θ Y ( D, P ( the dfference s non-neglgble only when the dataset s small there are better alternatve approxmatons Θ Y 31
32 The Laplace approxmaton ths s a method for approxmatng any dstrbuton P X (x conssts of approxmatng P X (x by a Gaussan centered at ts peak let s assume that 1 Z ( x g( x P X = where g(x s an unormalzed dstrbuton (g(x > 0, for all x and Z the normalzaton constant Z = g ( x dx we make a Taylor seres approxmaton of g(x at ts maxmum x 0 32
33 Laplace approxmaton the Taylor expanson s log g( x = log g( x c ( o x x K (the frst-order term s zero because x 0 s a maxmum wth 2 c = x 2 log g( x x= x 0 x 0 P X (x and we approxmate g(x by an unormalzed Gaussan { ( 2 } c x x g' ( x = g( xo exp 2 and then compute the normalzaton constant 0 Z = g( x o 2π c 33
34 Laplace approxmaton ths can obvously be extended to the multvarate case the approxmaton s T log g( x = log g( xo ( x x ( 2 0 A x x0 wth A the Hessan of g(x at x 0 A j = 2 x x j log g( x and the normalzaton constant Z = g( x o ( 2 d 2π A 1 x= x 0 n physcs ths s also called a saddle-pont approxmaton 34
35 Laplace approxmaton note that the approxmaton can be made for the predctve dstrbuton ( x D = G( x, x Α P X T *, X T or for the parameter posteror n whch case ( D G(, A P Θ T = MAP, Θ T P ( x D P ( x G(, A d X T X Θ MAP, Θ T = ths s clearly superor to the MAP approxmaton ( x D = P Θ ( x δ ( d P X T X Θ MAP 35
36 Other methods there are two other man alternatves, when ths s not enough varatonal approxmatons samplng methods (Markov Chan Monte Carlo varatonal approxmatons consst of boundng the ntractable functon searchng for the best bound samplng methods consst desgnng a Markov chan that has the desred dstrbuton as ts equlbrum dstrbuton sample from ths chan samplng methods converge to the true dstrbuton but convergence s slow and hard to detect 36
37 37
What is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12
14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationHow To Calculate The Accountng Perod Of Nequalty
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationPrediction of Disability Frequencies in Life Insurance
Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but
More informationHow To Find The Dsablty Frequency Of A Clam
1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng 1, Fran Weber 1, Maro V. Wüthrch 2 Abstract: For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet
More informationRealistic Image Synthesis
Realstc Image Synthess - Combned Samplng and Path Tracng - Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationSketching Sampled Data Streams
Sketchng Sampled Data Streams Florn Rusu, Aln Dobra CISE Department Unversty of Florda Ganesvlle, FL, USA frusu@cse.ufl.edu adobra@cse.ufl.edu Abstract Samplng s used as a unversal method to reduce the
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationLogistic Regression. Steve Kroon
Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems
More informationCharacterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University
Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationEvaluating credit risk models: A critique and a new proposal
Evaluatng credt rsk models: A crtque and a new proposal Hergen Frerchs* Gunter Löffler Unversty of Frankfurt (Man) February 14, 2001 Abstract Evaluatng the qualty of credt portfolo rsk models s an mportant
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang
More informationLatent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006
Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationRisk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008
Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationApproximating Cross-validatory Predictive Evaluation in Bayesian Latent Variables Models with Integrated IS and WAIC
Approxmatng Cross-valdatory Predctve Evaluaton n Bayesan Latent Varables Models wth Integrated IS and WAIC Longha L Department of Mathematcs and Statstcs Unversty of Saskatchewan Saskatoon, SK, CANADA
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationMARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS
MARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS Tmothy J. Glbrde Assstant Professor of Marketng 315 Mendoza College of Busness Unversty of Notre Dame Notre Dame, IN 46556
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationRegression Models for a Binary Response Using EXCEL and JMP
SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationNPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationLecture 5,6 Linear Methods for Classification. Summary
Lecture 5,6 Lnear Methods for Classfcaton Rce ELEC 697 Farnaz Koushanfar Fall 2006 Summary Bayes Classfers Lnear Classfers Lnear regresson of an ndcator matrx Lnear dscrmnant analyss (LDA) Logstc regresson
More information1 Example 1: Axis-aligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationStatistical Methods to Develop Rating Models
Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationHow To Understand The Results Of The German Meris Cloud And Water Vapour Product
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationEstimation of Dispersion Parameters in GLMs with and without Random Effects
Mathematcal Statstcs Stockholm Unversty Estmaton of Dsperson Parameters n GLMs wth and wthout Random Effects Meng Ruoyan Examensarbete 2004:5 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationSTATISTICAL DATA ANALYSIS IN EXCEL
Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 14-01-013 petr.nazarov@crp-sante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationTitle Language Model for Information Retrieval
Ttle Language Model for Informaton Retreval Rong Jn Language Technologes Insttute School of Computer Scence Carnege Mellon Unversty Alex G. Hauptmann Computer Scence Department School of Computer Scence
More informationExhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of Regresson-Based Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationOptimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account
Amercan J. of Engneerng and Appled Scences (): 8-6, 009 ISSN 94-700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a Day-Ahead Electrcty Market wth Rsk Management Taken nto Account
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationSingle and multiple stage classifiers implementing logistic discrimination
Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul - PUCRS Av. Ipranga,
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationBayesian Cluster Ensembles
Bayesan Cluster Ensembles Hongjun Wang 1, Hanhua Shan 2 and Arndam Banerjee 2 1 Informaton Research Insttute, Southwest Jaotong Unversty, Chengdu, Schuan, 610031, Chna 2 Department of Computer Scence &
More informationApplied Research Laboratory. Decision Theory and Receiver Design
Decson Theor and Recever Desgn Sgnal Detecton and Performance Estmaton Sgnal Processor Decde Sgnal s resent or Sgnal s not resent Nose Nose Sgnal? Problem: How should receved sgnals be rocessed n order
More informationHedging Interest-Rate Risk with Duration
FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationMachine Learning and Data Mining Lecture Notes
Machne Learnng and Data Mnng Lecture Notes CSC 411/D11 Computer Scence Department Unversty of Toronto Verson: February 6, 2012 Copyrght c 2010 Aaron Hertzmann and Davd Fleet CONTENTS Contents Conventons
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More informationSupport vector domain description
Pattern Recognton Letters 20 (1999) 1191±1199 www.elsever.nl/locate/patrec Support vector doman descrpton Davd M.J. Tax *,1, Robert P.W. Dun Pattern Recognton Group, Faculty of Appled Scence, Delft Unversty
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationData Broadcast on a Multi-System Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819-840 (2008) Data Broadcast on a Mult-System Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationBERNSTEIN POLYNOMIALS
On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationInverse Modeling of Tight Gas Reservoirs
Inverse Modelng of Tght Gas Reservors Der Fakultät für Geowssenschaften, Geotechnk und Bergbau der Technschen Unverstät Bergakademe Freberg engerechte Dssertaton Zur Erlangung des akademschen Grades Doktor-Ingeneur
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationAn Interest-Oriented Network Evolution Mechanism for Online Communities
An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationThe Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationRisk Model of Long-Term Production Scheduling in Open Pit Gold Mining
Rsk Model of Long-Term Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://www-bcf.usc.edu/
More informationTraffic-light a stress test for life insurance provisions
MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationChapter XX More advanced approaches to the analysis of survey data. Gad Nathan Hebrew University Jerusalem, Israel. Abstract
Household Sample Surveys n Developng and Transton Countres Chapter More advanced approaches to the analyss of survey data Gad Nathan Hebrew Unversty Jerusalem, Israel Abstract In the present chapter, we
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationVasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio
Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of
More informationHow To Solve A Problem In A Powerline (Powerline) With A Powerbook (Powerbook)
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationStress test for measuring insurance risks in non-life insurance
PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance
More information1 De nitions and Censoring
De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationAN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE
AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE Yu-L Huang Industral Engneerng Department New Mexco State Unversty Las Cruces, New Mexco 88003, U.S.A. Abstract Patent
More informationMethod for assessment of companies' credit rating (AJPES S.BON model) Short description of the methodology
Method for assessment of companes' credt ratng (AJPES S.BON model) Short descrpton of the methodology Ljubljana, May 2011 ABSTRACT Assessng Slovenan companes' credt ratng scores usng the AJPES S.BON model
More informationNON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationHow To Know The Components Of Mean Squared Error Of Herarchcal Estmator S
S C H E D A E I N F O R M A T I C A E VOLUME 0 0 On Mean Squared Error of Herarchcal Estmator Stans law Brodowsk Faculty of Physcs, Astronomy, and Appled Computer Scence, Jagellonan Unversty, Reymonta
More informationFace Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More informationWhen do data mining results violate privacy? Individual Privacy: Protect the record
When do data mnng results volate prvacy? Chrs Clfton March 17, 2004 Ths s jont work wth Jashun Jn and Murat Kantarcıoğlu Indvdual Prvacy: Protect the record Indvdual tem n database must not be dsclosed
More informationEfficient Reinforcement Learning in Factored MDPs
Effcent Renforcement Learnng n Factored MDPs Mchael Kearns AT&T Labs mkearns@research.att.com Daphne Koller Stanford Unversty koller@cs.stanford.edu Abstract We present a provably effcent and near-optmal
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More information