Learning the structure of Bayesian belief networks
|
|
|
- Michael Wood
- 5 years ago
- Views:
Transcription
1 Lectue 17 Leanng the stuctue of Bayesan belef netwoks Mlos Hauskecht 5329 Sennott Squae Leanng of BBN Leanng. Leanng of paametes of condtonal pobabltes Leanng of the netwok stuctue Vaables: Obsevable values pesent n evey data sample Hdden they values ae neve obseved n data Mssng values values sometmes pesent, sometmes not Next: All vaables ae obsevable 1. Leanng of paametes of BBN 2. Leanng of the model (BBN stuctue
2 Leanng of BBN paametes. Example. Example: Pneumona Pneumona?? HWBC Pneum Pn???? eve Hgh WBC Palen Pneum eve Pneum Pneum??? Leanng of BBN paametes. Example. Data D (dffeent patent cases: Pal ev Cou HWB Pneu eve Pneumona Hgh WBC
3 Estmates of paametes of BBN Much lke multple con toss o oll of a dce poblems. A smalle leanng poblem coesponds to the leanng of exactly one condtonal dstbuton Example: eve Pneumona Poblem: How to pck the data to lean? Leanng of BBN paametes. Example. Lean: eve Pneumona Step 1: Select data ponts wth Pneumona Pal ev Cou HWB Pneu eve Pneumona Hgh WBC
4 Leanng of BBN paametes. Example. Lean: Step 1: eve Pneumona Ignoe the est Pal ev Cou HWB Pneu eve Pneumona Hgh WBC Leanng of BBN paametes. Example. Lean: eve Pneumona Step 2: Select values of the andom vaable defnng the dstbuton of eve Pal ev Cou HWB Pneu eve Pneumona Hgh WBC
5 Leanng of BBN paametes. Example. Lean: eve Pneumona Step 2: Ignoe the est ev eve Pneumona Hgh WBC Leanng of BBN paametes. Example. Lean: eve Pneumona Step 3a: Leanng the ML estmate ev eve Pneumona Hgh WBC eve Pneumona
6 Leanng of BBN paametes. Bayesan leanng. Lean: eve Pneumona Step 3b: Leanng the Bayesan estmate Assume the po ev θ eve Pneumona ~ Beta(3,4 eve Pneumona Hgh WBC Posteo: Pneumona ~ Beta(6,6 θ eve Model selecton BBN has two components: Stuctue of the netwok (models condtonal ndependences A set of paametes (condtonal chld-paent dstbutons We aleady know how to lean the paametes fo the fxed stuctue But how to lean the stuctue of the BBN? Alam? Buglay Alam Quake Buglay Quake John May John May
7 Leanng the stuctue Ctea we can choose to scoe the stuctue S Magnal lkelhood maxmze P ( D S, ξ ξ - epesents the po knowledge Maxmum posteo pobablty maxmze P ( S D, ξ P ( S D, ξ P ( D S, ξ P ( S P ( D ξ ξ How to compute magnal lkelhood P ( D S, ξ? Leanng of BBNs Notaton: anges ove all possble vaables 1,..,n j1,..,q anges ove all possble paent combnatons k1,.., anges ove all possble vaable values - paametes of the BBN j s a vecto of epesentng paametes of the condtonal pobablty dstbuton; such that 1 N N j j - a numbe of nstances n the dataset whee paents of vaable X take on values j and X has value k N - po counts (paametes of Beta and Dchlet pos
8 Magnal lkelhood Integate ove all possble paamete settngs P ( D S, ξ D S,, ξ p( S, ξ d Usng the assumpton of paamete and sample ndependence P ( D S, ξ n q j 1 j 1 j j We can use log-lkelhood scoe nstead log D S, ξ n q j log + log j Scoe s decomposable along vaables!!! j j k om the d assumpton: D N h 1 1 h x paents, Let numbe of values that attbute x can take q numbe of possble paent combnatons N numbe of cases n D whee x has value k and paents wth values j. n n h x k paents n q j k Magnal lkelhood q j k P ( j, θ N N
9 om paamete ndependence Pos fo p( j ξ j ( j1,..., j s a vecto of paametes; we use a Dchlet dstbuton wth paametes to epesent t P ( j 1 ξ j1,..., j ξ Dchlet( j,..., j Magnal lkelhood n p( ξ p( ξ 1 j 1 q 1 j Combne thngs togethe: P ( D S P ( D S, P ( S d Γ n q ( N j k Γ n q ( j j Magnal lkelhood N d a j k 1 j j n q d
10 An altenatve way to compute the magnal lkelhood Integate ove all possble paamete settngs Posteo of paametes, gven data and the stuctue ck Gves the soluton P ( D ξ D, ξ p( ξ d D ξ D, ξ p( ξ p( D, ξ D ξ D, ξ p( ξ D ξ p( D, ξ n q j 1 j 1 j j Leanng the stuctue Lkelhood of data fo the BBN (stuctue and paametes D, ξ measues the goodness of ft of the BBN to data Magnal lkelhood (fo the stuctue only P ( D S, ξ Does not measue only a goodness of ft. It s: dffeent fo stuctues of dffeent complexty Incopoates pefeences towads smple stuctues, mplements Occam s azo!!!!
11 Occam s Razo Why thee s a pefeence towads smple stuctues? Rewte magnal lkelhood as D S, ξ We know that D S,, ξ p( S, ξ d p( S, ξ d p( ξ d 1 Intepetaton: n moe complex stuctues thee ae moe ways paametes can be set badly he numeato: count of good assgnments he denomnato: count of all assgnments Appoxmatons of pobablstc scoes Appoxmatons of the magnal lkelhood and posteo scoes Infomaton based measues Akake cteon Bayesan nfomaton cteon (BIC Mnmum descpton length (MDL Reflect the tadeoff between the ft to data and pefeence towads smple stuctues Example: Akake cteon. Maxmze: scoe( S log D ML, ξ compl(s Bayesan nfomaton cteon (BIC Maxmze: 1 scoe( S log D ML, ξ compl(s logn 2
12 Optmzng the stuctue ndng the best stuctue s a combnatoal optmzaton poblem A good featue: the scoe s decomposable along vaables: n q Γ Γ + ( j + ( N log P ( D S, ξ log log 1 j 1 Γ ( j j Γ ( Algothm dea: Seach the space of stuctues usng local changes (addtons and deletons of a lnk Advantage: we do not have to compute the whole scoe fom scatch Recompute the patal scoe fo the affected vaable Optmzng the stuctue. Algothms Geedy seach Stat fom the stuctue wth no lnks Add a lnk that yelds the best scoe mpovement Metopols algothm (wth smulated annealng Local addtons and deletons Avods beng tapped n local optmal
Learning Bayesian belief networks
Lectue 6 Leag Bayesa belef etwoks Mlos Hauskecht mlos@cs.ptt.edu 5329 Seott Squae Admstato Mdtem: Wedesday, Mach 7, 2004 I class Closed book Mateal coveed by Spg beak, cludg ths lectue Last yea mdtem o
Machine learning: Density estimation
CS 70 Foundatons of AI Lecture 3 Machne learnng: ensty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square ata: ensty estmaton {.. n} x a vector of attrbute values Objectve: estmate the model of
Learning Bayesian belief networks
Lecture 4 Learning Bayesian belief networks Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Administration Midterm: Monday, March 7, 2003 In class Closed book Material covered by Wednesday, March
CS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements
CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before
Bayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems
Dept. of Math. Unvesty of Oslo Statstcal Reseach Repot No 3 ISSN 0806 3842 June 2010 Bayesan Assessment of Avalabltes and Unavalabltes of Multstate Monotone Systems Bent Natvg Jøund Gåsemy Tond Retan June
Detection and Estimation Theory
ESE 54 Detecton and Etmaton Theoy Joeph A. O Sullvan Samuel C. Sach Pofeo Electonc Sytem and Sgnal Reeach Laboatoy Electcal and Sytem Engneeng Wahngton Unvety 411 Jolley Hall 314-935-4173 (Lnda anwe) jao@wutl.edu
Machine Learning. Spectral Clustering. Lecture 23, April 14, Reading: Eric Xing 1
Machne Leanng -7/5 7/5-78, 78, Spng 8 Spectal Clusteng Ec Xng Lectue 3, pl 4, 8 Readng: Ec Xng Data Clusteng wo dffeent ctea Compactness, e.g., k-means, mxtue models Connectvty, e.g., spectal clusteng
The Greatest Deviation Correlation Coefficient and its Geometrical Interpretation
By Rudy A. Gdeon The Unvesty of Montana The Geatest Devaton Coelaton Coeffcent and ts Geometcal Intepetaton The Geatest Devaton Coelaton Coeffcent (GDCC) was ntoduced by Gdeon and Hollste (987). The GDCC
Chapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
CIS587 - Artificial Intellgence. Bayesian Networks CIS587 - AI. KB for medical diagnosis. Example.
CIS587 - Artfcal Intellgence Bayesan Networks KB for medcal dagnoss. Example. We want to buld a KB system for the dagnoss of pneumona. Problem descrpton: Dsease: pneumona Patent symptoms (fndngs, lab tests):
P 365. r r r )...(1 365
SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty
an application to HRQoL
AlmaMate Studoum Unvesty of Bologna A flexle IRT Model fo health questonnae: an applcaton to HRQoL Seena Boccol Gula Cavn Depatment of Statstcal Scence, Unvesty of Bologna 9 th Intenatonal Confeence on
Space of ML Problems. CSE 473: Artificial Intelligence. Parameter Estimation and Bayesian Networks. Learning Topics
/7/7 CSE 73: Artfcal Intellgence Bayesan - Learnng Deter Fox Sldes adapted from Dan Weld, Jack Breese, Dan Klen, Daphne Koller, Stuart Russell, Andrew Moore & Luke Zettlemoyer What s Beng Learned? Space
Dirichlet Mixture Priors: Inference and Adjustment
Dchlet Mxtue Pos: Infeence and Adustment Xugang Ye (Wokng wth Stephen Altschul and Y Kuo Yu) Natonal Cante fo Botechnology Infomaton Motvaton Real-wold obects Independent obsevatons Categocal data () (2)
Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
The Backpropagation Algorithm
The Backpopagaton Algothm Achtectue of Feedfowad Netwok Sgmodal Thehold Functon Contuctng an Obectve Functon Tanng a one-laye netwok by teepet decent Tanng a two-laye netwok by teepet decent Copyght Robet
Tian Zheng Department of Statistics Columbia University
Haplotype Tansmsson Assocaton (HTA) An "Impotance" Measue fo Selectng Genetc Makes Tan Zheng Depatment of Statstcs Columba Unvesty Ths s a jont wok wth Pofesso Shaw-Hwa Lo n the Depatment of Statstcs at
Department 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/
Clustering Techniques
Clusteng Tehnques Refeenes: Beln Chen 2003. Moden Infomaton Reteval, haptes 5, 7 2. Foundatons of Statstal Natual Language Poessng, Chapte 4 Clusteng Plae smla obets n the same goup and assgn dssmla obets
If there are k binding constraints at x then re-label these constraints so that they are the first k constraints.
Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then
CS649 Sensor Networks IP Track Lecture 3: Target/Source Localization in Sensor Networks
C649 enso etwoks IP Tack Lectue 3: Taget/ouce Localaton n enso etwoks I-Jeng Wang http://hng.cs.jhu.edu/wsn06/ png 006 C 649 Taget/ouce Localaton n Weless enso etwoks Basc Poblem tatement: Collaboatve
Correspondence Analysis & Related Methods
Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense
Multistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 58-64 007 ISSN 549-3644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm
EM and Structure Learning
EM and Structure Learnng Le Song Machne Learnng II: Advanced Topcs CSE 8803ML, Sprng 2012 Partally observed graphcal models Mxture Models N(μ 1, Σ 1 ) Z X N N(μ 2, Σ 2 ) 2 Gaussan mxture model Consder
Khintchine-Type Inequalities and Their Applications in Optimization
Khntchne-Type Inequaltes and The Applcatons n Optmzaton Anthony Man-Cho So Depatment of Systems Engneeng & Engneeng Management The Chnese Unvesty of Hong Kong ISDS-Kolloquum Unvestaet Wen 29 June 2009
CIS526: 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
MULTILAYER PERCEPTRONS
Last updated: Nov 26, 2012 MULTILAYER PERCEPTRONS Outline 2 Combining Linea Classifies Leaning Paametes Outline 3 Combining Linea Classifies Leaning Paametes Implementing Logical Relations 4 AND and OR
8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
4 SingularValue Decomposition (SVD)
/6/00 Z:\ jeh\self\boo Kannan\Jan-5-00\4 SVD 4 SngulaValue Decomposton (SVD) Chapte 4 Pat SVD he sngula value decomposton of a matx s the factozaton of nto the poduct of thee matces = UDV whee the columns
A. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
Optimization Algorithms for System Integration
Optmzaton Algothms fo System Integaton Costas Papadmtou 1, a and Evaggelos totsos 1,b 1 Unvesty of hessaly, Depatment of Mechancal and Industal Engneeng, Volos 38334, Geece a costasp@uth.g, b entotso@uth.g
Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork
Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html June June To be dscussed on Tuesday of Week-II. Classwok. Deve all
Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
INTRODUCTION. consider the statements : I there exists x X. f x, such that. II there exists y Y. such that g y
INRODUCION hs dssetaton s the eadng of efeences [1], [] and [3]. Faas lemma s one of the theoems of the altenatve. hese theoems chaacteze the optmalt condtons of seveal mnmzaton poblems. It s nown that
The geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
2E 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
Announcements. Stereo (Part 3) Summary of Stereo Constraints. Features on same epipolar line. Stereo matching. Truco Fig. 7.5
Announcements Steeo (Pat ) Homewok s due Nov, :59 PM Readng: Chapte 7: Steeopss CSE 5A Lectue 0 Featues on same eppola lne Summay of Steeo Constants CONSRAIN BRIEF DESCRIPION -D Eppola Seach Abtay mages
Using DP for hierarchical discretization of continuous attributes. Amit Goyal (31 st March 2008)
Usng DP fo heachcal dscetzaton of contnos attbtes Amt Goyal 31 st Mach 2008 Refeence Chng-Cheng Shen and Yen-Lang Chen. A dynamc-pogammng algothm fo heachcal dscetzaton of contnos attbtes. In Eopean Jonal
PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
3. A Review of Some Existing AW (BT, CT) Algorithms
3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms
Clustering. Outline. Supervised vs. Unsupervised Learning. Clustering. Clustering Example. Applications of Clustering
Clusteng CS478 Mahne Leanng Spng 008 Thosten Joahms Conell Unvesty Outlne Supevsed vs. Unsupevsed Leanng Heahal Clusteng Heahal Agglomeatve Clusteng (HAC) Non-Heahal Clusteng K-means EM-Algothm Readng:
Integral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
PHY126 Summer Session I, 2008
PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment
Backward Haplotype Transmission Association (BHTA) Algorithm. Tian Zheng Department of Statistics Columbia University. February 5 th, 2002
Backwad Haplotype Tansmsson Assocaton (BHTA) Algothm A Fast ult-pont Sceenng ethod fo Complex Tats Tan Zheng Depatment of Statstcs Columba Unvesty Febuay 5 th, 2002 Ths s a jont wok wth Pofesso Shaw-Hwa
Set of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
The Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
Event Shape Update. T. Doyle S. Hanlon I. Skillicorn. A. Everett A. Savin. Event Shapes, A. Everett, U. Wisconsin ZEUS Meeting, October 15,
Event Shape Update A. Eveett A. Savn T. Doyle S. Hanlon I. Skllcon Event Shapes, A. Eveett, U. Wsconsn ZEUS Meetng, Octobe 15, 2003-1 Outlne Pogess of Event Shapes n DIS Smla to publshed pape: Powe Coecton
Complex atoms and the Periodic System of the elements
Complex atoms and the Peodc System of the elements Non-cental foces due to electon epulson Cental feld appoxmaton electonc obtals lft degeneacy of l E n l = R( hc) ( n δ ) l Aufbau pncple Lectue Notes
SURVEY OF APPROXIMATION ALGORITHMS FOR SET COVER PROBLEM. Himanshu Shekhar Dutta. Thesis Prepared for the Degree of MASTER OF SCIENCE
SURVEY OF APPROXIMATION ALGORITHMS FOR SET COVER PROBLEM Hmanshu Shekha Dutta Thess Pepaed fo the Degee of MASTER OF SCIENCE UNIVERSITY OF NORTH TEXAS Decembe 2009 APPROVED: Fahad Shahokh, Mao Pofesso
A Brief Guide to Recognizing and Coping With Failures of the Classical Regression Assumptions
A Bef Gude to Recognzng and Copng Wth Falues of the Classcal Regesson Assumptons Model: Y 1 k X 1 X fxed n epeated samples IID 0, I. Specfcaton Poblems A. Unnecessay explanatoy vaables 1. OLS s no longe
A NOTE ON ELASTICITY ESTIMATION OF CENSORED DEMAND
Octobe 003 B 003-09 A NOT ON ASTICITY STIATION OF CNSOD DAND Dansheng Dong an Hay. Kase Conell nvesty Depatment of Apple conomcs an anagement College of Agcultue an fe Scences Conell nvesty Ithaca New
Multilayer neural networks
Lecture Multlayer neural networks Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Mdterm exam Mdterm Monday, March 2, 205 In-class (75 mnutes) closed book materal covered by February 25, 205 Multlayer
RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S
Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets
MACHINE LEARNING. Mistake and Loss Bound Models of Learning
Iowa State Unvesty MACHINE LEARNING Vasant Honava Bonfomatcs and Computatonal Bology Pogam Cente fo Computatonal Intellgence, Leanng, & Dscovey Iowa State Unvesty honava@cs.astate.edu www.cs.astate.edu/~honava/
The 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
6.6 The Marquardt Algorithm
6.6 The Mqudt Algothm lmttons of the gdent nd Tylo expnson methods ecstng the Tylo expnson n tems of ch-sque devtves ecstng the gdent sech nto n tetve mtx fomlsm Mqudt's lgothm utomtclly combnes the gdent
Physics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
Finite 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
Weighted Infinite Relational Model for Network Data
Jounal of Communcatons Vol. 10, No. 6, June 2015 Weghted Infnte Relatonal Model fo Netwo Data Xaojuan Jang and Wensheng Zhang Insttute of Automaton, Unvesty of Chnese Academy of Scences, Bejng 100190,
Optimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis
Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each
APPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
Momentum is conserved if no external force
Goals: Lectue 13 Chapte 9 v Employ consevation of momentum in 1 D & 2D v Examine foces ove time (aka Impulse) Chapte 10 v Undestand the elationship between motion and enegy Assignments: l HW5, due tomoow
UNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
xp(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
Temporal-Difference Learning
.997 Decision-Making in Lage-Scale Systems Mach 17 MIT, Sping 004 Handout #17 Lectue Note 13 1 Tempoal-Diffeence Leaning We now conside the poblem of computing an appopiate paamete, so that, given an appoximation
= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
Error Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
N = N t ; t 0. N is the number of claims paid by the
Iulan MICEA, Ph Mhaela COVIG, Ph Canddate epatment of Mathematcs The Buchaest Academy of Economc Studes an CECHIN-CISTA Uncedt Tac Bank, Lugoj SOME APPOXIMATIONS USE IN THE ISK POCESS OF INSUANCE COMPANY
Markov 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
State Estimation. Ali Abur Northeastern University, USA. Nov. 01, 2017 Fall 2017 CURENT Course Lecture Notes
State Estmaton Al Abu Notheasten Unvesty, USA Nov. 0, 07 Fall 07 CURENT Couse Lectue Notes Opeatng States of a Powe System Al Abu NORMAL STATE SECURE o INSECURE RESTORATIVE STATE EMERGENCY STATE PARTIAL
Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010
Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton
Course 395: Machine Learning - Lectures
Course 395: Machne Learnng - Lectures Lecture 1-2: Concept Learnng (M. Pantc Lecture 3-4: Decson Trees & CC Intro (M. Pantc Lecture 5-6: Artfcal Neural Networks (S.Zaferou Lecture 7-8: Instance ased Learnng
CSC321 Tutorial 9: Review of Boltzmann machines and simulated annealing
CSC321 Tutoral 9: Revew of Boltzmann machnes and smulated annealng (Sldes based on Lecture 16-18 and selected readngs) Yue L Emal: yuel@cs.toronto.edu Wed 11-12 March 19 Fr 10-11 March 21 Outlne Boltzmann
Machine Learning 4771
Machne Leanng 4771 Instucto: Tony Jebaa Topc 6 Revew: Suppot Vecto Machnes Pmal & Dual Soluton Non-sepaable SVMs Kenels SVM Demo Revew: SVM Suppot vecto machnes ae (n the smplest case) lnea classfes that
Life-long Informative Paths for Sensing Unknown Environments
Lfe-long Infomatve Paths fo Sensng Unknown Envonments Danel E. Solteo Mac Schwage Danela Rus Abstact In ths pape, we have a team of obots n a dynamc unknown envonment and we would lke them to have accuate
Lecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
Optimal System for Warm Standby Components in the Presence of Standby Switching Failures, Two Types of Failures and General Repair Time
Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Optmal System fo Wam Standby omponents n the esence of Standby Swtchng Falues, Two Types of Falues and Geneal Repa Tme Mohamed Salah EL-Shebeny
Additional File 1 - Detailed explanation of the expression level CPD
Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor
Rotational Motion. Lecture 6. Chapter 4. Physics I. Course website:
Lectue 6 Chapte 4 Physics I Rotational Motion Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi Today we ae going to discuss: Chapte 4: Unifom Cicula Motion: Section 4.4 Nonunifom Cicula
On Maneuvering Target Tracking with Online Observed Colored Glint Noise Parameter Estimation
Wold Academy of Scence, Engneeng and Technology 6 7 On Maneuveng Taget Tacng wth Onlne Obseved Coloed Glnt Nose Paamete Estmaton M. A. Masnad-Sha, and S. A. Banan Abstact In ths pape a compehensve algothm
Energy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
A Tutorial on Low Density Parity-Check Codes
A Tutoal on Low Densty Paty-Check Codes Tuan Ta The Unvesty of Texas at Austn Abstact Low densty paty-check codes ae one of the hottest topcs n codng theoy nowadays. Equpped wth vey fast encodng and decodng
THE REGRESSION MODEL OF TRANSMISSION LINE ICING BASED ON NEURAL NETWORKS
The 4th Intenatonal Wokshop on Atmosphec Icng of Stuctues, Chongqng, Chna, May 8 - May 3, 20 THE REGRESSION MODEL OF TRANSMISSION LINE ICING BASED ON NEURAL NETWORKS Sun Muxa, Da Dong*, Hao Yanpeng, Huang
Efficiency of the principal component Liu-type estimator in logistic
Effcency of the pncpal component Lu-type estmato n logstc egesson model Jbo Wu and Yasn Asa 2 School of Mathematcs and Fnance, Chongqng Unvesty of Ats and Scences, Chongqng, Chna 2 Depatment of Mathematcs-Compute
Expectation 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
Problem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
Bayesian Uncertainty Quantification and Propagation in Large-Order Finite Element Models using CMS Techniques
EACS 5 th Euopean Confeence on Stuctual Contol Genoa, Italy 8- June Pape No. # 9 Bayesan Uncetanty Quantfcaton and Popagaton n Lage-Ode Fnte Element Models usng CMS Technques Costas PAPADIMITRIOU*, Dmta-Chstna
Lecture 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
To Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
Order Reduction of Continuous LTI Systems using Harmony Search Optimization with Retention of Dominant Poles
Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles a Akhlesh
The M 2 -tree: Processing Complex Multi-Feature Queries with Just One Index
The M -tee: Pocessng Complex Mult-Featue Quees wth Just ne Index Paolo Cacca, Maco Patella DEIS - CSITE-CNR, Unvesty of Bologna, Italy fpcacca,mpatellag@des.unbo.t Abstact Motvated by the needs fo effcent
Maximum Likelihood Estimation
Maxmum Lkelhood Estmaton INFO-2301: Quanttatve Reasonng 2 Mchael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quanttatve Reasonng 2 Paul and Boyd-Graber Maxmum Lkelhood Estmaton 1 of 9 Why MLE?
An Experiment/Some Intuition (Fall 2006): Lecture 18 The EM Algorithm heads coin 1 tails coin 2 Overview Maximum Likelihood Estimation
An Experment/Some Intuton I have three cons n my pocket, 6.864 (Fall 2006): Lecture 18 The EM Algorthm Con 0 has probablty λ of heads; Con 1 has probablty p 1 of heads; Con 2 has probablty p 2 of heads
Random Variables and Probability Distribution Random Variable
Random Vaiables and Pobability Distibution Random Vaiable Random vaiable: If S is the sample space P(S) is the powe set of the sample space, P is the pobability of the function then (S, P(S), P) is called
Information Retrieval
Clusteng Technques fo Infomaton Reteval Beln Chen Depatment t of Compute Scence & Infomaton Engneeng Natonal Tawan Nomal Unvesty Refeences:. Chstophe D. Mannng, Pabhaa Raghavan and Hnch Schütze, Intoducton
Maximum Likelihood Directed Enumeration Method in Piecewise-Regular Object Recognition
Maxmum Lkelhood Dected Enumeaton Method n Pecewse-Regula Obect Recognton Andey Savchenko Abstact We exploe the poblems of classfcaton of composte obect (mages, speech sgnals wth low numbe of models pe
Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
Vector Control. Application to Induction Motor Control. DSP in Motion Control - Seminar
Vecto Contol Application to Induction Moto Contol Vecto Contol - Pinciple The Aim of Vecto Contol is to Oient the Flux Poducing Component of the Stato Cuent to some Suitable Flux Vecto unde all Opeating