Bayes Nets. CS 188: Artificial Intelligence Spring Example: Alarm Network. Building the (Entire) Joint
|
|
- Felicia Clare Nichols
- 6 years ago
- Views:
Transcription
1 C 188: Aificial Inelligence ping 2008 Bayes Nes 2/5/08, 2/7/08 Dan Klein UC Bekeley Bayes Nes A Bayes ne is an efficien encoding of a pobabilisic model of a domain Quesions we can ask: Infeence: given a fixed BN, wa is P( e)? epesenaion: given a fixed BN, wa kinds of disibuions can i encode? Modeling: wa BN is mos appopiae fo a given domain? Bayes Ne Bayes Ne emanics A Bayes ne: A se of nodes, one pe vaiable A dieced, acyclic gap A condiional disibuion of eac vaiable condiioned on is paens (e paameesθ) A 1 A n emanics: A BN defines a join pobabiliy disibuion ove is vaiables: Building e (Enie) Join : Alam Newok We can ake a Bayes ne and build any eny fom e full join disibuion i encodes ypically, ee s no eason o build ALL of i We build wa we need on e fly o empasize: evey BN ove a domain implicily epesens some join disibuion ove a domain, bu is specified by local pobabiliies 1
2 ize of a Bayes Ne How big is a join disibuion ove N Boolean vaiables? How big is an N-node ne if nodes ave k paens? Bo give you e powe o calculae BNs: Huge space savings! Also easie o elici local CPs Also uns ou o be fase o answe queies (nex class) Bayes Nes o fa: ow a Bayes ne encodes a join disibuion Nex: ow o answe queies abou a disibuion Key idea: condiional independence Las class: assembled BNs using an inuiive noion of condiional independence as causaliy oday: fomalize ese ideas Main goal: answe queies abou condiional independence and influence Afe a: ow o answe numeical queies (infeence) Condiional Independence eminde: independence and ae independen if : Independence Fo is gap, you can fiddle wi θ (e CPs) all you wan, bu you won be able o epesen any disibuion in wic e flips ae dependen! and ae condiionally independen given Z 1 2 (Condiional) independence is a popey of a disibuion All disibuions opology Limis Disibuions Independence in a BN Given some gap opology G, only ceain join disibuions can be encoded e gap sucue guaanees ceain (condiional) independences (ee mig be moe independence) Adding acs inceases e se of disibuions, bu as seveal coss Z Z Z Impoan quesion abou a BN: Ae wo nodes independen given ceain evidence? If yes, can calculae using algeba (eally edious) If no, can pove wi a coune example : Z Quesion: ae and Z independen? Answe: no necessaily, we ve seen examples oewise: low pessue causes ain wic causes affic. can influence Z, Z can influence (via ) Addendum: ey could be independen: ow? 2
3 Causal Cains Common Cause is configuaion is a causal cain Z : Low pessue : ain Z: affic Anoe basic configuaion: wo effecs of e same cause Ae and Z independen? Ae and Z independen given? Z Is independen of Z given?! Evidence along e cain blocks e influence Obseving e cause blocks influence beween effecs.! : Pojec due : Newsgoup busy Z: Lab full Common Effec e Geneal Case Las configuaion: wo causes of one effec (v-sucues) Ae and Z independen? : emembe e ballgame and e ain causing affic, no coelaion? ill need o pove ey mus be (omewok) Ae and Z independen given? No: emembe a seeing affic pu e ain and e ballgame in compeiion? is is backwads fom e oe cases Obseving e effec enables influence beween effecs. Z : aining Z: Ballgame : affic Any complex example can be analyzed using ese ee canonical cases Geneal quesion: in a given BN, ae wo vaiables independen (given evidence)? oluion: gap seac! eacabiliy eacabiliy (e Bayes Ball) ecipe: sade evidence nodes Aemp 1: if wo nodes ae conneced by an undieced pa no blocked by a saded node, ey ae condiionally independen Almos woks, bu no quie Wee does i beak? Answe: e v-sucue a doesn coun as a link in a pa unless saded D L B Coec algoim: ade in evidence a a souce node y o eac age by seac aes: pai of (node, pevious sae ) uccesso funcion: unobseved: o any cild o any paen if coming fom a cild obseved: Fom paen o paen If you can eac a node, i s condiionally independen of e sa node given evidence 3
4 L B D Causaliy? Vaiables: : aining : affic D: oof dips : I m sad Quesions: D Wen Bayes nes eflec e ue causal paens: Ofen simple (nodes ave fewe paens) Ofen easie o ink abou Ofen easie o elici fom expes BNs need no acually be causal omeimes no causal ne exiss ove e domain E.g. conside e vaiables affic and Dips End up wi aows a eflec coelaion, no causaion Wa do e aows eally mean? opology may appen o encode causal sucue opology only guaaneed o encode condiional independencies : affic Basic affic ne Le s muliply ou e join : evese affic evese causaliy? 1/4 3/4 3/4 1/4 3/16 1/16 6/16 6/16 9/16 7/16 1/3 2/3 3/16 1/16 6/16 6/16 1/2 1/2 1/7 6/7 4
5 : Coins Alenae BNs Exa acs don peven epesening independence, jus allow non-independence ummay Infeence Bayes nes compacly encode join disibuions Guaaneed independencies of disibuions can be deduced fom BN gap sucue A Bayes ne may ave oe independencies a ae no deecable unil you inspec is specific disibuion e Bayes ball algoim (aka d-sepaaion) ells us wen an obsevaion of one vaiable can cange belief abou anoe vaiable Infeence: calculaing some saisic fom a join pobabiliy disibuion s: Poseio pobabiliy: Mos likely explanaion: D L B Infeence by Enumeaion eminde: Alam Newok P(sun)? P(sun wine)? P(sun wine, wam)? summe summe summe summe wine wine wine wine wam wam cold cold wam wam cold cold sun ain sun ain sun ain sun ain P
6 Infeence by Enumeaion Given unlimied ime, infeence in BNs is easy ecipe: ae e maginal pobabiliies you need Figue ou ALL e aomic pobabiliies you need Calculae and combine em : Wee did we use e BN sucue? We didn! Nomalizaion ick In is simple meod, we only need e BN o synesize e join enies Nomalize Infeence by Enumeaion Infeence by Enumeaion? Geneal case: Evidence vaiables: Quey vaiables: Hidden vaiables: All vaiables We wan: Fis, selec e enies consisen wi e evidence econd, sum ou H: Finally, nomalize e emaining enies o condiionalize Obvious poblems: Wos-case ime complexiy O(d n ) pace complexiy O(d n ) o soe e join disibuion 6
7 Nesing ums Vaiable Eliminaion: Idea Aomic infeence is exemely slow! ligly cleve way o save wok: Move e sums as fa ig as possible : Los of edundan wok in e compuaion ee We can save ime if we cace all paial esuls is is e basic idea beind vaiable eliminaion ampling Pio ampling Basic idea: Daw N samples fom a sampling disibuion Compue an appoximae poseio pobabiliy ow is conveges o e ue pobabiliy P Cloudy Ouline: ampling fom an empy newok ejecion sampling: ejec samples disageeing wi evidence Likeliood weiging: use evidence o weig samples pinkle WeGass ain Pio ampling is pocess geneaes samples wi pobabiliy i.e. e BN s join pobabiliy Le e numbe of samples of an even be en I.e., e sampling pocedue is consisen We ll ge a bunc of samples fom e BN: c, s,, w c, s,, w Cloudy C c, s,, w pinkle ain c, s,, w c, s,, w WeGass W If we wan o know P(W) We ave couns <w:4, w:1> Nomalize o ge P(W) = <w:0.8, w:0.2> is will ge close o e ue disibuion wi moe samples Can esimae anying else, oo Wa abou P(C )? P(C, w)? 7
8 ejecion ampling Likeliood Weiging Le s say we wan P(C) No poin keeping all samples aound Jus ally couns of C oucomes Le s say we wan P(C s) ame ing: ally C oucomes, bu ignoe (ejec) samples wic don ave =s is is ejecion sampling I is also consisen (coec in e limi) pinkle Cloudy C WeGass W c, s,, w c, s,, w c, s,, w c, s,, w c, s,, w ain Poblem wi ejecion sampling: If evidence is unlikely, you ejec a lo of samples ou don exploi you evidence as you sample Conside P(B a) Buglay Idea: fix evidence vaiables and sample e es Buglay Alam Alam Poblem: sample disibuion no consisen! oluion: weig by pobabiliy of evidence given paens Likeliood ampling Likeliood Weiging ampling disibuion if z sampled and e fixed evidence Cloudy Cloudy C pinkle ain Now, samples ave weigs W ain WeGass ogee, weiged sampling disibuion is consisen Likeliood Weiging Noe a likeliood weiging doesn solve all ou poblems ae evidence is aken ino accoun fo downseam vaiables, bu no upseam ones A bee soluion is Makov-cain Mone Calo (MCMC), moe advanced We ll eun o sampling fo obo localizaion and acking in dynamic BNs Cloudy C W ain 8
Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example
C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional
More informationProbabilistic Models. CS 188: Artificial Intelligence Fall Independence. Example: Independence. Example: Independence? Conditional Independence
C 188: Aificial Inelligence Fall 2007 obabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Lecue 15: Bayes Nes 10/18/2007 Given a join disibuion, we can eason abou unobseved vaiables
More informationCS 188: Artificial Intelligence Fall Probabilistic Models
CS 188: Aificial Inelligence Fall 2007 Lecue 15: Bayes Nes 10/18/2007 Dan Klein UC Bekeley Pobabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Given a join disibuion, we can
More informationAnnouncements. CS 188: Artificial Intelligence Fall Example Bayes Net. Bayes Nets. Example: Traffic. Bayes Net Semantics
CS 188: Artificial Intelligence Fall 2008 ecture 15: ayes Nets II 10/16/2008 Announcements Midterm 10/21: see prep page on web Split rooms! ast names A-J go to 141 McCone, K- to 145 winelle One page note
More informationBayes Nets. CS 188: Artificial Intelligence Fall Example: Alarm Network. Bayes Net Semantics. Building the (Entire) Joint. Size of a Bayes Net
CS 188: Artificial Intelligence Fall 2010 Lecture 15: ayes Nets II Independence 10/14/2010 an Klein UC erkeley A ayes net is an efficient encoding of a probabilistic model of a domain ayes Nets Questions
More informationReinforcement learning
Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback
More information[ ] 0. = (2) = a q dimensional vector of observable instrumental variables that are in the information set m constituents of u
Genealized Mehods of Momens he genealized mehod momens (GMM) appoach of Hansen (98) can be hough of a geneal pocedue fo esing economics and financial models. he GMM is especially appopiae fo models ha
More informationBayes Nets: Independence
Bayes Nets: Independence [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http://ai.berkeley.edu.] Bayes Nets A Bayes
More informationAnnouncements. CS 188: Artificial Intelligence Spring Probability recap. Outline. Bayes Nets: Big Picture. Graphical Model Notation
CS 188: Artificial Intelligence Spring 2010 Lecture 15: Bayes Nets II Independence 3/9/2010 Pieter Abbeel UC Berkeley Many slides over the course adapted from Dan Klein, Stuart Russell, Andrew Moore Current
More informationCS 188: Artificial Intelligence Fall Announcements
C 188: Atificial Intelligence Fall 2006 Lectue 14: oaility 10/17/2006 Dan Klein UC Bekeley Announcements Gades: Check midtem, p1.1, and p1.2 gades in glookup Let us know if thee ae polems, so we can calculate
More informationAnnouncements. CS 188: Artificial Intelligence Fall Today. Uncertainty. Random Variables. Probabilities. Lecture 14: Probability 10/17/2006
C 188: Atificial Intelligence all 2006 Lectue 14: oaility 10/17/2006 Announcements Gades: Check midtem, p1.1, and p1.2 gades in glookup Let us know if thee ae polems, so we can calculate useful peliminay
More informationOutline. CSE 473: Artificial Intelligence Spring Bayes Nets: Big Picture. Bayes Net Semantics. Hidden Markov Models. Example Bayes Net: Car
CSE 473: rtificial Intelligence Spring 2012 ayesian Networks Dan Weld Outline Probabilistic models (and inference) ayesian Networks (Ns) Independence in Ns Efficient Inference in Ns Learning Many slides
More informationComparison between the Discrete and Continuous Time Models
Comparison beween e Discree and Coninuous Time Models D. Sulsky June 21, 2012 1 Discree o Coninuous Recall e discree ime model Î = AIS Ŝ = S Î. Tese equaions ell us ow e populaion canges from one day o
More informationln y t 2 t c where c is an arbitrary real constant
SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies
More informationSTUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE
More informationLecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain
Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as
More informationLecture 17: Kinetics of Phase Growth in a Two-component System:
Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien
More informationAdditional Exercises for Chapter What is the slope-intercept form of the equation of the line given by 3x + 5y + 2 = 0?
ddiional Eercises for Caper 5 bou Lines, Slopes, and Tangen Lines 39. Find an equaion for e line roug e wo poins (, 7) and (5, ). 4. Wa is e slope-inercep form of e equaion of e line given by 3 + 5y +
More informationComputer Propagation Analysis Tools
Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion
More informationMeasures the linear dependence or the correlation between r t and r t-p. (summarizes serial dependence)
. Definiions Saionay Time Seies- A ime seies is saionay if he popeies of he pocess such as he mean and vaiance ae consan houghou ime. i. If he auocoelaion dies ou quickly he seies should be consideed saionay
More informationCSE 473: Artificial Intelligence Autumn 2011
CSE 473: Artificial Intelligence Autumn 2011 Bayesian Networks Luke Zettlemoyer Many slides over the course adapted from either Dan Klein, Stuart Russell or Andrew Moore 1 Outline Probabilistic models
More informationArtificial Intelligence Bayes Nets: Independence
Artificial Intelligence Bayes Nets: Independence Instructors: David Suter and Qince Li Course Delivered @ Harbin Institute of Technology [Many slides adapted from those created by Dan Klein and Pieter
More informationPHYS PRACTICE EXAM 2
PHYS 1800 PRACTICE EXAM Pa I Muliple Choice Quesions [ ps each] Diecions: Cicle he one alenaive ha bes complees he saemen o answes he quesion. Unless ohewise saed, assume ideal condiions (no ai esisance,
More informationCombinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions
Inenaional Mahemaical Foum, Vol 8, 03, no 0, 463-47 HIKARI Ld, wwwm-hikaicom Combinaoial Appoach o M/M/ Queues Using Hypegeomeic Funcions Jagdish Saan and Kamal Nain Depamen of Saisics, Univesiy of Delhi,
More informationKalman Filter: an instance of Bayes Filter. Kalman Filter: an instance of Bayes Filter. Kalman Filter. Linear dynamics with Gaussian noise
COM47 Inoducion o Roboics and Inelligen ysems he alman File alman File: an insance of Bayes File alman File: an insance of Bayes File Linea dynamics wih Gaussian noise alman File Linea dynamics wih Gaussian
More informationAnnouncements. CS 188: Artificial Intelligence Fall Causality? Example: Traffic. Topology Limits Distributions. Example: Reverse Traffic
CS 188: Artificial Intelligence Fall 2008 Lecture 16: Bayes Nets III 10/23/2008 Announcements Midterms graded, up on glookup, back Tuesday W4 also graded, back in sections / box Past homeworks in return
More informationPseudosteady-State Flow Relations for a Radial System from Department of Petroleum Engineering Course Notes (1997)
Pseudoseady-Sae Flow Relaions fo a Radial Sysem fom Deamen of Peoleum Engineeing Couse Noes (1997) (Deivaion of he Pseudoseady-Sae Flow Relaions fo a Radial Sysem) (Deivaion of he Pseudoseady-Sae Flow
More informationTwo-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch
Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion
More informationCS 5522: Artificial Intelligence II
CS 5522: Artificial Intelligence II Bayes Nets: Independence Instructor: Alan Ritter Ohio State University [These slides were adapted from CS188 Intro to AI at UC Berkeley. All materials available at http://ai.berkeley.edu.]
More informationBayesian Networks. Vibhav Gogate The University of Texas at Dallas
Bayesian Networks Vibhav Gogate The University of Texas at Dallas Intro to AI (CS 6364) Many slides over the course adapted from either Dan Klein, Luke Zettlemoyer, Stuart Russell or Andrew Moore 1 Outline
More informationAn Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants
An Open cycle and losed cycle Gas ubine Engines Mehods o impove he pefomance of simple gas ubine plans I egeneaive Gas ubine ycle: he empeaue of he exhaus gases in a simple gas ubine is highe han he empeaue
More informationGeneral Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security
1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,
More information, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t
Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission
More informationF (u) du. or f(t) = t
8.3 Topic 9: Impulses and dela funcions. Auor: Jeremy Orloff Reading: EP 4.6 SN CG.3-4 pp.2-5. Warmup discussion abou inpu Consider e rae equaion d + k = f(). To be specific, assume is in unis of d kilograms.
More informationProbabilistic Models. Models describe how (a portion of) the world works
Probabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables All models
More informationBayesian Networks. Vibhav Gogate The University of Texas at Dallas
Bayesian Networks Vibhav Gogate The University of Texas at Dallas Intro to AI (CS 4365) Many slides over the course adapted from either Dan Klein, Luke Zettlemoyer, Stuart Russell or Andrew Moore 1 Outline
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationOverview. Overview Page 1 of 8
COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIORNIA DECEMBER 2001 COMPOSITE BEAM DESIGN AISC-LRD93 Tecnical Noe Compac and Noncompac Requiemens Tis Tecnical Noe descibes o e pogam cecks e AISC-LRD93 specificaion
More informationLecture 22 Electromagnetic Waves
Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should
More informationRecap: Bayes Nets. CS 473: Artificial Intelligence Bayes Nets: Independence. Conditional Independence. Bayes Nets. Independence in a BN
CS 473: Artificial Intelligence ayes Nets: Independence A ayes net is an efficient encoding of a probabilistic model of a domain ecap: ayes Nets Questions we can ask: Inference: given a fixed N, what is
More informationAsymptotic Equipartition Property - Seminar 3, part 1
Asympoic Equipariion Propery - Seminar 3, par 1 Ocober 22, 2013 Problem 1 (Calculaion of ypical se) To clarify he noion of a ypical se A (n) ε and he smalles se of high probabiliy B (n), we will calculae
More informationMolecular Evolution and Phylogeny. Based on: Durbin et al Chapter 8
Molecula Evoluion and hylogeny Baed on: Dubin e al Chape 8. hylogeneic Tee umpion banch inenal node leaf Topology T : bifucaing Leave - N Inenal node N+ N- Lengh { i } fo each banch hylogeneic ee Topology
More informationr P + '% 2 r v(r) End pressures P 1 (high) and P 2 (low) P 1 , which must be independent of z, so # dz dz = P 2 " P 1 = " #P L L,
Lecue 36 Pipe Flow and Low-eynolds numbe hydodynamics 36.1 eading fo Lecues 34-35: PKT Chape 12. Will y fo Monday?: new daa shee and daf fomula shee fo final exam. Ou saing poin fo hydodynamics ae wo equaions:
More informationRisk tolerance and optimal portfolio choice
Risk oleance and opimal pofolio choice Maek Musiela BNP Paibas London Copoae and Invesmen Join wok wih T. Zaiphopoulou (UT usin) Invesmens and fowad uiliies Pepin 6 Backwad and fowad dynamic uiliies and
More informationSections 3.1 and 3.4 Exponential Functions (Growth and Decay)
Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens
More informationRelative and Circular Motion
Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you
More informationCircular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.
1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule
More informationExponential and Logarithmic Equations and Properties of Logarithms. Properties. Properties. log. Exponential. Logarithmic.
Eponenial and Logaihmic Equaions and Popeies of Logaihms Popeies Eponenial a a s = a +s a /a s = a -s (a ) s = a s a b = (ab) Logaihmic log s = log + logs log/s = log - logs log s = s log log a b = loga
More informationÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s
MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationLow-complexity Algorithms for MIMO Multiplexing Systems
Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :
More information02. MOTION. Questions and Answers
CLASS-09 02. MOTION Quesions and Answers PHYSICAL SCIENCE 1. Se moves a a consan speed in a consan direcion.. Reprase e same senence in fewer words using conceps relaed o moion. Se moves wi uniform velociy.
More informationThe Production of Polarization
Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview
More informationBayesian networks. Soleymani. CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018
Bayesian networks CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018 Soleymani Slides have been adopted from Klein and Abdeel, CS188, UC Berkeley. Outline Probability
More informationProduct rule. Chain rule
Probability Recap CS 188: Artificial Intelligence ayes Nets: Independence Conditional probability Product rule Chain rule, independent if and only if: and are conditionally independent given if and only
More informationVariance and Covariance Processes
Vaiance and Covaiance Pocesses Pakash Balachandan Depamen of Mahemaics Duke Univesiy May 26, 2008 These noes ae based on Due s Sochasic Calculus, Revuz and Yo s Coninuous Maingales and Bownian Moion, Kaazas
More informationMATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH
Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias
More informationReinforcement learning
CS 75 Mchine Lening Lecue b einfocemen lening Milos Huskech milos@cs.pi.edu 539 Senno Sque einfocemen lening We wn o len conol policy: : X A We see emples of bu oupus e no given Insed of we ge feedbck
More informationSection 5: Chain Rule
Chaper The Derivaive Applie Calculus 11 Secion 5: Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of
More informationControl Volume Derivation
School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass
More informationToday - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations
Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy
More informationLecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light
Lecue 5 Chape 3 lecomagneic Theo, Phoons, and Ligh Gauss s Gauss s Faada s Ampèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d qv A q A J ds ds In mae fields ae defined hough ineacion
More informationOutline. CSE 573: Artificial Intelligence Autumn Bayes Nets: Big Picture. Bayes Net Semantics. Hidden Markov Models. Example Bayes Net: Car
CSE 573: Artificial Intelligence Autumn 2012 Bayesian Networks Dan Weld Many slides adapted from Dan Klein, Stuart Russell, Andrew Moore & Luke Zettlemoyer Outline Probabilistic models (and inference)
More informationAnnouncements. CS 188: Artificial Intelligence Spring Bayes Net Semantics. Probabilities in BNs. All Conditional Independences
CS 188: Artificial Intelligence Spring 2011 Announcements Assignments W4 out today --- this is your last written!! Any assignments you have not picked up yet In bin in 283 Soda [same room as for submission
More informationSupport Vector Machines
Suppo Veco Machine CSL 3 ARIFICIAL INELLIGENCE SPRING 4 Suppo Veco Machine O, Kenel Machine Diciminan-baed mehod olean cla boundaie Suppo veco coni of eample cloe o bounday Kenel compue imilaiy beeen eample
More informationNumerical solution of fuzzy differential equations by Milne s predictor-corrector method and the dependency problem
Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Numeical soluion o uzz dieenial equaions b Milne s pedico-coeco meod and e dependenc poblem Kanagaajan K Indakuma S Muukuma
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationUnderstanding the asymptotic behaviour of empirical Bayes methods
Undersanding he asympoic behaviour of empirical Bayes mehods Boond Szabo, Aad van der Vaar and Harry van Zanen EURANDOM, 11.10.2011. Conens 2/20 Moivaion Nonparameric Bayesian saisics Signal in Whie noise
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationThe sudden release of a large amount of energy E into a background fluid of density
10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy
More informationOur main purpose in this section is to undertake an examination of the stock
3. Caial gains ax and e sock rice volailiy Our main urose in is secion is o underake an examinaion of e sock rice volailiy by considering ow e raional seculaor s olding canges afer e ax rae on caial gains
More informationSection P.1 Notes Page 1 Section P.1 Precalculus and Trigonometry Review
Secion P Noe Pge Secion P Preclculu nd Trigonomer Review ALGEBRA AND PRECALCULUS Eponen Lw: Emple: 8 Emple: Emple: Emple: b b Emple: 9 EXAMPLE: Simplif: nd wrie wi poiive eponen Fir I will flip e frcion
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]
ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationTHE CATCH PROCESS (continued)
THE CATCH PROCESS (coninued) In our previous derivaion of e relaionsip beween CPUE and fis abundance we assumed a all e fising unis and all e fis were spaially omogeneous. Now we explore wa appens wen
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationTesting What You Know Now
Tesing Wha You Know Now To bes each you, I need o know wha you know now Today we ake a well-esablished quiz ha is designed o ell me his To encourage you o ake he survey seriously, i will coun as a clicker
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationDynamic Estimation of OD Matrices for Freeways and Arterials
Novembe 2007 Final Repo: ITS Dynamic Esimaion of OD Maices fo Feeways and Aeials Auhos: Juan Calos Heea, Sauabh Amin, Alexande Bayen, Same Madana, Michael Zhang, Yu Nie, Zhen Qian, Yingyan Lou, Yafeng
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationYou must fully interpret your results. There is a relationship doesn t cut it. Use the text and, especially, the SPSS Manual for guidance.
POLI 30D SPRING 2015 LAST ASSIGNMENT TRUMPETS PLEASE!!!!! Due Thursday, December 10 (or sooner), by 7PM hrough TurnIIn I had his all se up in my mind. You would use regression analysis o follow up on your
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationThe Quantum Theory of Atoms and Molecules: The Schrodinger equation. Hilary Term 2008 Dr Grant Ritchie
e Quanum eory of Aoms and Molecules: e Scrodinger equaion Hilary erm 008 Dr Gran Ricie An equaion for maer waves? De Broglie posulaed a every paricles as an associaed wave of waveleng: / p Wave naure of
More informationTopic 4a Introduction to Root Finding & Bracketing Methods
/8/18 Couse Instucto D. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: cumpf@utep.edu Topic 4a Intoduction to Root Finding & Backeting Methods EE 4386/531 Computational Methods in EE Outline
More informationAnnouncements. Inference. Mid-term. Inference by Enumeration. Reminder: Alarm Network. Introduction to Artificial Intelligence. V22.
Introduction to Artificial Intelligence V22.0472-001 Fall 2009 Lecture 15: Bayes Nets 3 Midterms graded Assignment 2 graded Announcements Rob Fergus Dept of Computer Science, Courant Institute, NYU Slides
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationOnline Completion of Ill-conditioned Low-Rank Matrices
Online Compleion of Ill-condiioned Low-Rank Maices Ryan Kennedy and Camillo J. Taylo Compue and Infomaion Science Univesiy of Pennsylvania Philadelphia, PA, USA keny, cjaylo}@cis.upenn.edu Laua Balzano
More informationSpecial Subject SC434L Digital Video Coding and Compression. ASSIGNMENT 1-Solutions Due Date: Friday 30 August 2002
SC434L DVCC Assignmen Special Sujec SC434L Digial Vieo Coing an Compession ASSINMENT -Soluions Due Dae: Fiay 30 Augus 2002 This assignmen consiss of wo pages incluing wo compulsoy quesions woh of 0% of
More informationLecture 4 Notes (Little s Theorem)
Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,
More informationBayes Nets III: Inference
1 Hal Daumé III (me@hal3.name) Bayes Nets III: Inference Hal Daumé III Computer Science University of Maryland me@hal3.name CS 421: Introduction to Artificial Intelligence 10 Apr 2012 Many slides courtesy
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationProblem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
More informationHomework 4 SOLUTION EE235, Summer 2012
Homework 4 SOLUTION EE235, Summer 202. Causal and Sable. These are impulse responses for LTI sysems. Which of hese LTI sysem impulse responses represen BIBO sable sysems? Which sysems are causal? (a) h()
More informationTwo Popular Bayesian Estimators: Particle and Kalman Filters. McGill COMP 765 Sept 14 th, 2017
Two Popular Bayesian Esimaors: Paricle and Kalman Filers McGill COMP 765 Sep 14 h, 2017 1 1 1, dx x Bel x u x P x z P Recall: Bayes Filers,,,,,,, 1 1 1 1 u z u x P u z u x z P Bayes z = observaion u =
More informationMEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING
MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens
More informationThe fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation
Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or
More informationME 141. Engineering Mechanics
ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics
More information