Prob obab abil ilit ity y an and d Tim Time: Ma Marko kov v Mo Mode dels ls Com omputer Science c cpsc sc322, Lecture 3 31 (Te Text xtboo ook k Chpt 6.5.1) June, 2 20, 2 2017 6/21/2017 CPSC322 Summer 2017 Slide 1
Lectu ture re Ov Overv rvie iew • Recap p • Te Tempo poral l Prob obabi bilistic ic Mo Mode dels ls • Start Markov Models • Markov Chain • Markov Chains in Natural Language Processing 6/21/2017 CPSC322 Summer 2017 Slide 2
Bi Big g Pi Pict ctur ure: e: R&R &R sy syst stem ems Environ onment Sto tochas asti tic Dete terministi tic Prob oblem Arc Consiste tency Se Sear arch Constr trai aint t Var ars + Sat atisfac acti tion Constr trai aints ts SLS Sta tati tic Belief N Nets ts Logics Query Qu Var ar. Eliminat ation Sear arch Decision Nets ts Sequenti Se tial al STRIPS Var ar. Eliminat ation Planning Mar arkov Pr v Processes Sear arch Representa tati tion Val alue Ite terat ation Reas asoning 6/21/2017 CPSC322 Summer 2017 Slide 3 Technique
Answerin ing Query u y unde der Un Uncertai ainty Probab abil ility ity Theory ry Dynamic mic Bayesia ian n Network rk Sta tati tic B Belief Netw twork & V Variab able le Elimi mina nation ion Hidden n Markov ov Models St Student t Tracing ng in Monitori oring ng tutorin ing g Sy Systems ms BioInforma rmati tics cs (e.g credit t cards) Markov ov Chains Natural al Language age Diagno nosti stic c Processin ssing Systems Sy ms (e.g., ., medici cine ne) Email il spam m filters rs 6/21/2017 CPSC322 Summer 2017 Slide 4
Lectu ture re Ov Overv rvie iew • Recap p • Te Tempo poral l Prob obabi bilistic ic Mo Mode dels ls • Start Markov Models • Markov Chain • Markov Chains in Natural Language Processing 6/21/2017 CPSC322 Summer 2017 Slide 5
Mod odell llin ing g st stat atic ic Envi viro ronme ments ts So far we have used Bnets to perform inference in st static environ onments s • For instance, the system keeps collecting evidence to diagnose the cause of a fault in a system (e.g., a car). • The environment (values of the evidence, the true cause) does not change as I gather new evidence • What does change? The system’s beliefs over possible causes 6/21/2017 CPSC322 Summer 2017 Slide 6
Mode deli ling Ev Evolv lvin ing E Envir ironments • Often we need to make inferences about evolving environments . • Represent the state of the world at each specific point in time via a series of snapshots, or time sl slices , Solve veProblem t-1 Solve veProblem t Knows-Subtrac action t-1 Knows-Subtrac action t t Moral ale t-1 Moral ale t Tutor Tu oring g sy syst stem tracing student knowledge and morale 6/21/2017 CPSC322 Summer 2017 Slide 7
Lectu ture re Ov Overv rvie iew • Recap p • Te Tempo poral l Prob obabi bilistic ic Mo Mode dels ls • Start Markov Models • Mar arko kov v Ch Chai ain • Markov Chains in Natural Language Processing 6/21/2017 CPSC322 Summer 2017 Slide 8
Si Simp mple lest st Pos ossi sible le DBN BN able for each time slice : let’s assume S t • On One ran andom va variab represents the state at time t . with domain { v 1 … v n } • Eac ach ran andom va variab able depends only o y on th the previ vious one • Thus • Intuitively S t conveys all of the information about the history that can affect the future states. • “The future is independent of the past given the present.” 6/21/2017 CPSC322 Summer 2017 Slide 9
Simplest Possible DBN (cont’) • How many CPTs do we need to specify? A. 1 A. C . . 2 D. 3 D. B. 4 • Stationary process assumption: the mechanism that regulates how state variables change overtime is st station onary, that is it can be described by a single transition model • P(S t |S t-1 ) 6/21/2017 CPSC322 Summer 2017 Slide 10
Stat atio ionar ary Ma y Markov Ch v Chai ain ( (SMC MC) A stationary Markov Chain : for all t >0 • P ( S t+1 | S 0 ,…, S t ) = P ( S t+1 | S t ) and • P ( S t +1 | S t ) is the same We only need to specify and • Simple Model, easy to specify • Often the natural model • The network can extend indefinitely • Var ariat ations of SMC ar are at at th the c core o of man any y Nat atural al L Lan anguag age Processing (NLP) ap applicat ations! 6/21/2017 CPSC322 Summer 2017 Slide 1 1
Stat atio ionar ary Ma y Markov Ch v Chai ain ( (SMC MC) A stationary Markov Chain : for all t >0 • P ( S t+1 | S 0 ,…, S t ) = P ( S t+1 | S t ) and • P ( S t +1 | S t ) is the same So we only need to specify? A. A. P ( S t +1 | S t ) and P ( S 0 ) B. P ( S 0 ) D. D. P ( S t | S t+1 ) C . . P ( S t +1 | S t ) 6/21/2017 CPSC322 Summer 2017 Slide 12
Sta Stati tion onar ary y Mar arko kov-Ch Chai ain: Exa xamp mple le t .6 Domain of variable S i is {t , q, p, a, h, e} q .4 p Probability of initial state P ( S 0 ) a P ( S t+1 | S t ) Stochastic Transition Matrix h e Which of these two is a possible STM? S S t 1 t 1 t q p a h e t q p a h e 1 0 0 0 0 0 t 0 .3 0 .3 .4 0 t 0 1 0 0 0 0 q q .4 0 .6 0 0 0 .3 0 1 0 0 0 p p 0 0 1 0 0 0 S S t t a 0 0 0 1 0 0 0 0 .4 .6 0 0 a 0 0 0 0 0 1 h 0 0 0 0 0 1 h 0 0 0 .2 0 1 e e 1 0 0 0 0 0 A. A.Left one only B. Right one only 6/21/2017 CPSC322 Summer 2017 Slide 13 D. D. None C . . Both
Sta Stati tion onar ary y Mar arko kov-Ch Chai ain: Exa xamp mple le Domain of variable S i is {t , q, p, a, h, e} t .6 We only need to specify… q .4 P ( S 0 ) p a Probability of initial state h e Stochastic Transition Matrix S t 1 t q p a h e P ( S t+1 | S t ) t 0 .3 0 .3 .4 0 .4 0 .6 0 0 0 q 0 0 1 0 0 0 p S t a 0 0 .4 .6 0 0 h 0 0 0 0 0 1 1 0 0 0 0 0 e 6/21/2017 Slide 14 CPSC322 Summer 2017
Mark rkov ov-Chai ain: Infe fere rence Probability of a sequence of states S 0 … S T P ( S ,..., S ) 0 T P ( S t+1 | S t ) t q p a h e P ( S 0 ) 0 .3 0 .3 .4 0 t t .6 .4 0 .6 0 0 0 q q .4 p 0 0 1 0 0 0 Exa xample: p 0 0 0 .4 .6 0 0 a a 0 P ( t , q , p ) h 0 0 0 0 0 1 0 h e 0 e 1 0 0 0 0 0 6/21/2017 CPSC322 Summer 2017 Slide 15
Lectu ture re Ov Overv rvie iew • Recap p • Te Tempo poral l Prob obabi bilistic ic Mo Mode dels ls • Marko kov Mod odels ls • Mar arko kov v Ch Chai ain • Markov Chains in Natural Language Processing 6/21/2017 CPSC322 Summer 2017 Slide 16
Ke Key pro roble lems s in in NLP P ( w ,.., w ) ? “Book me a room near UBC” 1 n Assign a probability to a sentence • Part-of-speech tagging Summariza zation on, Machine • Word-sense disambiguation, Translation…..... • Probabilistic Parsing Predict the next word • Speech recognition • Hand-writing recognition • Augmentative communication for the disabled P ( w ,.., w ) ? Impo possib ible le t to 1 n ate estim imat 6/21/2017 CPSC322 Summer 2017 17
P ( w ,.., w ) ? Impo possib ible le t to e estim imat ate! 1 n Assuming 10 5 words and average sentence contains 10 words ……. Google Go le la languag age re repo posit itory y (22 Sept. 2006) contained “only”: 95,119,665,584 sentences once Mos ost se sentences w s will not ot appear or or appear on only on 6/21/2017 CPSC322 Summer 2017 18
What at can an w we do? o? Make a strong simplifying assumption! Sentences are generated by a Markov Chain n P ( w ,.., w ) P ( w | S ) P ( w | w ) 1 n 1 k k 1 k 2 P(Th The b big g red dog og barks ks)= P(Th The|<S>) * * 6/21/2017 CPSC322 Summer 2017 19
Est stim imat ates es fo for r Bi Bigr gram ams Silly language repositories with on only two o se sentences: “<S> The big red dog barks against the big pink dog” “<S> The big pink dog is much smaller” C ( big , red ) N P ( big , red ) C ( big , red ) pairs P ( red | big ) C ( big ) P ( big ) C ( big ) N words 6/21/2017 CPSC322 Summer 2017 20
Bigrams in practice… If you have 10 5 words in your dictionary P ( w i w | ) i 1 will contain this many numbers.. ?? A. A.2 *10 5 B. 10 10 B. . 5 * 10 5 D.2 *10 10 D. C. 6/21/2017 CPSC322 Summer 2017 Slide 21
Learning Goals for today’s class Yo You c can an: • Specify a Markov Chain and compute the probability of a sequence of states • Justify and apply Markov Chains to compute the probability of a Natural Language sentence 6/21/2017 CPSC322 Summer 2017 Slide 22
Mar arko kov v Mod odel els Simplest Possible Markov Chains Dynamic Bnet We cannot observe directly what we care Hidden Markov Model about Add Actions and Values Markov Decision (Rewards) Processes (MDPs) 6/21/2017 CPSC322 Summer 2017 Slide 23
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