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CS B553: ALGORITHMS FOR OPTIMIZATION AND LEARNING Temporal sequences: Hidden Markov Models and Dynamic Bayesian Networks MOTIVATION Observing a stream of data Monitoring (of people, computer systems, etc) Surveillance, tracking Finance & economics Science Questions: Modeling & forecasting Unobserved variables TIME SERIES MODELING Time occurs in steps t=0,1,2,… Time step can be seconds, days, years, etc State variable Xt, t=0,1,2,… For partially observed problems, we see observations Ot, t=1,2,… and do not see the X’s X’s are hidden variables (aka latent variables) MODELING TIME Arrow of time Causes Effects Causality => Bayesian networks are natural models of time series PROBABILISTIC MODELING For now, assume fully observable case X0 X1 X2 X3 X1 X2 X3 What parents? X0 MARKOV ASSUMPTION Assume Xt+k is independent of all Xi for i<t P(Xt+k | X0,…,Xt+k-1) = P(Xt+k | Xt,…,Xt+k-1) K-th order Markov Chain Order 0 X0 X1 X2 X3 Order 1 X0 X1 X2 X3 Order 2 X0 X1 X2 X3 Order 3 X0 X1 X2 X3 1ST ORDER MARKOV CHAIN MC’s of order k>1 can be converted into a 1st order MC on the variable Yt = {Xt,…,Xt+k-1} So w.o.l.o.g., “MC” refers to a 1st order MC X0 X1 X2 X3 X0 X1’ X2’ X3’ X1 X2 X3 X4 Y0 Y1 Y2 Y3 INFERENCE IN MC What independence relationships can we read from the BN? X0 X1 X2 X3 Observe X1 X0 independent of X2, X3, … P(Xt|Xt-1) known as transition model INFERENCE IN MC Prediction: the probability of future state? P(Xt) = S =S = S x0,…,xt-1P x0,…,xt-1P (X0) xt-1P(Xt|Xt-1) (X0,…,Xt) P x1,…,xt P(Xt-1) P(Xi|Xi-1) [Recursive approach] Approach: maintain a belief state bt(X)=P(Xt), use above equation to advance to bt+1(X) Equivalent to VE algorithm in sequential order BELIEF STATE EVOLUTION P(Xt) = Sxt-1P(Xt|Xt-1) P(Xt-1) “Blurs” over time, and (typically) approaches a stationary distribution as t grows Limited prediction power Rate of blurring known as mixing time STATIONARY DISTRIBUTIONS For discrete variables Val(X)={1,…,n}: Transition matrix Tij = P(Xt=i|Xt-1=j) Belief bt(X) is just a vector bt,i=P(Xt=i) Belief update equation: bt+1 = T*bt A stationary distribution b is one in which b = Tb => b is an eigenvector of T with eigenvalue 1 => b is in the null space of (T-I) HISTORY DEPENDENCE In Markov models, the state must be chosen so that the future is independent of history given the current state Often this requires adding variables that cannot be directly observed minimum essentials “the bare” market wipes himself with the rabbit Are these people walking toward you or away from you? What comes next? PARTIAL OBSERVABILITY Hidden Markov Model (HMM) X0 X1 X2 X3 Hidden state variables O1 O2 O3 Observed variables P(Ot|Xt) called the observation model (or sensor model) INFERENCE IN HMMS Filtering Prediction Smoothing, aka hindsight Most likely explanation X0 X1 X2 X3 O1 O2 O3 INFERENCE IN HMMS Filtering Prediction Smoothing, aka hindsight Most likely explanation Query variable X0 X1 X2 O1 O2 FILTERING Name comes from signal processing P(Xt|o1:t) = S xt-1 P(xt-1|o1:t-1) P(Xt|xt-1,ot) P(Xt|Xt-1,ot) = P(ot|Xt-1,Xt)P(Xt|Xt-1)/P(ot|Xt-1) = a P(ot|Xt)P(Xt|Xt-1) Query variable X0 X1 X2 O1 O2 FILTERING P(Xt|o1:t) = a Sxt-1P(xt-1|o1:t-1) P(ot|Xt)P(Xt|xt-1) Forward recursion If we keep track of belief state bt(X) = P(Xt|o1:t) => O(|Val(X)|2) updates for each t! Query variable X0 X1 X2 O1 O2 PREDICT-UPDATE INTERPRETATION Given old belief state bt-1(X) Predict: First compute MC update bt’(Xt)=P(Xt|o1:t-1) = a Sxbt-1(x) P(Xt|Xt-1=x) Update: Re-weight to account for observation probabilities: bt(x) = bt’(x)P(ot|Xt=x) X0 Query variable X1 X2 O1 O2 INFERENCE IN HMMS Filtering Prediction Smoothing, aka hindsight Most likely explanation Query X0 X1 X2 X3 O1 O2 O3 PREDICTION P(Xt+k|o1:t) 2 steps: P(Xt|o1:t), then P(Xt+k|Xt) Filter to time t, then predict as with standard MC Query X0 X1 X2 X3 O1 O2 O3 INFERENCE IN HMMS Filtering Prediction Smoothing, aka hindsight Most likely explanation Query X0 X1 X2 X3 O1 O2 O3 SMOOTHING P(Xk|o1:t) for k < t P(Xk|o1:k,ok+1:t) = P(ok+1:t|Xk,o1:k)P(Xk|o1:k)/P(ok+1:t|o1:k) = a P(ok+1:t|Xk)P(Xk|o1:k) Standard filtering to time k Query X0 X1 X2 X3 O1 O2 O3 SMOOTHING Computing P(ok+1:t|Xk) P(ok+1:t|Xk) = S =S = S xk+1P(ok+1:t|Xk,xk+1) xk+1P(ok+1:t|xk+1) P(xk+1|Xk) P(xk+1|Xk) xk+1P(ok+2:t|xk+1)P(ok+1|xk+1)P(xk+1|Xk) Backward recursion Given prior states X0 X1 O1 X2 O2 X3 O3 What’s the probability of this sequence? INTERPRETATION Filtering/prediction: Smoothing: Equivalent to forward variable elimination / belief propagation Equivalent to forward VE/BP up to query variable, then backward VE/BP from last observation back to query variable Running BP to completion gives the smoothed estimates for all variables (forward-backward algorithm) INFERENCE IN HMMS Filtering Prediction Smoothing, aka hindsight Most likely explanation Subject of next lecture X0 X1 X2 X3 O1 O2 O3 Query returns a path through state space x0,…,x3 APPLICATIONS OF HMMS IN NLP Speech recognition Hidden phones (e.g., ah eh ee th r) Observed, noisy acoustic features (produced by signal processing) PHONE OBSERVATION MODELS Phonet Model defined to be robust over variations in accent, speed, pitch, noise Signal processing Features (24,13,3,59) Featurest PHONE TRANSITION MODELS Phonet Phonet+1 Good models will capture (among other things): Featurest Pronunciation of words Subphone structure Coarticulation effects Triphone models = order 3 Markov chain WORD SEGMENTATION Words run together when pronounced Unigrams P(wi) Bigrams P(wi|wi-1) Trigrams P(wi|wi-1,wi-2) Random 20 word samples from R&N using N-gram models Logical are as confusion a may right tries agent goal the was diesel more object then informationgathering search is Planning purely diagnostic expert systems are very similar computational approach would be represented compactly using tic tac toe a predicate Planning and scheduling are integrated the success of naïve bayes model is just a possible prior source by that time WHAT ABOUT MODELS WITH MANY VARIABLES? Say X has n binary variables, O has m binary variables Naively, a distribution over Xt may be intractable to represent (2n entries) Transition models P(Xt |Xt-1) require 22n entries Observation models P(Ot |Xt) require 2n+m entries Is there a better way? EXAMPLE: FAILURE DETECTION Consider a battery meter sensor Battery = true level of battery BMeter = sensor reading Transient failures: send garbage at time t Persistent failures: send garbage forever EXAMPLE: FAILURE DETECTION Consider a battery meter sensor Battery = true level of battery BMeter = sensor reading Transient failures: send garbage at time t 5555500555… Persistent failures: sensor is broken 5555500000… DYNAMIC BAYESIAN NETWORK Template model relates variables on prior time step to the next time step (2-TBN) “Unrolling” the template for all t gives the ground Bayesian network Batteryt-1 Batteryt BMetert BMetert ~ N(Batteryt,s) DYNAMIC BAYESIAN NETWORK Batteryt-1 Batteryt BMetert BMetert ~ N(Batteryt,s) Transient failure model P(BMetert=0 | Batteryt=5) = 0.03 RESULTS ON TRANSIENT FAILURE Meter reads 55555005555… E(Batteryt) Transient failure occurs Without model With model RESULTS ON PERSISTENT FAILURE Meter reads 5555500000… E(Batteryt) Persistent failure occurs With transient model PERSISTENT FAILURE MODEL Brokent-1 Brokent Batteryt-1 Batteryt BMetert BMetert ~ N(Batteryt,s) P(BMetert=0 | Batteryt=5) = 0.03 P(BMetert=0 | Brokent) = 1 RESULTS ON PERSISTENT FAILURE Meter reads 5555500000… E(Batteryt) Persistent failure occurs With persistent failure model With transient model HOW TO PERFORM INFERENCE ON DBN? Exact inference on “unrolled” BN E.g. Variable Elimination Typical order: eliminate sequential time steps so that the network isn’t actually constructed Unrolling is done only implicitly Br0 Br1 Ba0 Br2 Br3 Br4 Ba1 Ba2 Ba3 Ba4 BM1 BM2 BM3 BM4 ENTANGLEMENT PROBLEM After n time steps, all n variables in the belief state become dependent! Unless 2-TBN can be partitioned into disjoint subsets (rare) Lost sparsity structure APPROXIMATE INFERENCE IN DBNS Limited history updates Assumed factorization of belief state Particle filtering INDEPENDENT FACTORIZATION Idea: assume belief state P(Xt) factors across individual attributes P(Xt) = P(X1,t)*…*P(Xn,t) Filtering: only maintain factored distributions P(X1,t|O1:t),…,P(Xn,t|O1:t) Filtering update: P(Xk,t|O1:t) = Sxt-1P(Xk,t|Ot,Xt-1) P(Xt-1|O1:t-1) = marginal probability query over 2TBN X1,t-1 X1,t Xn,t-1 Xn,t O1,t Om,t NEXT TIME Viterbi algorithm Read K&F 13.2 for some context Kalman and particle filtering Read K&F 15.3-4