Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Sequence Data

1. Conditional Random Fields: Probabilistic Modelsfor Segmenting and Labeling Sequence Data J. Lafferty, A. McCallum, F. Pereira Presentation: Inna Weiner Learning Seminar, 2004

2. Outline • Labeling sequence data problem • Classification with probabilistic models: Generative and Discriminative • Why HMMs and MEMMs are not good enough • Conditional Field Model • Experimental Results Learning Seminar, 2004

3. x1 x2 x3 X: Thinking noun verb is being noun Y: y1 y2 y3 Labeling Sequence Data Problem • X is a random variable over data sequences • Y is a random variable over label sequences • Yi is assumed to range over a finite label alphabet A • The problem: • Learn how to give labels from a closed set Y to a data sequence X Learning Seminar, 2004

4. Labeling Sequence Data Problem • The lab setup: Let a monkey do some behavioral task while recording movement and neural activity • Motor task: Reach to target • Goal:Map neural activityto behavior • In our notation: • X: Neural Data • Y: Hand movements Learning Seminar, 2004

5. Generative Probabilistic Models • Learning problem: Choose Θ to maximize jointlikelihood: L(Θ)= Σ log pΘ(yi,xi) • The goal: maximization of the joint likelihood of training examples y = argmax p*(y|x) = argmax p*(y,x)/p(x) • Needs to enumerate all possible observation sequences Learning Seminar, 2004

6. Markov Model • A Markov process or model assumes that we can predict the future based just on the present (or on a limited horizon into the past): • Let {X1,…,XT} be a sequence of random variables taking values {1,…,N} then the Markov properties are: • Limited Horizon: P(Xt+1|X1,…,Xt) = P(Xt+1|Xt) = • Time invariant (stationary): = P(X2|X1) Learning Seminar, 2004

7. Describing a Markov Chain • A Markov Chain can be described by the transition matrix A and the initial probabilities Q: Aij = P(Xt+1=j|Xt=i) qi = P(X1=i) Learning Seminar, 2004

8. Hidden Markov Model • In a Hidden Markov Model (HMM) we do not observe the sequence that the model passed through (X) but only some probabilistic function of it (Y). Thus, it is a Markov model with the addition of emission probabilities: Bik = P(Yt = k|Xt = i) Learning Seminar, 2004

9. The Three Problems of HMM • Likelihood: Given a series of observations y and a model λ = {A,B,q}, compute the likelihood p(y| λ) • Inference: Given a series of observations y and a model lambda compute the most likely series of hidden states x. • Learning: Given a series of observations, learn the best model λ Learning Seminar, 2004

10. Likelihood in HMMs • Given a model λ = {A,B,q}, we can compute the likelihood by P(y) = p(y| λ) = Σp(x)p(y|x) = = q(x1)ΠA(xt+1|xt) ΠB(yt|xt) • But … this computation complexity is O(NT), when |xi| = N  impossible in practice Learning Seminar, 2004

11. Forward-Backward algorithm • To compute likelihood: • Need to enumerate over all paths in the lattice (all possible instantiations of X1…XT). But … some starting subpath(blue) is common to many continuing paths (blue+red) • The idea: • using dynamic programming, calculate a path in terms of shorter sub-paths Learning Seminar, 2004

12. Forward-Backward algorithm (cont’d) • We build a matrix of the probability of being at time t at state i: αt(i)=P(xt=i,y1y2…yt). This is a function of the previous column (forward procedure): Learning Seminar, 2004

13. Forward-Backward algorithm (cont’d) We can similarly define a backwards procedure for filling the matrix βt(i) = P(yt+1…yT|xt=i) Learning Seminar, 2004

14. Forward-Backward algorithm (cont’d) • And we can easily combine: P(y,xt=i) = P(xt=i,y1y2…yt)* P(yt+1…yT|xt=i)= =αt(i)βt(i) • And then we get: P(y) = ΣP(y,xt=i) = Σαt(i)βt(i) • Summary: we presented a polynomial algorithm for computing likelihood in HMMs. Learning Seminar, 2004

15. HMM – why not? • Advantages: • Estimation very easy. • Closed form solution • The parameters can be estimated with relatively high confidence from small samples • But: • The model represents all possible (x,y) sequences and defines joint probability over all possible observation and label sequences  needless effort Learning Seminar, 2004

16. Discriminative Probabilistic Models Generative Discriminative “Solve the problem you need to solve”: The traditional approach inappropriately uses a generative joint model in order to solve a conditional problem in which the observations are given. To classify we need p(y|x) – there’s no need to implicitly approximate p(x). Learning Seminar, 2004

17. Discriminative Models - Estimation • Choose Θy to maximize conditional likelihood: L(Θy)= Σ log pΘy(yi|xi) • Estimation usually doesn’t have closed form • Example – MinMI discriminative approach (2nd week lecture) Learning Seminar, 2004

18. Maximum Entropy Markov Model • MEMM: • a conditional model that represents the probability of reaching a state given an observation and the previous state • These conditional probabilities are specified by exponential models based on arbitrary observation features Learning Seminar, 2004

19. The Label Bias Problem • The mass that arrives at the state must be distributed among the possible successor states • Potential victims: Discriminative Models Learning Seminar, 2004

20. The Label Bias Problem: Solutions • Determinization of the Finite State Machine • Not always possible • May lead to combinatorial explosion • Start with a fully connected model and let the training procedure to find a good structure • Prior structural knowledge has proven to be valuable in information extraction tasks Learning Seminar, 2004

21. Random Field Model: Definition • Let G = (V, E) be a finite graph, and let A be a finite alphabet. • The configuration spaceΩ is the set of all labelings of the vertices in V by letters in A. If C is a part of V and ω is an element of Ω is a configuration, the ωc denotes the configuration restricted to C. • A random field on G is a probability distribution on Ω. Learning Seminar, 2004

22. Random Field Model: The Problem • Assume that a finite number of features can define a class • The features fi(w) are given and fixed. • The goal: estimating λ to maximize likelihood for training examples Learning Seminar, 2004

23. Conditional Random Field: Definition • X – random variable over data sequences • Y - random variable over label sequences • Yi is assumed to range over a finite label alphabet A • Discriminative approach: we construct a conditional model p(y|x) and do not explicitly model marginal p(x) Learning Seminar, 2004

24. CRF - Definition • Let G = (V, E) be a finite graph, and let A be a finite alphabet • Y is indexed by the vertices of G • Then (X,Y) is a conditional random field if the random variables Yv, conditioned on X, obey the Markov property with respect to the graph: p(Y|X,Yw,w≠v) = p(Yv|X,Yw,w~v), where w~v means that w and v are neighbors in G Learning Seminar, 2004

25. CRF on Simple Chain Graph • We will handle the case when G is a simple chain: G = (V = {1,…,m}, E={ (I,i+1) }) HMM (Generative) MEMM (Discriminative) CRF Learning Seminar, 2004

26. Fundamental Theorem of Random Fields (Hammersley & Clifford) • Assumption: • G structure is a tree, of which simple chain is a private case Learning Seminar, 2004

27. CRF – the Learning Problem • Assumption: the features fk and gk are given and fixed. • For example, a boolean feature gk is TRUE if the word Xi is upper case and the label Yi is a “noun”. • The learning problem • We need to determine the parameters Θ = (λ1, λ2, . . . ; µ1, µ2, . . .) from training data D = {(x(i), y(i))} with empirical distribution p~(x, y). Learning Seminar, 2004

28. CRF – Estimation • And we return to the log-likelihood maximization problem, this time – we need to find Θ that maximizes the conditional log-likelihood: Learning Seminar, 2004

29. CRF – Estimation • From now on we assume that the dependencies of Y, conditioned on X, form a chain. • To simplify some expressions, we add special start and stop states Y0 = start and Yn+1 = stop. Learning Seminar, 2004

30. CRF – Estimation • Suppose that p(Y|X) is a CRF. For each position i in the observation sequence X, we define the |Y|*|Y| matrix random variable Mi(x) = [Mi(y', y|x)] by: ei is the edge with labels (Yi-1,Yi) and vi is the vertex with label Yi Learning Seminar, 2004

31. CRF – Estimation • The normalization function Z(x) is • The conditional probability of a label sequence y is written as Learning Seminar, 2004

32. Parameter Estimation for CRFs • The parameter vector Θ that maximizes the log-likelihood is found using a iterative scaling algorithm. • We define standard HMM-like forward and backward vectors α and β, which allow polynomial calculations. • For example: Learning Seminar, 2004

33. Experimental Results – Set 1 • Set 1: modeling label bias • Data was generated from a simple HMM which encodes a noisy version of the finite-state network (“rib/ rob”) • Each state emits its designated symbol with probability 29/32 and any of the other symbols with probability 1/32 • We train both an MEMM and a CRF • The observation features are simply the identity of the observation symbols. • 2, 000 training and 500 test samples were used • Results: • CRF error: 4.6% • MEMM error: 42% • Conclusion: • MEMM fails to discriminate between the two branches and we get the label bias problem Learning Seminar, 2004

34. Experimental Results – Set 2 • Set 2: modeling mixed order sources • Data was generated from a mixed-order HMM with state transition probabilities given by p(yi|yi-1, yi-2) = α p2(yi|yi-1, yi-2) + (1 - α) p1(yi|yi-1) • Similarly, emission probabilities given by p(xi|yi, xi-1) = α p2(xi|yi, xi-1)+(1- α) p1(xi|yi) • Thus, for α = 0 we have a standard first-order HMM. • For each randomly generated model, a sample of 1,000 sequences of length 25 is generated for training and testing. Learning Seminar, 2004

35. Experimental Results – Set 2 Learning Seminar, 2004

36. Experimental Results – Set 3 • Set 3: Part-Of-Speech tagging experiments Learning Seminar, 2004

37. Conclusions • Conditional random fields offer a unique combination of properties: • discriminatively trained models for sequence segmentation and labeling • combination of arbitrary and overlapping observation features from both the past and future • efficient training and decoding based on dynamic programming for a simple chain graph • parameter estimation guaranteed to find the global optimum • CRFs main current limitation is the slow convergence of the training algorithm relative to MEMMs, let alone to HMMs, for which training on fully observed data is very efficient. Learning Seminar, 2004

38. Thank you Learning Seminar, 2004