Corpora and Statistical Methods Lecture 8

1. Corpora and Statistical MethodsLecture 8 Albert Gatt

2. Part 2 Markov and Hidden Markov Models: Conceptual Introduction

3. In this lecture • We focus on (Hidden) Markov Models • conceptual intro to Markov Models • relevance to NLP • Hidden Markov Models • algorithms

4. Acknowledgement • Some of the examples in this lecture are taken from a tutorial on HMMs by Wolgang Maass

5. Talking about the weather • Suppose we want to predict tomorrow’s weather. The possible predictions are: • sunny • foggy • rainy • We might decide to predict tomorrow’s outcome based on earlier weather • if it’s been sunny all week, it’s likelier to be sunny tomorrow than if it had been rainy all week • how far back do we want to go to predict tomorrow’s weather?

6. Statistical weather model • Notation: • S: the state space, a set of possible values for the weather: {sunny, foggy, rainy} • (each state is identifiable by an integer i) • X: a sequence of random variables, each taking a value from S • these model weather over a sequence of days • t is an integer standing for time • (X1, X2, X3, ... XT) models the value of a series of random variables • each takes a value from S with a certain probability P(X=si) • the entire sequence tells us the weather over T days

7. Statistical weather model • If we want to predict the weather for day t+1, our model might look like this: • E.g. P(weather tomorrow = sunny), conditional on the weather in the past t days. • Problem: the larger t gets, the more calculations we have to make.

8. Markov Properties I: Limited horizon • The probability that we’re in state si at time t+1 only depends on where we were at time t: • Given this assumption, the probability of any sequence is just:

9. Markov Properties II: Time invariance • The probability of being in state si given the previous state does not change over time:

10. Concrete instantiation This is essentially a transition matrix, which gives us probabilities of going from one state to the other. We can denote state transition probabilities as aij(prob. of going from state i to state j)

11. Graphical view • Components of the model: • states (s) • transitions • transition probabilities • initial probability distribution for states Essentially, a non-deterministic finite state automaton.

12. Example continued • If the weather today (Xt) is sunny, what’s the probability that tomorrow (Xt+1) is sunny and the day after (Xt+2) is rainy? Markov assumption

13. Formal definition • A Markov Model is a triple (S, , A) where: • S is the set of states •  are the probabilities of being initially in some state • A are the transition probabilities

14. Hidden Markov Models

15. A slight variation on the example • You’re locked in a room with no windows • You can’t observe the weather directly • You only observe whether the guy who brings you food is carrying an umbrella or not • Need a model telling you the probability of seeing the umbrella, given the weather • distinction between observations and their underlying emitting state. • Define: • Otas an observation at time t • K = {+umbrella, -umbrella} as the possible outputs • We’re interested in P(Ot=k|Xt=si) • i.e. p. of a given observation at t given that the underlying weather state at t is si

16. Symbol emission probabilities This is the hidden model, telling us the probability that Ot = k given that Xt = si We assume that each underlying state Xt = si emits an observation with a given probability.

17. Using the hidden model • Model gives:P(Ot=k|Xt=si) • Then, by Bayes’ Rule we can compute: P(Xt=si|Ot=k) • Generalises easily to an entire sequence

18. Circles indicate states Arrows indicate probabilistic dependencies between states HMM in graphics

19. HMM in graphics • Green nodes are hidden states • Each hidden state depends only on the previous state (Markov assumption)

20. Why HMMs? • HMMs are a way of thinking of underlying events probabilistically generating surface events. • Example: Parts of speech • a POS is a class or set of words • we can think of language as an underlying Markov Chain of parts of speech from which actual words are generated (“emitted”) • So what are our hidden states here, and what are the observations?

21. ADJ N V HMMs in POS Tagging DET • Hidden layer (constructed through training) • Models the sequence of POSs in the training corpus

22. ADJ N V tall lady is HMMs in POS Tagging DET the • Observations are words. • They are “emitted” by their corresponding hidden state. • The state depends on its previous state.

23. Why HMMs • There are efficient algorithms to train HMMs using Expectation Maximisation • General idea: • training data is assumed to have been generated by some HMM (parameters unknown) • try and learn the unknown parameters in the data • Similar idea is used in finding the parameters of some n-gram models, especially those that use interpolation.

24. Formalisation of a Hidden Markov model

25. Crucial ingredients (familiar) • Underlying states: S = {s1,…,sN} • Output alphabet (observations): K = {k1,…,kM} • State transition probabilities: A = {aij}, i,jЄ S • State sequence: X = (X1,…,XT+1) + a function mapping each Xt to a state s • Output sequence: O = (O1,…,OT) • where each otЄ K

26. Crucial ingredients (additional) • Initial state probabilities: Π = {πi}, i Є S (tell us the initial probability of each state) • Symbol emission probabilities: B = {bijk}, i,jЄ S, k Є K (tell us the probability b of seeing observation Ot=k, given that Xt=si and Xt+1 = sj)

27. Trellis diagram of an HMM a1,1 s1 a1,2 s2 a1,3 s3

28. Trellis diagram of an HMM a1,1 s1 a1,2 s2 a1,3 s3 o1 Obs. seq: o3 o2 t1 t3 time: t2

29. Trellis diagram of an HMM b1,1,k b1,1,k a1,1 s1 a1,2 b1,2,k s2 a1,3 b1,3,k s3 o1 Obs. seq: o3 o2 t1 t3 time: t2

30. The fundamental questions for HMMs • Given a model μ = (A, B, Π), how do we compute the likelihood of an observation P(O| μ)? • Given an observation sequence O, and model μ, which is the state sequence (X1,…,Xt+1) that best explains the observations? • This is the decoding problem • Given an observation sequence O, and a space of possible models μ = (A, B, Π), which model best explains the observed data?

31. Application of question 1 (ASR) • Given a model μ = (A, B, Π), how do we compute the likelihood of an observation P(O| μ)? • Input of an ASR system: a continuous stream of sound waves, which is ambiguous • Need to decode it into a sequence of phones. • is the input the sequence [n iy d] or [n iy]? • which sequence is the most probable?

32. Application of question 2 (POS Tagging) • Given an observation sequence O, and model μ, which is the state sequence (X1,…,Xt+1) that best explains the observations? • this is the decoding problem • Consider a POS Tagger • Input observation sequence: • I can read • need to find the most likely sequence of underlying POS tags: • e.g. is can a modal verb, or the noun? • how likely is it that can is a noun, given that the previous word is a pronoun?

33. Summary • HMMs are a way of representing: • sequences of observations arising from • sequences of states • states are the variables of interest, giving rise to the observations • Next up: • algorithms for answering the fundamental questions about HMMs