Albert Gatt Corpora and Statistical Methods Lecture 8

Markov and Hidden Markov Models: Conceptual Introduction Part 2

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

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

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?

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 (X 1, X 2, X 3,... X T ) models the value of a series of random variables each takes a value from S with a certain probability P(X=s i ) the entire sequence tells us the weather over T days

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.

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

Markov Properties II: Time invariance The probability of being in state s i given the previous state does not change over time:

Concrete instantiation Day tDay t+1 sunnyrainyfoggy sunny rainy foggy 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 a ij (prob. of going from state i to state j)

Graphical view Components of the model: 1. states (s) 2. transitions 3. transition probabilities 4. initial probability distribution for states Essentially, a non-deterministic finite state automaton.

Example continued If the weather today (X t ) is sunny, what’s the probability that tomorrow (X t+1 ) is sunny and the day after (X t+2 ) is rainy? Markov assumption

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

Hidden Markov Models

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: O t as an observation at time t K = {+umbrella, -umbrella} as the possible outputs We’re interested in P(O t =k|X t =s i ) i.e. p. of a given observation at t given that the underlying weather state at t is s i

Symbol emission probabilities weatherProbability of umbrella sunny0.1 rainy0.8 foggy0.3 This is the hidden model, telling us the probability that O t = k given that X t = s i We assume that each underlying state X t = s i emits an observation with a given probability.

Using the hidden model Model gives:P(O t =k|X t =s i ) Then, by Bayes’ Rule we can compute: P(X t =s i |O t =k) Generalises easily to an entire sequence

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

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

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?

HMMs in POS Tagging ADJNV DET  Hidden layer (constructed through training)  Models the sequence of POSs in the training corpus

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

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.

Formalisation of a Hidden Markov model

Crucial ingredients (familiar) Underlying states: S = {s 1,…,s N } Output alphabet (observations): K = {k 1,…,k M } State transition probabilities: A = {a ij }, i,j Є S State sequence: X = (X 1,…,X T+1 ) + a function mapping each X t to a state s Output sequence: O = (O 1,…,O T ) where each o t Є K

Crucial ingredients (additional) Initial state probabilities: Π = { π i }, i Є S (tell us the initial probability of each state) Symbol emission probabilities: B = {b ijk }, i,j Є S, k Є K (tell us the probability b of seeing observation O t =k, given that X t =s i and X t+1 = s j )

Trellis diagram of an HMM s1s1 s2s2 s3s3 a 1,1 a 1,2 a 1,3

Trellis diagram of an HMM s1s1 s2s2 s3s3 a 1,1 a 1,2 a 1,3 o1o1 o2o2 o3o3 Obs. seq: time: t1t1 t2t2 t3t3

Trellis diagram of an HMM s1s1 s2s2 s3s3 a 1,1 a 1,2 a 1,3 o1o1 o2o2 o3o3 Obs. seq: time: t1t1 t2t2 t3t3 b 1,1,k b 1,2,k b 1,3,k

The fundamental questions for HMMs 1. Given a model μ = (A, B, Π ), how do we compute the likelihood of an observation P(O| μ )? 2. Given an observation sequence O, and model μ, which is the state sequence (X 1,…,X t+1 ) that best explains the observations? This is the decoding problem 3. Given an observation sequence O, and a space of possible models μ = (A, B, Π ), which model best explains the observed data?

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?

Application of question 2 (POS Tagging) Given an observation sequence O, and model μ, which is the state sequence (X 1,…,X t+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?

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