Hidden markov model for NLP applications

Define formally the HMM, Hidden Markov Model and its usage in Natural language processing, Example HMM, Formal definition of HMM

Hidden Markov Model

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Hidden Markov Model (HMM) is a simple sequence labeling model. It is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (i.e. hidden) states. By relating the observed events (Example - words in a sentence) with the hidden states (Example - part of speech tags), it helps us in finding the most probable hidden state sequence (Example – most relevant POS tag sequence for the given input sentence).

HMM can be defined formally as a 5-tuple (Q, A, O, B, π) where each component can be defined as follows;

Component	Detailed components	Description
Q	q1, q2, q3, …, qN	Set of N hidden states
A	a11, a12, …, ann	Set of transition probabilities A is the state transition probability matrix Each aij in A represents a transition probability value of moving from state i to state j. Sum of transition probability values from a single state to all other states should be 1. That is,
O	o1, o2, …, oT	A sequence of T observations
B	bi(ot)	A sequence of observation likelihoods (emission probabilities) Each bi(ot) represents the emission probability. That is, the probability of an observation ot which  is generated from a state i.
π	π1, π2, …, πN	Set of initial probabilities. π1 is the probability that the Markov chain will start in state i. if π1 = 0, it implies that the state i cannot be an initial state. The sum of all initial probabilities should be 1. That is,

Understanding Hidden Markov Model - Example:

These components are explained with the following HMM. In this example, the states are related to the weather conditions (Hot, Wet, Cold) and observations are related to the fabrics that we wear (Cotton, Nylon, Wool).

As per the given HMM,

• Q = set of states = {Hot, Wet, Cold}

• A = transition probability matrix

o   Transition probability matrix

	Current state
Previous state		Hot	Wet	Cold
	Hot	0.6	0.3	0.1
	Wet	0.4	0.4	0.2
	Cold	0.1	0.4	0.5

o   How to read this matrix? In this matrix, for example, a_ij is a transition probability from state i to state j [which is represented as conditional probability P(j|i)];

a_ij = a₁₁ = P(Hot|Hot) = 0.6

a_ij = a₂₃ = P(Cold|Wet) = 0.2

a_ij = a₃₁ = P(Hot|Cold) = 0.1

o   Sum of transition probability from a single state to all the other states = 1. In other words, we would say that the total weights of arcs (or edges) going out of a state should be equal to 1. In our example;

P(Hot|Hot)+P(Wet|Hot)+P(Cold|Hot) = 0.6+0.3+0.1 = 1

• O = sequence of observations = {Cotton, Nylon, Wool}

• B = Emission probability matrix

o   Emission probability matrix

	Cotton	Nylon	Wool
Hot	0.8	0.5	0.05
Wet	0.15	0.4	0.2
Cold	0.05	0.1	0.75

o   The above said matrix consists of emission probability values represented as b_i(o_t). b_i(o_t) is the probability of an observation o_t generated from a state b_i.  For example, P(Nylon | Hot) = 0.5, P(Wool | Cold) = 0.75 etc.

• π = [π₁, π₂, …, π_N] = set of prior probabilities = [0.6, 0.3, 0.1]. Here, the values refer to the prior probabilities P(Hot) = 0.6, P(Wet) = 0.3, and P(Cold) = 0.1

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Hidden Markov Model (HMM)

Define formally the HMM

What is transition and emission probabilities?

Applications of HMM in NLP

Hidden markov model example