# What is Hidden Markov Models

##### Hidden Markov Models: Understanding the Basics

Hidden Markov Models (HMMs) are statistical models used to represent and analyze sequences of data. They are commonly used in artificial intelligence, machine learning, and natural language processing applications. HMMs are powerful tools for modeling sequential data, such as speech recognition, bioinformatics, and image processing.

In this article, we will cover the basics of HMMs, how they work, and their applications.

##### Understanding Markov Models

Before we dive into the workings of HMMs, we first need to understand Markov models. A Markov model is a mathematical model used to represent systems that change over time. These systems are assumed to be "Markovian," meaning that the probability of the system transitioning from one state to another depends only on its current state, and not on any previous states.

For example, consider a simple weather model. We can represent the weather as a Markov process where the weather on any given day is dependent on the weather on the previous day. If it is sunny today, the probability of it being sunny tomorrow is higher than if it were rainy today.

Markov models are often used in language modeling, where the probability of a word occurring in a sentence is dependent only on the previous words in that sentence.

##### Introducing Hidden Markov Models

While Markov models are useful for modeling systems that are observable and easily quantifiable, many real-world systems have hidden or unobserved states. This is where HMMs come in.

An HMM is a statistical model that extends Markov models to handle unobservable or hidden states. In an HMM, we assume that there is a hidden state sequence that generates the observable data that we see.

For example, consider a speech recognition system. We can model the speech as a sequence of sounds, and we assume that there is a hidden sequence of phonemes that generated the sound sequence. In an HMM, the goal is to learn the parameters of the model that best represent the relationship between the hidden state sequence and the observable data.

##### How an HMM Works

An HMM is characterized by three components:

• The set of hidden states
• The set of observable symbols
• The parameters that define the probabilities of transitioning between states and generating observable symbols

Let's break down each of these components.

###### The Set of Hidden States

The set of hidden states in an HMM represents the unobservable variables that generate the observable data. For example, in a speech recognition system, the hidden states could represent the sequence of phonemes that generate the sounds.

Each hidden state is associated with a probability distribution that describes the likelihood of generating each observable symbol. This is known as the emission probability distribution.

###### The Set of Observable Symbols

The set of observable symbols in an HMM represents the observed variables that are generated by the hidden states. For example, in a speech recognition system, the observable symbols could represent the sequence of sounds that the speaker produces.

###### The Parameters of the Model

The parameters of an HMM consist of the probabilities of transitioning between hidden states and the probabilities of generating observable symbols from each hidden state.

The probabilities of transitioning between hidden states are represented by the transition probability matrix, A. This matrix contains the probability of transitioning from one hidden state to another.

The probabilities of generating observable symbols from each hidden state are represented by the emission probability distribution, B. This distribution contains the probability of emitting each observable symbol from each hidden state.

Finally, the initial distribution, π, contains the probabilities of starting in each hidden state.

###### The Forward Algorithm

The forward algorithm is an algorithm used to compute the likelihood of a sequence of observable symbols given an HMM.

The algorithm works as follows:

1. Initialize the forward variable α(1) to be the initial distribution, π.
2. For each observation t in the input sequence, update the forward variable as follows:
• Multiply the previous forward variable, α(t-1), by the transition matrix, A.
• Multiply the result by the emission probability distribution, B.
• Sum the result over all possible hidden states.
3. Return the probability of the input sequence, which is the sum of the final forward variable.

The forward algorithm is used to evaluate the likelihood of a sequence of observable symbols given an HMM. It is also used as a subroutine in many other algorithms, such as the Viterbi algorithm and the Baum-Welch algorithm.

##### Learning HMM Parameters

Learning the parameters of an HMM is an important task in many applications. The parameters define the PDFs and transition probability matrix that describe the underlying system.

The Baum-Welch algorithm is an iterative algorithm used to estimate the parameters of an HMM from a set of sequences. The algorithm works as follows:

1. Initialize the parameters randomly.
2. Compute the forward and backward variables for each sequence.
3. Compute the intermediate variable γ(i,t) and ξ(i,j,t) for each sequence, where γ(i,t) is the probability of being in hidden state i at time t, and ξ(i,j,t) is the probability of transitioning from hidden state i to hidden state j at time t.
4. Compute the new values for the transition matrix, A, the emission matrix, B, and the initial distribution, π, using the intermediate variables γ(i,t) and ξ(i,j,t).
5. Repeat steps 2-4 until convergence.

The Baum-Welch algorithm is a form of expectation-maximization (EM) algorithm, which is a general-purpose algorithm for maximum likelihood estimation that is widely used in machine learning and artificial intelligence. The EM algorithm works by iteratively computing the expected value of the complete data likelihood (the E-step) and then maximizing this expected value with respect to the parameters (the M-step).

##### Applications of Hidden Markov Models

HMMs are widely used in various fields, including:

In speech recognition, HMMs are used to model the relationship between speech sounds and their phonetic representations. In natural language processing, they are used to model the relationship between words and their grammatical roles.

In bioinformatics, HMMs are used to identify genes and other functional elements in DNA sequences. In image alignment and tracking, they are used to track the motion of objects in video sequences. In financial modeling, they are used to model the probabilities of market movements and to predict asset prices.

##### Conclusion

Hidden Markov Models are powerful statistical models used to represent and analyze sequences of data. They are widely used in various fields, including speech recognition, natural language processing, bioinformatics, and image processing. HMMs are particularly useful for modeling sequential data and systems with hidden or unobservable states.

In this article, we covered the basics of HMMs, how they work, and their applications. We hope that this article has provided you with a clear understanding of what HMMs are and how they can be used.