# Hidden Markov Models

## What are Hidden Markov Models?

Hidden Markov Models (HMMs) are a class of probabilistic models used to represent systems that evolve over time and exhibit both observable and hidden (or latent) variables. HMMs are based on the Markov assumption, which states that the future state of the system depends only on the current state, not on the past states. HMMs are widely used in various applications, such as speech recognition, natural language processing, bioinformatics, and finance.

## How do Hidden Markov Models work?

Hidden Markov Models work by modeling a system as a sequence of hidden states, with each state emitting an observable output based on a certain probability distribution. The model consists of three main components:

1. State transition probabilities: The probabilities of transitioning from one hidden state to another.
2. Emission probabilities: The probabilities of observing a particular output given a hidden state.
3. Initial state probabilities: The probabilities of starting in each hidden state.

Given a sequence of observed outputs, HMMs can be used to solve various tasks, such as finding the most likely sequence of hidden states (decoding), estimating the model parameters (learning), or computing the likelihood of the observed outputs (evaluation).

## Example of Hidden Markov Models in Python

To work with Hidden Markov Models in Python, you can use the `hmmlearn` library:

``````\$ pip install hmmlearn
``````

Here’s a simple example of using an HMM for sequence prediction:

``````import numpy as np
from hmmlearn import hmm

# Create a synthetic dataset
states = ["sunny", "rainy"]
obs = ["walk", "shop", "clean"]
state_seq = np.array([0, 0, 1, 1, 0, 1]).reshape(-1, 1)
obs_seq = np.array([0, 1, 1, 2, 0, 2]).reshape(-1, 1)

# Set up the HMM model
model = hmm.MultinomialHMM(n_components=2)
model.startprob_ = np.array([0.6, 0.4])
model.transmat_ = np.array([[0.7, 0.3],
[0.4, 0.6]])
model.emissionprob_ = np.array([[0.6, 0.3, 0.1],
[0.1, 0.3, 0.6]])

# Train the HMM model
model.fit(obs_seq)

# Predict the hidden state sequence for a new observation sequence
new_obs_seq = np.array([0, 1, 1, 0, 2]).reshape(-1, 1)
logprob, state_pred = model.decode(new_obs_seq)
print("Predicted state sequence:", state_pred)
``````

In this example, we create a synthetic dataset representing the hidden states (sunny and rainy) and observations (walk, shop, and clean). We then set up an HMM with multinomial emissions and train it on the observed sequence. Finally, we use the trained model to predict the hidden state sequence for a new observation sequence.

## Additional resources on Hidden Markov Models

To learn more about Hidden Markov Models, you can explore the following resources: