Moving from Markov to Hidden Markov Models: A Practical Guide

When dealing with systems where the underlying process cannot be directly observed, models like the Hidden Markov Model (HMM) become indispensable tools for analysis. This article explores why and how we shift from a traditional Markov model to an HMM for modeling sequential data in scenarios where only partial, noisy observations are available.

Introduction to Markov and Hidden Markov Models

A Markov Model is a probabilistic model that describes a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. This makes it a powerful tool for modeling scenarios where the future state is influenced only by the current state, such as weather prediction. However, its simplicity also comes with limitations. In many practical scenarios, especially in natural language processing and bioinformatics, the true state might not be directly observable. This is where the Hidden Markov Model (HMM) comes in.

Why Move to an HMM?

The primary motivation for moving from a Markov model to an HMM is the ability to make use of sequential observations that are not directly indicative of the hidden states. Consider a scenario where you can only observe the humidity levels on a window and want to infer the true state, which could be rainy or sunny. In scenarios such as this, the HMM provides a more accurate model by incorporating the dynamics of hidden states that are not directly observable, yet influence the observed data.

Practical Examples of HMM

Natural Language Processing (NLP): In NLP, the true mood of a speaker might not be directly observable, but we can observe behaviors like the frequency of laughter or the length of pauses. HMMs can model these indirect signals to infer the true mood.

Bioinformatics: In the study of HIV/AIDS, the true state of a patient (e.g., the level of the virus in the blood) cannot be directly observed, but we have indirect measures like white blood cell counts. HMMs can help in understanding the dynamics of the disease by analyzing these observable measures.

Technical Details

An HMM assumes that the observations are functions of some underlying Markov chain that we do not directly observe. This means that while the past states are not directly observable, their influence contributes to the characteristics of the current observation. This 'memory' of past states is what makes HMMs more powerful in inferential tasks compared to traditional Markov models.

Mathematically, an HMM is defined by the following components:

Hidden States: The underlying states that we are trying to infer. Observations: The data that we can directly observe, which are functions of the hidden states. Transition Probabilities: The probabilities of moving from one hidden state to another. Emission Probabilities: The probabilities of observing a particular symbol given a hidden state.

The key advantage of HMMs lies in their ability to balance the need for incorporating past information while maintaining a manageable number of parameters. Unlike the Markov chain with an exponential increase in parameters for higher-order models, HMMs are generally more efficient and scalable.

Conclusion

The transition from a Markov model to an HMM is a strategic move in scenarios where the underlying state is unknown or not directly observable. By leveraging the HMM, we can effectively analyze and model complex, sequential data with partial observations, making it a vital tool in fields ranging from NLP to bioinformatics.