What Is the Difference Between Markov Chains and Hidden Markov Models?

Markov Chains and Hidden Markov Models (HMMs) are fundamental concepts in the field of probability theory and statistics, with extensive applications ranging from economics and finance to biology and computer science. Understanding these models can provide significant insights into the dynamics of various stochastic processes.

In this article, we going to explore the fundamentals of Markov Chains and Hidden Markov Models, and their differences.

Choosing between Markov Chains and HMMs largely depends on the specifics of the application. Markov Chains are well-suited to scenarios where state transitions are clear and directly observable, such as in simple prediction games or basic weather modeling. On the other hand, HMMs excel in complex environments where the underlying states cannot be directly observed, such as in speech recognition, genetic sequence analysis, or any context where the system's internal states are inferred from the outputs they produce.

What is a Markov Chain?

A Markov Chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. This property is known as the Markov property or memorylessness. Markov Chains are characterized by:

• States: The distinct conditions or positions that the system can be in.
• Transition Probabilities: The probabilities of moving from one state to another, which are typically represented in a matrix form.

Markov Chains are used to model and predict a variety of systems such as inventory levels in business, queue lengths in operations research, and even to some extent the behavior patterns in social sciences.

What is a Hidden Markov Model?

While a Markov Chain assumes that the underlying states are directly visible to the observer, a Hidden Markov Model deals with situations where the states are hidden and not directly observable. Instead, certain observations related to the states are visible. The components of an HMM include:

• Hidden States: The actual states of the model, which are not directly observable.
• Observations: The visible outputs, which are influenced by the hidden states.
• Transition Probabilities: As in Markov Chains, these describe the chance of transitioning from one hidden state to another.
• Emission Probabilities: These probabilities model the likelihood of observing a particular visible state given a hidden state.
• Initial State Probabilities: These are the probabilities of starting in a particular state.

HMMs are particularly useful in areas like speech recognition, where the spoken words (hidden states) are inferred from the sound signals (observations). Other applications include biological sequence analysis and financial market analysis.

Comparative Analysis: Markov Chains vs. Hidden Markov Models

Direct Comparison of Markov Chains and Hidden Markov Models

1. State Visibility:

• Markov Chains: All states in a Markov Chain are completely observable. This transparency allows for direct computation of probabilities associated with state transitions.
• Hidden Markov Models: States in HMMs are not observable (hidden); what is observed is a set of outputs influenced by these states. The true state is inferred through these observations, which adds a layer of complexity to the model.

2. Model Complexity:

• Markov Chains: Simpler in nature, involving fewer parameters — primarily the transition probabilities between observable states.
• Hidden Markov Models: More complex, requiring additional parameters such as emission probabilities (linking hidden states to observable outputs) and initial state probabilities.

3. Parameter Estimation:

• Markov Chains: Transition probabilities can be estimated directly from the sequence of observed states.
• Hidden Markov Models: Estimation involves algorithms like the Baum-Welch algorithm, which iteratively adjusts the model parameters to maximize the likelihood of the observed sequence given the model.

4. Dependence on Observations:

• Markov Chains: The model's future state predictions depend solely on the current state, not on the path taken to reach that state.
• Hidden Markov Models: Predictions depend on observed outputs, which indirectly give information about the path of hidden states.

Visual Diagrams

1. Markov Chain Diagram:

• Shows a simple chain of states with arrows indicating transition probabilities.
• Example: A weather model where states are "Sunny", "Rainy", "Cloudy", and arrows show the probability of moving from one state to another from one day to the next.

2. Hidden Markov Model Diagram:

• Similar to the Markov Chain diagram but includes a second layer representing the observable outputs.
• Arrows from hidden states to observed outputs indicate emission probabilities.
• Example: A speech recognition model where hidden states might represent phonemes and observed outputs are digital sound signals.

Discussion on Visibility of States

• Markov Chains: The complete visibility of states simplifies both the understanding and implementation of the model. Applications typically involve straightforward processes where the state directly affects the outcome, like predicting the next letter in a sequence or customer behavior in a store.
• Hidden Markov Models: The hidden nature of states makes HMMs ideal for scenarios where the process being modeled is not directly observable or is influenced by non-visible factors. Applications include decoding encrypted information, speech recognition, or biological phenomena, where the observed sequences (like DNA) provide clues about underlying states (genetic traits).

Summary of Markov Chains vs. Hidden Markov Models

Technical Differences: Markov Chains vs. Hidden Markov Models

The technical differences between Markov Chains and Hidden Markov Models (HMMs) stem primarily from their mathematical structure and the way they utilize probabilities to model systems and processes. Here’s an in-depth look at these distinctions:

Mathematical Formulation

1. Markov Chains:
   • Formulation: A Markov Chain is defined by a set of states S and a transition matrix P where each element P_{ij} represents the probability of transitioning from state i to state j. Mathematically, it is expressed as:
     • P_{ij} = Pr(X_{n+1} =j | X_n = i)
   • State Sequence: The future state depends only on the current state, reflecting the Markov property:
     • Pr(X_{n+1}| X_n, X_{n-1}, ... , X_1) = Pr(X_{n+1}|X_n)
2. Hidden Markov Models:
   • Formulation: An HMM involves hidden states and observable outputs. It is defined by:
   • Pr(X_1 = i) = \pi_i , Pr(X_{n+1} = j | X_n = i) = A_{ij} , Pr(Y_n =k| X_n = j) = B_{jk}
     • Transition probabilities A, where A_{ij}​ is the probability of transitioning from hidden state i to j.
     • Emission probabilities B, where B_{ij}​ is the probability of observing output j from hidden state i.
     • Initial state probabilities \pi , where \pi is the probability of the system starting in state i.
   • State and Observation Sequence: The observed outputs depend on the current hidden state, and the hidden state sequence follows the Markov property, but is not directly observable.

Usage of Transition Probabilities

• Markov Chains: Transition probabilities directly determine the likelihood of moving from one observable state to another. These probabilities are crucial for predicting future states and are estimated from directly observable data.
• Hidden Markov Models: Transition probabilities in HMMs determine the likelihood of moving between hidden states, which cannot be directly observed. The estimation of these probabilities must be inferred indirectly through observed outputs using algorithms like the Baum-Welch algorithm.