Hidden Markov Model (HMM)

The model operates under the first-order Markov assumption: the probability of transitioning to the next hidden state depends only on the current state, not on the full history of previous states. This memoryless property simplifies computation and is formalized as:

• Transition Probability Matrix (A): Defines P(q_t | q_{t-1}), the probability of moving from state i to state j.
• This assumption enables efficient recursive algorithms but limits the model's ability to capture long-range dependencies without higher-order extensions.

The system's true state is latent and unobservable. You never see the actual state sequence; you only see a sequence of emissions that are probabilistically generated by those hidden states.

• Finite State Space: The model has a fixed set of N discrete hidden states, e.g., {Browsing, Comparing, Ready-to-Purchase}.
• Initial State Distribution (π): A probability vector defining the likelihood of the sequence starting in each hidden state.
• Inference involves decoding the most likely hidden state sequence given the observations.

Each hidden state governs a probability distribution over the observable outputs. The Emission Matrix (B) defines P(o_t | q_t), the likelihood of observing a specific symbol given the current hidden state.

• Discrete Emissions: States emit symbols from a finite alphabet (e.g., product categories viewed).
• Continuous Emissions: States can emit continuous values using Gaussian Mixture Models (GMM-HMM), common in speech recognition where emissions are acoustic feature vectors.
• The emission distribution is the sole link between the latent dynamics and the visible data.

Every application of an HMM is built on solving three fundamental problems, each with a dedicated dynamic programming algorithm:

• Evaluation (The Forward Algorithm): Given a model λ = (A, B, π) and an observation sequence O, compute P(O|λ)—the likelihood that the model generated the sequence. Used for scoring and classification.
• Decoding (The Viterbi Algorithm): Find the single most likely sequence of hidden states Q* that explains the observation sequence. Used for intent prediction and state labeling.
• Learning (The Baum-Welch Algorithm): Given an observation sequence, adjust the model parameters λ to maximize P(O|λ). This is a special case of Expectation-Maximization (EM) for unsupervised training.

HMMs are fundamentally generative models. They learn the joint probability distribution P(X, Y) of the observations and the hidden state sequence, then use Bayes' rule to infer the conditional P(Y|X).

• Generative Strength: They model the complete data generation process, making them robust for sequence synthesis and anomaly detection.
• Contrast with CRFs: Conditional Random Fields (CRFs) are discriminative and directly model the conditional boundary P(Y|X), often outperforming HMMs on sequence labeling tasks where complex, overlapping features are critical.
• In practice, HMMs excel when the underlying state dynamics are physically meaningful and data is limited.

Standard HMMs exhibit an implicit geometric state duration distribution. The probability of remaining in a state for d time steps decays exponentially.

• Limitation: This is a poor fit for real-world behaviors with fixed or non-exponential durations (e.g., a mandatory 30-second video ad).
• Explicit Duration HMMs: An advanced variant that explicitly models state duration with a parameterized distribution (e.g., Poisson or Gamma), significantly improving performance in applications like user dwell time modeling and activity recognition where state persistence matters.