Hidden Markov Models (HMM)

The fundamental assumption that the future state depends only on the current state, not on the full sequence history. In a clinical context, this means a patient's next health state transition is conditionally independent of their entire prior trajectory given their present state.

• First-order Markov chain: P(q_t | q_{t-1}, ..., q_1) = P(q_t | q_{t-1})
• Simplifies computation for longitudinal patient modeling
• Enables tractable inference over long observation sequences

HMMs maintain a strict separation between the latent health states (disease stage, underlying phenotype) and the emission observations (lab values, vital signs, symptoms). The true biological condition is never directly measured—only its noisy clinical manifestations are observed.

• Hidden states: Unobserved disease progression stages
• Emission probabilities: Likelihood of observing specific clinical measurements given a hidden state
• Models the diagnostic uncertainty inherent in clinical practice

A stochastic matrix A where each element a_ij represents the probability of transitioning from hidden state i to hidden state j. In patient stratification, this captures the natural history of disease progression.

• Rows sum to 1.0 (stochastic constraint)
• Can model irreversible progression by setting certain transitions to zero
• Learned from longitudinal cohort data via the Baum-Welch algorithm
• Example: A 3-state model (Stable → Progressive → Severe) with estimated transition rates between each

The observation model B defines how hidden states generate visible clinical data. For continuous biomarkers like blood pressure, Gaussian emissions are typical; for categorical symptoms, multinomial distributions are used.

• Continuous emissions: Model lab values with mean μ_k and variance σ²_k per state
• Discrete emissions: Model symptom presence/absence with categorical probabilities
• Enables soft assignment of patients to states based on observed measurements
• Critical for distinguishing phenotypically similar but mechanistically distinct subgroups

HMMs are defined by three computational problems that must be solved for any clinical application:

1. Evaluation (Forward-Backward Algorithm): Given a model and an observation sequence, compute the likelihood P(O|λ). Used to score how well a patient fits a disease progression model.
2. Decoding (Viterbi Algorithm): Find the most likely sequence of hidden states given observations. Used to estimate a patient's disease trajectory.
3. Learning (Baum-Welch/EM): Estimate model parameters (A, B, π) from training data. Used to discover disease state transitions from cohort data.

The prior probability vector π specifies the likelihood of a patient starting in each hidden state at time t=0. In disease modeling, this often reflects population-level prevalence of different disease stages at baseline.

• Constrains π_i ≥ 0 and Σ π_i = 1
• Can incorporate clinical priors (e.g., screening data) to improve initialization
• Influences early trajectory inference before sufficient longitudinal data accumulates
• Example: A cohort with 70% early-stage, 25% mid-stage, 5% late-stage disease at enrollment

A foundational probabilistic model that assumes data is generated from a mixture of Gaussian distributions. Unlike HMMs, GMMs treat observations as independent and identically distributed, ignoring temporal sequence. In patient stratification, GMMs provide soft cluster assignments where each patient has a probability of belonging to each subgroup. This contrasts with HMMs, which model transitions between latent states over time. GMMs are often used as the emission distributions within HMMs to model continuous clinical variables.

A computational method that orders patients or cells along a continuous pseudotemporal path based on molecular profiles. While HMMs model discrete state transitions, trajectory inference captures smooth, continuous disease progression. Techniques like Monocle and Slingshot construct minimum spanning trees in reduced-dimensional space. This is critical for understanding dynamic processes like tumor evolution or immune response, where rigid discrete states fail to capture the underlying biology.

A class of models, including the Dirichlet Process Mixture and the Hierarchical Dirichlet Process HMM, that allow the number of latent states to grow with the data rather than being fixed a priori. This solves a critical limitation of standard HMMs: the need to pre-specify the number of health states. In patient stratification, this enables the discovery of novel disease subtypes without imposing artificial boundaries, letting the data determine the optimal number of clinical trajectories.

A resampling-based methodology that aggregates results from multiple clustering runs to identify robust patient subgroups. When applied to HMM-derived state sequences, consensus clustering validates whether discovered health state transitions are stable and reproducible across data perturbations. This addresses a key challenge in longitudinal stratification: distinguishing true biological signal from algorithmic noise. The consensus matrix quantifies the co-occurrence probability of patients in the same cluster.

The process of identifying distinct disease subtypes defined by a specific functional or pathobiological mechanism. HMMs contribute to endotype discovery by modeling the temporal evolution of molecular markers, revealing not just static differences but distinct mechanistic trajectories. For example, in asthma, HMMs can differentiate patients with persistent eosinophilic inflammation from those with intermittent neutrophilic flares, guiding targeted biologic therapy selection.

A computational method for integrating diverse patient data types by constructing and fusing patient similarity networks. When combined with HMMs, SNF can create a unified similarity matrix from static omics data, which then informs the initial state probabilities or transition dynamics of the HMM. This hybrid approach leverages the temporal modeling power of HMMs while benefiting from SNF's ability to find coherent patterns across heterogeneous data modalities like genomics, imaging, and clinical labs.