Hidden Markov Model (HMM) Framework

• Hidden Markov Model (HMM) is a statistical framework that models sequential data by inferring latent states and generating observable outputs.
• The NMF-based approach decomposes high-order empirical statistics to estimate state-dependent emission distributions and perform data-driven model order selection.
• HMM techniques are applied in fields such as speech recognition, computational biology, and finance, offering robust parameter estimation for complex time series.

A hidden Markov model (HMM) is a statistical framework in which an observable sequence is generated by an underlying, unobserved Markov process. This framework has become a foundational tool for modeling sequential data in fields such as computational biology, speech recognition, and machine learning, enabling efficient inference and parameter estimation for systems characterized by latent dynamics and observable outputs.

1. Fundamental Structure of Hidden Markov Models

An HMM consists of a finite set of hidden states $\{S_1, \ldots, S_N\}$, with transitions governed by a first-order Markov process and probabilistic emissions generating observable outputs (from a set of $M$ possible symbols or a continuous space). The model is parametrized by an initial state distribution $\pi$, a state transition matrix $A$, and a set of emission distributions $B$. For a sequence of observations $O_1, \ldots, O_T$, the joint likelihood is computed as: $$P(O_{1:T}, S_{1:T}\mid \lambda) = \pi_{S_1} \prod_{t=2}^T a_{S_{t-1},S_t} \prod_{t=1}^T b_{S_t}(O_t),$$ where $\lambda = (\pi, A, B)$. The classic inference tasks involve decoding the most probable sequence of hidden states (e.g., via the Viterbi algorithm) and parameter learning (e.g., via the Baum–Welch or EM algorithm).

2. NMF-Based HMM Learning: Matrix Factorization of High-Order Statistics

The paper "Learning Hidden Markov Models using Non-Negative Matrix Factorization" (0809.4086) introduces an alternative HMM learning technique that departs from the Expectation–Maximization tradition. The core innovation is to represent empirical high-order Markov statistics through matrices such as $R^{(p,s)}$ and $F^{(p,s)}$, constructed from the observed data $O_{1:T}$. Here:

• $R^{(p,s)}$ is a contingency histogram counting frequencies of length-$p$ prefixes and length-$s$ suffixes,
• $F^{(p,s)}$ is $R^{(p,s)}$ normalized row-wise, yielding conditional probabilities $$F^{(p,s)}_{u,v} \approx P(\text{suffix } v \mid \text{prefix } u)$$.

A key theoretical insight is that, in the presence of an underlying HMM with $N$ recurrent states, $F^{(p,s)}$ is approximately factorizable as $F^{(p,s)} \approx C D$, where $C$ and $D$ are non-negative matrices with:

• $C_{u,k} \approx P(S_k \mid U, p, \lambda)$ collects posterior state probabilities given the prefix,
• $D_{k,v} \approx P(v \mid S_k, s, \lambda)$ gives the emission distribution of suffix $v$ from state $S_k$.

The factorization is obtained by solving: $\min_{C,D} D_{ID}(F^{(p,s)} \Vert C D),$ where $D_{ID}$ denotes the I-divergence. This approach enables a mixture decomposition for each observed conditional distribution, using the statewise emission templates as mixture components.

3. Distinctions from the Baum–Welch Algorithm

The NMF-based approach diverges from classical Baum–Welch/EM strategies in several ways:

• Parameter estimation is performed via matrix factorization on aggregated statistics $F^{(p,s)}$, as opposed to iterative marginalization over full state trajectories.
• The input to learning is the sufficient statistics matrix, providing computational compression and robustness, particularly with long or noisy sequences.
• Decomposition via NMF permits an explicit convex mixture representation of output statistics, directly connecting prefixes to state-dependent generative templates.
• Unlike the EM scheme, in which all parameters are updated via local likelihood maximization, the NMF-based learning globally adjusts mixture matrices to best match empirical conditional distributions.

A limitation of the NMF approach is that, due to the NP-hardness of exact nonnegative matrix factorization, only approximate (locally optimal) solutions are generally available.

4. Estimation of the Number of Recurrent States

The method identifies the correct number of recurrent (ergodic) states $N$ by examining the spectral properties of $F^{(p,s)}$. The “positive rank” or “prank” of the matrix equates to the minimal number of required hidden states in noiseless conditions. In practice:

• One computes the singular value decomposition of $F^{(p,s)}$ or a weighted version,
• A significant gap after the $N$th singular value identifies $N$ as the positive rank,
• The relation $$\mathrm{rank}(F^{(p,s)}) \leq N \leq \mathrm{prank}(F^{(p,s)})$$ provides a data-driven lower and upper bound on the minimal number of states,
• Spectral analysis using SVD thus provides a principled, empirical means for model order selection.

This approach is robust to overfitting induced by noise, as high-order empirical matrices often show a clear spectral gap even with moderate data lengths.

5. Iterative Refinement and Algorithmic Workflow

The learning process iterates between NMF of high-order statistics and estimation of HMM parameters:

1. Compute $F^{(p,s)}$ from the observed data and obtain initial NMF factors $C_0$, $D_0$.
2. Extract state-dependent emission distributions from $D_0$; compute transition matrices by solving a linear system relating suffix distributions (e.g., using equations akin to $$P(v_1 V^{(s-1)}\mid S_i, s, \lambda) = \sum_j a_{ij}(1)P(V^{(s-1)}\mid S_j, s-1, \lambda)$$).
3. Given estimated transitions and emissions, regenerate $D_0'$ using the newly computed HMM, and refactor $C_0'$ via a linear programming problem subject to constraint enforcement.
4. Repeat the NMF and parameter estimation cycle, refining transition and emission probability matrices to better match $F^{(p,s)}$.

This iterative loop aims to minimize the divergence between the model’s synthetic high-order statistics and those observed empirically.

6. Empirical Validation and Application Examples

The paper validates the method on synthetic and semi-synthetic models:

• In a deterministic HMM (e.g., the “even process”), the approach accurately infers the SVD rank and reconstructs ground-truth transition matrices after two iterations.
• For models with degenerate statistical behavior (e.g., HMMs indistinguishable from deterministic HMMs), the learned models yield observation statistics almost identical to the source process, with extremely small I-divergence rates.
• In genuinely stochastic models where no deterministic HMM exists, the procedure requires higher-order statistics but provides strong agreement between fitted and true models.
• An example where $\operatorname{rank}(F) < \operatorname{prank}(F)$ is included to illustrate complications when factorization structures cannot always be mapped to valid HMMs, reinforcing the role of positive rank in characterizing HMM complexity.

7. Applicability and Methodological Implications

The NMF-based framework is particularly well-suited for:

• Speech recognition, e.g., for compressed or online inference from long audio streams,
• Pattern recognition and natural language processing, where long-range dependencies must be captured,
• Robust time series modeling in finance and anomaly detection,
• Computational biology, including gene or protein sequence analysis.

The approach provides an alternative to EM-based procedures by emphasizing global, compressed data statistics and spectral/model order inference. It offers a direct route to model complexity selection, is compatible with standard nonnegative matrix optimization routines, and leverages connections to automata-theoretic results (e.g., the Myhill–Nerode theorem).

In summary, learning HMMs via non-negative matrix factorization of higher-order empirical Markov matrices constitutes a robust and interpretable alternative to traditional EM methods, supporting both parameter estimation and model order selection in a unified, data-driven workflow (0809.4086).