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On the Relation of State Space Models and Hidden Markov Models

Aydin Ghojogh, M. Hadi Sepanj, Benyamin Ghojogh

TL;DR

This paper analyzes the relationship between HMMs, linear Gaussian SSMs, Kalman filtering, and modern NLP state space models (e.g., S4, Mamba) from a probabilistic graphical modeling perspective. It presents parallel formulations, inference procedures (forward–backward for HMMs vs Kalman filtering/smoothing for LG-SSMs) and learning algorithms (EM for HMMs/SSMs vs gradient-based training for deterministic NLP SSMs), and it clarifies when models are equivalent and how deterministically updated NLP SSMs relate to classical probabilistic ones. By contrasting latent semantics, uncertainty interpretation, and training paradigms, the work articulates a conceptual bridge between control theory, probabilistic modeling, and deep learning. The results highlight that HMMs, LG-SSMs, and deterministic NLP SSMs occupy a spectrum along which identical temporal structures yield different modeling assumptions and inference strategies, with S4/Mamba representing a deterministic specialization within this spectrum and EM remaining central for probabilistic variants.

Abstract

State Space Models (SSMs) and Hidden Markov Models (HMMs) are foundational frameworks for modeling sequential data with latent variables and are widely used in signal processing, control theory, and machine learning. Despite their shared temporal structure, they differ fundamentally in the nature of their latent states, probabilistic assumptions, inference procedures, and training paradigms. Recently, deterministic state space models have re-emerged in natural language processing through architectures such as S4 and Mamba, raising new questions about the relationship between classical probabilistic SSMs, HMMs, and modern neural sequence models. In this paper, we present a unified and systematic comparison of HMMs, linear Gaussian state space models, Kalman filtering, and contemporary NLP state space models. We analyze their formulations through the lens of probabilistic graphical models, examine their inference algorithms -- including forward-backward inference and Kalman filtering -- and contrast their learning procedures via Expectation-Maximization and gradient-based optimization. By highlighting both structural similarities and semantic differences, we clarify when these models are equivalent, when they fundamentally diverge, and how modern NLP SSMs relate to classical probabilistic models. Our analysis bridges perspectives from control theory, probabilistic modeling, and modern deep learning.

On the Relation of State Space Models and Hidden Markov Models

TL;DR

This paper analyzes the relationship between HMMs, linear Gaussian SSMs, Kalman filtering, and modern NLP state space models (e.g., S4, Mamba) from a probabilistic graphical modeling perspective. It presents parallel formulations, inference procedures (forward–backward for HMMs vs Kalman filtering/smoothing for LG-SSMs) and learning algorithms (EM for HMMs/SSMs vs gradient-based training for deterministic NLP SSMs), and it clarifies when models are equivalent and how deterministically updated NLP SSMs relate to classical probabilistic ones. By contrasting latent semantics, uncertainty interpretation, and training paradigms, the work articulates a conceptual bridge between control theory, probabilistic modeling, and deep learning. The results highlight that HMMs, LG-SSMs, and deterministic NLP SSMs occupy a spectrum along which identical temporal structures yield different modeling assumptions and inference strategies, with S4/Mamba representing a deterministic specialization within this spectrum and EM remaining central for probabilistic variants.

Abstract

State Space Models (SSMs) and Hidden Markov Models (HMMs) are foundational frameworks for modeling sequential data with latent variables and are widely used in signal processing, control theory, and machine learning. Despite their shared temporal structure, they differ fundamentally in the nature of their latent states, probabilistic assumptions, inference procedures, and training paradigms. Recently, deterministic state space models have re-emerged in natural language processing through architectures such as S4 and Mamba, raising new questions about the relationship between classical probabilistic SSMs, HMMs, and modern neural sequence models. In this paper, we present a unified and systematic comparison of HMMs, linear Gaussian state space models, Kalman filtering, and contemporary NLP state space models. We analyze their formulations through the lens of probabilistic graphical models, examine their inference algorithms -- including forward-backward inference and Kalman filtering -- and contrast their learning procedures via Expectation-Maximization and gradient-based optimization. By highlighting both structural similarities and semantic differences, we clarify when these models are equivalent, when they fundamentally diverge, and how modern NLP SSMs relate to classical probabilistic models. Our analysis bridges perspectives from control theory, probabilistic modeling, and modern deep learning.
Paper Structure (21 sections, 11 equations, 1 figure, 1 table)

This paper contains 21 sections, 11 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Introduction to State Space Models and Hidden Markov Models
  3. Continuous-Time State Space Model
  4. Discrete-Time State Space Model
  5. Linear Gaussian State Space Model and Kalman Filter
  6. Hidden Markov Model
  7. Inference and Learning in SSM and HMM
  8. Joint Likelihood Distribution
  9. Expectation--Maximization in HMM
  10. E-Step in HMM
  11. M-Step in HMM
  12. Expectation--Maximization in SSM
  13. E-Step in SSM
  14. M-Step in SSM
  15. Comparison of Hidden Markov Models and State Space Models
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The Probabilistic Graphical Model (PGM) of the hidden Markov models and state space models. Note that state space models can also have input signal $\boldsymbol{x}_k$ in its graphical model, but it is not included in this figure for the sake of easier comparison of hidden Markov models and state space models.