# Bayesian Structural Inference for Hidden Processes

arXiv: https://arxiv.org/abs/1309.1392 · HTML: https://mlsanity.com/paper/1309.1392/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2013.

## Built on by (most cited first, 5 of 5)

- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Topology, Convergence, and Reconstruction of Predictive States](https://mlsanity.com/paper/2109.09203/) (2021)
- [Exploring Predictive States via Cantor Embeddings and Wasserstein Distance](https://mlsanity.com/paper/2206.04198/) (2022)
- [Ultimate limit on learning non-Markovian behavior: Fisher information rate and excess information](https://mlsanity.com/paper/2310.03968/) (2023)
- [Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes](https://mlsanity.com/paper/2505.19275/) (2025)

## Abstract (arXiv)

We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian Structural Inference (BSI) relies on a set of candidate unifilar HMM (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of topological epsilon-machines. (A sequel then removes the topological restriction.) This subset of the uHMM topologies has the added benefit that inferred models are guaranteed to be epsilon-machines, irrespective of estimated transition probabilities. Properties of epsilon-machines and uHMMs allow for the derivation of analytic expressions for estimating transition probabilities, inferring start states, and comparing the posterior probability of candidate model topologies, despite process internal structure being only indirectly present in data. We demonstrate BSI's effectiveness in estimating a process's randomness, as reflected by the Shannon entropy rate, and its structure, as quantified by the statistical complexity. We also compare using the posterior distribution over candidate models and the single, maximum a posteriori model for point estimation and show that the former more accurately reflects uncertainty in estimated values. We apply BSI to in-class examples of finite- and infinite-order Markov processes, as well to an out-of-class, infinite-state hidden process.