Hidden Markov Models (HMMs) are foundational tools for modeling sequential data with latent Markovian structure, yet fitting them to real-world data remains computationally challenging. In this work, we show that pre-trained large language models (LLMs) can effectively model data generated by HMMs via in-context learning (ICL)--their ability to infer patterns from examples within a prompt. On a diverse set of synthetic HMMs, LLMs achieve predictive accuracy approaching the theoretical optimum. We uncover novel scaling trends influenced by HMM properties, and offer theoretical conjectures for these empirical observations. We also provide practical guidelines for scientists on using ICL as a diagnostic tool for complex data. On real-world animal decision-making tasks, ICL achieves competitive performance with models designed by human experts. To our knowledge, this is the first demonstration that ICL can learn to predict HMM-generated sequences--an advance that deepens our understanding of in-context learning in LLMs and establishes its potential as a powerful tool for uncovering hidden structure in complex scientific data.

Hidden Markov Models (HMMs) are fundamental tools for modeling sequential data with latent states that follow Markovian dynamics. However, they present significant challenges in model fitting and computational efficiency on real-world datasets. In this work, we demonstrate that pre-trained large language models (LLMs) can effectively model data generated by HMMs through in-context learning (ICL) -- their ability to learn patterns from examples within the input context. We evaluate LLMs' performance on diverse synthetic HMMs, showing that their prediction accuracy converges to the theoretical optimum. We discover novel scaling trends influenced by HMM properties and provide theoretical conjectures for these empirical observations.

Neural Information Processing Systems http://nips.cc/

The correlation between events is ubiquitous and important for temporal events modelling. In many cases, the correlation exists between not only events' emitted observations, but also their arrival times. State space models (e.g., hidden Markov model) and stochastic interaction point process models (e.g., Hawkes process) have been studied extensively yet separately for the two types of correlations in the past. In this paper, we propose a Bayesian nonparametric approach that considers both types of correlations via unifying and generalizing the hidden semiMarkov model and interaction point process model. The proposed approach can simultaneously model both the observations and arrival times of temporal events, and automatically determine the number of latent states from data.

We study the basic problem of learning overcomplete HMMs---those that have many hidden states but a small output alphabet. Despite having significant practical importance, such HMMs are poorly understood with no known positive or negative results for efficient learning. In this paper, we present several new results---both positive and negative---which help define the boundaries between the tractable-learning setting and the intractable setting. We show positive results for a large subclass of HMMs whose transition matrices are sparse, well-conditioned and have small probability mass on short cycles. We also show that learning is impossible given only a polynomial number of samples for HMMs with a small output alphabet and whose transition matrices are random regular graphs with large degree. We also discuss these results in the context of learning HMMs which can capture long-term dependencies.

Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an $m$-state hidden Markov model (HMM) with only smoothness assumptions, such as H\olderian conditions, on the emission densities. By leveraging some recent advances in continuous linear algebra and numerical analysis, we develop a computationally efficient spectral algorithm for learning nonparametric HMMs. Our technique is based on computing an SVD on nonparametric estimates of density functions by viewing them as \emph{continuous matrices}. We derive sample complexity bounds via concentration results for nonparametric density estimation and novel perturbation theory results for continuous matrices. We implement our method using Chebyshev polynomial approximations. Our method is competitive with other baselines on synthetic and real problems and is also very computationally efficient.