# Differentiable Filtering for Learning Hidden Markov Models

arXiv: https://arxiv.org/abs/2511.10571 · HTML: https://mlsanity.com/paper/2511.10571/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [How to Train Your Differentiable Filter](https://mlsanity.com/paper/2012.14313/) (2020)
- [On Limitation of Transformer for Learning HMMs](https://mlsanity.com/paper/2406.04089/) (2024)
- [Learning Overcomplete HMMs](https://mlsanity.com/paper/1711.02309/) (2017)
- [Hidden Markov Neural Networks](https://mlsanity.com/paper/2004.06963/) (2020)
- [Regime Learning for Differentiable Particle Filters](https://mlsanity.com/paper/2405.04865/) (2024)
- [DKFNet: Differentiable Kalman Filter for Field Inversion and Machine Learning](https://mlsanity.com/paper/2509.07474/) (2025)
- [From Small to Large Language Models: Revisiting the Federalist Papers](https://mlsanity.com/paper/2503.01869/) (2025)

## Built on by (most cited first, 2 of 2)

- [Transformer-like Inference from Optimal Control](https://mlsanity.com/paper/2605.15608/) (2026)
- [Accuracy and Uncertainty in Machine Deception Detection (Research White Paper)](https://mlsanity.com/paper/2610.15314/) (2026)

## Abstract (arXiv)

Hidden Markov Models (HMMs) are fundamental for modeling sequential data, yet learning their parameters from observations remains challenging. Classical methods like the Baum-Welch algorithm are computationally intensive and prone to local optima, while modern spectral algorithms offer provable guarantees but may produce probability outputs outside valid ranges. This work introduces Belief Net, a differentiable filtering framework that learns HMM parameters by formulating the forward filter as a structured neural network and optimizing it with stochastic gradient descent. This architecture recursively updates the belief state, which represents the posterior probability distribution over hidden states based on the observation history. Unlike black-box transformer models, Belief Net's learnable weights are explicitly the logits of the initial distribution, transition matrix, and emission matrix, ensuring full interpretability. The model processes observation sequences using a decoder-only (causal) architecture and is trained end-to-end with standard autoregressive next-observation prediction loss. On synthetic HMM data, Belief Net achieves faster convergence than Baum-Welch while successfully recovering parameters in both undercomplete and overcomplete settings, whereas spectral methods prove ineffective in the latter. Comparisons with transformer-based models are also presented on real-world language data.