Differentiable Filtering for Learning Hidden Markov Models

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.