Backpropagation dominates modern machine learning, yet it is not the only principled method for optimizing dynamical systems. We propose Kalman World Models (KWM), a class of learned state-space models trained via recursive Bayesian filtering rather than reverse-mode automatic differentiation. Instead of gradient descent updates, we replace parameter learning with Kalman-style gain adaptation. Training becomes online filtering; error signals become innovations. We further extend this framework to transformer-based large language models (LLMs), where internal activations are treated as latent dynamical states corrected via innovation terms. This yields a gradient-free training and adaptation paradigm grounded in control theory. We derive stability conditions, analyze computational complexity, and provide empirical results on sequence modeling tasks demonstrating competitive performance with improved robustness and continual adaptation properties.