The Transformer is the foundational building block of modern AI, yet offers no principled handling of \emph{uncertainty}, which is prevalent in real applications: cold-start tokens with sparse histories in sequential recommendation, heterogeneous signal quality in language models, and attention sinks induced by unconstrained softmax. Every token is treated with uniform confidence. We show this uniformity is a degenerate case of our \emph{Bayesian Filtering Transformer} (BFT): attention becomes precision-weighted kriging, the residual connection becomes a Kalman update with adaptive gain, and the FFN becomes a dynamics model propagating precision via a Jacobian--plus--process-noise rule. Observation precision comes from a parameter-free Restricted Maximum Likelihood (REML) estimator with a conjugate Bayesian prior. BFT replaces any Transformer layer with negligible overhead. On sequential recommendation, BFT applied to three major architectures yields significant gains on six benchmarks, with the largest improvements on cold-start users and rare items where uncertainty is highest. On supervised fine-tuning of large language models with noisy data, BFT improves robustness in two regimes: noisy supervision (token-label corruption in question answering) and noisy context (retrieval-augmented QA with real RAG distractors). A single principled modification -- restoring precision -- unlocks substantial headroom across both classical sequence-modeling and modern LLM regimes.