Transformer Architectures as Complete Bayes Processes: A Formal Proof in the Measure-Theoretic Kernel Framework

Prior-data fitted networks (PFNs) have recently emerged as a powerful approach for Bayesian prediction tasks, approximating the posterior predictive distribution (PPD) through in-context learning. Despite their strong empirical performance and ability to go beyond point predictions, theoretical understandings of the algorithmic capability of transformers to learn distributions in context are still lacking. Focusing on Gaussian process regression problems, we show by construction that transformers can implement a gradient descent algorithm targeting the posterior predictive mean and variance, followed by nonlinear mappings that yield binned probabilities of PPD. We study the error bounds of the approximated PPD in terms of attention depth and bin resolution. Based on these results, we further demonstrate the key role of normalization and the choice of attention depth in enabling the extrapolation abilities of transformers beyond the pretraining sample size range. We conduct simulations that corroborate our findings, providing insight into the expressivity of PFNs targeting PPDs and how architectural choices may influence generalization capabilities.

We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\widetilde{O}(\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.

Standard transformer architectures apply a single attention mechanism uniformly across all tokens and sequence positions, irrespective of local context or computational budget. We propose Meta-Attention, a framework that dynamically routes each token to the most appropriate attention strategy -- full softmax attention, linear (kernel) attention, or sliding-window local attention -- via a Bayesian Meta-Controller. Unlike prior routing approaches that use deterministic or prior-free learned routing, the Meta-Controller treats per-token mechanism selection as posterior inference under a compute-aware Dirichlet prior: routing weights are the output of an amortised variational posterior q(alpha | x_t; phi) trained with an Evidence Lower Bound (ELBO) objective that jointly encodes task performance and attention-mechanism cost. This design produces principled routing uncertainty estimates that govern the soft-to-hard routing transition, mitigates routing collapse without ad hoc load-balancing losses, and yields better compute-performance trade-offs than deterministic or prior-free learned routing at negligible overhead. Phase 1 empirical results on a Tiny LM benchmark confirm core predictions: the Bayesian controller's learned routing distribution implies a projected normalised FLOP cost of 25.1% under hard routing, vs. 59.3% for the prior-free baseline (-34.2 pp), and reduces routing entropy from 55.8% to 43.3% (-12.5 pp), demonstrating that the Dirichlet prior prevents routing collapse while the non-Bayesian model defaults to full attention. We present the Bayesian architecture, ELBO training objective, and a Phase 1 PyTorch prototype validating forward-pass correctness, posterior diversity, and a controlled ablation against a prior-free baseline. Code available at: https://github.com/KFEAL/meta-attention