Neural expectation-maximization and meta-models

Transformers are the dominant architecture in AI, yet why they work remains poorly understood. This paper offers a precise answer: a transformer is a Bayesian network. We establish this in five ways. First, we prove that every sigmoid transformer with any weights implements weighted loopy belief propagation on its implicit factor graph. One layer is one round of BP. This holds for any weights -- trained, random, or constructed. Formally verified against standard mathematical axioms. Second, we give a constructive proof that a transformer can implement exact belief propagation on any declared knowledge base. On knowledge bases without circular dependencies this yields provably correct probability estimates at every node. Formally verified against standard mathematical axioms. Third, we prove uniqueness: a sigmoid transformer that produces exact posteriors necessarily has BP weights. There is no other path through the sigmoid architecture to exact posteriors. Formally verified against standard mathematical axioms. Fourth, we delineate the AND/OR boolean structure of the transformer layer: attention is AND, the FFN is OR, and their strict alternation is Pearl's gather/update algorithm exactly. Fifth, we confirm all formal results experimentally, corroborating the Bayesian network characterization in practice. We also establish the practical viability of loopy belief propagation despite the current lack of a theoretical convergence guarantee. We further prove that verifiable inference requires a finite concept space. Any finite verification procedure can distinguish at most finitely many concepts. Without grounding, correctness is not defined. Hallucination is not a bug that scaling can fix. It is the structural consequence of operating without concepts. Formally verified against standard mathematical axioms.

Transformer networks have achieved remarkable empirical success across a wide range of applications, yet their theoretical expressive power remains insufficiently understood. In this paper, we study the expressive capabilities of Transformer architectures. We first establish an explicit approximation of maxout networks by Transformer networks while preserving comparable model complexity. As a consequence, Transformers inherit the universal approximation capability of ReLU networks under similar complexity constraints. Building on this connection, we develop a framework to analyze the approximation of continuous piecewise linear functions by Transformers and quantitatively characterize their expressivity via the number of linear regions, which grows exponentially with depth. Our analysis establishes a theoretical bridge between approximation theory for standard feedforward neural networks and Transformer architectures. It also yields structural insights into Transformers: self-attention layers implement max-type operations, while feedforward layers realize token-wise affine transformations.

Sequence modelling requires determining which past tokens are causally relevant from the context and their importance: a process inherent to the attention layers in transformers, yet whose underlying learned mechanisms remain poorly understood. In this work, we formalize the task of estimating token importance as an in-context learning problem by introducing a framework based on Mixture of Transition Distributions, where a latent variable determines the influence of past tokens on the next. The distribution over this latent variable is parameterized by unobserved mixture weights that transformers must learn in-context. We demonstrate that transformers can implement Mirror Descent to learn these weights from the context. Specifically, we give an explicit construction of a three-layer transformer that exactly implements one step of Mirror Descent and prove that the resulting estimator is a first-order approximation of the Bayes-optimal predictor. Corroborating our construction and its learnability via gradient descent, we empirically show that transformers trained from scratch learn solutions consistent with our theory: their predictive distributions, attention patterns, and learned transition matrix closely match the construction, while deeper models achieve performance comparable to multi-step Mirror Descent.