# Expectation–Maximisation

Transformer attention is interpreted as max-product belief propagation on a Markov random field that varies from instance to instance, while the parameter change between layers acts as the parameter-update step of EM, a view that groups self-attention, mixture-of-experts, gated units and selective state-space models under one meta-model principle in which the input generates the parameters.

Transformer attention is interpreted as max-product belief propagation on a Markov random field that varies from instance to instance, while the parameter change between layers acts as the parameter-update step of EM, a view that groups self-attention, mixture-of-experts, gated units and selective state-space models under one meta-model principle in which the input generates the parameters.

The study proposes HierEM, a hierarchical expectation-maximization framework that treats each site's observed prostate lesion annotation as a noisy observation of a latent clean lesion mask, alternating between inferring a voxel-wise posterior over that latent mask and training a CNN with the posterior as a soft target while estimating site- and case-level sensitivity and specificity under a logistic-normal hierarchical prior; on three-site data it reaches per-site mean DSC of 29.50%–39.69% in pooled held-out evaluation and 27.91%–32.67% in leave-one-site-out generalization, with statistically significant improvements over comparison methods (p < 0.039) and interpretable per-site label-quality estimates (sensitivity α of 31.5%–47.3% at specificity β ≈ 0.99).

The study proposes HierEM, a hierarchical expectation-maximization framework that treats each site's observed prostate lesion annotation as a noisy observation of a latent clean lesion mask, alternating between inferring a voxel-wise posterior over that latent mask and training a CNN with the posterior as a soft target while estimating site- and case-level sensitivity and specificity under a logistic-normal hierarchical prior; on three-site data it reaches per-site mean DSC of 29.50%–39.69% in pooled held-out evaluation and 27.91%–32.67% in leave-one-site-out generalization, with statistically significant improvements over comparison methods (p < 0.039) and interpretable per-site label-quality estimates (sensitivity α of 31.5%–47.3% at specificity β ≈ 0.99).