Simulation of fish-school gradient tracking shows that speed modulation plus simple social forces let the school be read as distributed Bayesian inference over a darkness gradient

The school-level posterior recovers darkness-gradient directions, with uncertainty depending on sampling geometry and residual correlations. Prior cognitive-language descriptions of collective sensing were largely phenomenological and did not explicitly identify the inference mechanism implemented by the group; this work provides an explicit school-level generative model and posterior. In simulations based on Berdahl et al. 2013, seven group sizes doubling from smallest to largest with 64 seeds each, mean patch cosine varies across group sizes and school cosine declines from smallest to largest, indicating that agreement with the local field can coexist with substantial directional variation in a large school.

Posterior calibration varies with environmental structure: the error-to-variance ratio rises above one as the school grows, indicating increasing overconfidence. The work links calibration diagnostics to a correlated likelihood (RBF working covariance), showing that pairwise speed differences share observations and residuals remain correlated after centering. The error-to-variance ratio reaches about 2 at the largest group size, and environmental sweeps show the ratio varies with spatial decay length and contrast, with school cosine reduced at the shortest decay length.

A heading model combining the inferred gradient with persistence predicts collective motion, and its directional estimates relate by a positive affine regression to the observed rise of the darkness-exposure ratio with group size. Prior work often characterized group computation through information-theoretic signatures rather than explicitly linking a group-level generative model to the group-size dependence of sensing performance. Affine coefficients fitted on training seeds and applied to a complementary validation split yield R² of approximately 0.9; permuting speeds within local patches reduces directional accuracy to near zero.

Under stated conditions, the inferred gradient's contribution to the heading update takes the precision-weighted form of a one-step expected-free-energy action. The derivation relates the fitted coupling variance to a multiplier determined by preferences, observations, and the unit-length constraint, supplying an action hypothesis linking the school posterior to movement toward darkness. The derivation relies on local linearity, a quasi-static field, and a homoscedastic likelihood, with conditions given in the appendix; the empirical test measures displacement alignment with the full predicted heading rather than directly testing EFE optimization.

The account applies to simulations of fish-school gradient tracking based on the Berdahl et al. 2013 speed-light mechanism, under local linearity, approximately Gaussian residuals, and the selected likelihood parameters. It offers researchers a route to map collective dynamics onto distributed inference: with the model and parameters held fixed, one can test gradient direction, uncertainty calibration, and motion prediction, and extend partial pooling or line-of-sight partitions to other collective systems. For engineering readers, its precision-weighted posterior-gradient drive parallels the speed-modulation principle used in distributed source-seeking controllers.

Current tests compare an estimated posterior with simulated gradients and behavior; they do not yet directly identify collective physical quantities corresponding to belief parameters and examine whether their evolution follows the proposed approximate-inference updates. The likelihood and patch partition are modeling choices, and local field curvature, overlapping observations, and temporal correlations affect inferred uncertainty, with school-level pooling remaining overconfident in the supplementary analysis. The authors note the need for fresh simulations with the selected likelihood and fitted parameters held fixed to test predictive performance independently, and for tests in other collective systems to determine how far the account extends. The EFE derivation's neglect of field change, curvature, future information gathering, and social and turning constraints, as well as global minimality, still require separate checks.