Far apart inside, the same outside

Exact Bayesian mixing Φq and the radial filter Rq act on the same input zq. Lower the switch probability q and watch the internal gap grow while the predictions stay the same.



1. Inside: the models disagree more and moreHow far apart the two models' internal scores are. Solid: actual gap. Dashed: the minimum the paper proves.

2. Outside: their predictions agreeHow different the two predicted probabilities are (KL divergence; 0 = identical).

3. The predictions themselves, at this qProbability each model gives each hidden state. The bars match.

internal gap Dq(zq), nats of logit
KL(exact ‖ radial), nats
gap ÷ LK(q); theorem limit c* = 0.0379

Computed exactly from Theorem 1 (Definition 1 maps at the witness zq = L(¼, −¼, 0, …)). It illustrates the map-level statement only: the gap grows like LK(q), which is logarithmic in 1/q, and the KL need not fall monotonically at moderate q. The path-level result (Theorem 2) is not shown here.