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Life sciences · Preprint
arXiv · September 10, 2026
Raises a question worth testing. It does not answer one.
This is a theoretical analysis proving that in stationary symmetric Gaussian hidden Markov models, the Kullback–Leibler divergence between exact Bayesian posterior updates and a deterministic radial filter can vanish while the update maps themselves diverge unboundedly. The result challenges the assumption that internal approximation error necessarily translates to predictive failure, but is restricted to a specific model class and does not establish when this gap becomes harmful in practice.
Theoretical analysis with numerical illustration. Intervention: Deterministic radial filter. Compared with: Exact Bayesian mixing (posterior update).
Unbounded gap between update maps coexists with vanishing predictive KL for every fixed finite K≥2 in stationary symmetric Gaussian HMM Separation of filters in centered logits grows at least linearly in confidence scale L_K(q) as q→0+ Expected terminal KL between filtered posteriors converges to zero at H(q)=⌈−log(q)/c⌉+1 along stationary HMM trajectories
Construction is fixed in K and does not provide a universal criterion for when internal gaps become harmful in decision-making or prediction tasks.
The source did not state who this applies to in practice.
A theoretical analysis proving a counterintuitive mathematical property in a specific model class, without empirical validation or clinical application.
As stated by the source record.
Quoted from the source exactly as published.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
Accurate posterior prediction need not require accurate approximation of Bayesian updates. We prove that an unbounded gap between the update maps can coexist with vanishing predictive KL for every fixed finite $K\ge2$ in a stationary symmetric Gaussian HMM. Exact Bayesian mixing and an explicit deterministic radial filter act on the same $K-1$ belief coordinates. As $q\to0^+$, their separation in centered logits in the worst case grows at least linearly in the natural confidence scale $L_K(q)$, while their categorical $D_{\mathrm{KL}}(\mathrm{exact}\|\mathrm{radial})$ vanishes at the same explicit witness. Along stationary HMM trajectories, the expected terminal KL between filtered posteriors also converges to zero at $H(q)=\lceil-\log(q)/c\rceil+1$. Typical blocks without switches drive both filters into a common confidence cone, where softmax curvature suppresses their disagreement; a single Gaussian maximal event controls adaptive noise. A sweep with equally spaced Gaussians over $K\in\{2,4,8\}$ illustrates the opposing trends, and binary controls at long horizons compare saturating and nonsaturating recurrences. The result isolates two missing links between internal update gaps and predictive cost: the contribution of separating states to expected loss and decoder sensitivity. Thus even an unbounded internal update gap does not by itself certify predictive failure. The construction is fixed in $K$ and does not provide a universal criterion for when compression is harmless or characterize when internal gaps must incur task loss.
Taken from the source record, never inferred. Follow any of these and new work involving them reaches your briefing.