Operator Distinguishability Collapse and Quadratic Fisher Scaling in Parametrized Markov Kernels
Operator Distinguishability Collapse and Quadratic Fisher Scaling in Parametrized Markov Kernels
We consider finite-state Markov kernels whose parameter dependence enters through a scalar gate applied to a fixed kernel perturbation. In this setting, policy distinguishability admits an exact reduction to the gate difference. For logistic gates, we derive a closed-form Fisher information along the parameter axis and show that, in the high-friction regime, Fisher information scales quadratically with operator distinguishability. Thus operator collapse and Fisher degeneracy are structurally linked. We further prove that the Fisher–Rao distance to the infinite-friction limit is finite. The results rely only on finite-state Markov structure and exponential-family behavior.
Fisher information, Total variation distance, Markov kernels, Operator theory, Spectral collapse, Exponential family, Parametrized processes, Information geometry
Fisher information, Total variation distance, Markov kernels, Operator theory, Spectral collapse, Exponential family, Parametrized processes, Information geometry
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