Statistical Inference for Stochastic-Belief Extended Realities

We develop a rigorous asymptotic-statistical theory for the inference of stochastic-beliefextended realities (SBxRs), the path-space probabilistic objects introduced in [1] for the modellingof perceptual experience in extended-reality systems. Given n independent percepttrajectories drawn from an SBxR with an unknown parameter θ0 ∈ Θ ⊂ Rd and a knownmodality-weight structure, we construct an importance-weighted plug-in estimator ˆgn ofthe aggregated Fisher–Rao metric g, prove its strong consistency and asymptotic normalityunder standard mixing-and-moment conditions on the per-modality Itˆo diffusions, and establishoperator-norm concentration via matrix Bernstein. The aggregated maximum-likelihoodestimator ˆθn is shown to be consistent and asymptotically efficient with limiting covarianceg(θ0)−1, and the plug-in estimator ˆgn(ˆθn) inherits a delta-method-style central limit theorem.We then address the joint identifiability of the aggregation exponent p ∈ (1,∞]from the Lp-Fisher–Rao Finsler family of [4], proving that p is qualitatively identifiableunder a non-collinearity condition on the per-modality Fisher matrices but that the Fisherinformation for p saturates as p → ∞, with a closed-form per-direction saturation pointp⋆(ξ) = 1/ log(r1(ξ)/r2(ξ)) in terms of the typical ratio of largest to second-largest weightedper-modality Fisher contributions. The estimator ˆpK achieves the parametric rate K−1/2uniformly on a bounded interval [1, peff ] determined by this saturation point but degradesto non-parametric rates beyond it. We provide a Le Cam two-point minimax lower boundmatching the upper bound up to constants, extend the framework to non-Gaussian perceptprocesses (jump-diffusions via Jacod–Shiryaev; fractional-Brownian percepts via Shimizu–Nakajima 2024), and prove the validity of an i.i.d. path-bootstrap. A JAX implementationand eight numerical experiments validate the theory on the trichromatic-color worked exampleof [1] and on synthetic non-Gaussian percept models.Keywords: Fisher information; Fisher–Rao metric; SBxR; path-