Conjecture: No reasonable priors recover the uniform density by tail averaging in the cosine-based model

Prove that, for the one-parameter family of densities f_θ(x) ∝ (1 + cos(θ x)) 1_[0,1](x) with θ ≥ 0, any prior on θ whose density is continuous and eventually decreasing cannot average tail oscillatory densities to recover the true uniform density f_0 on [0,1].

Background

The authors contrast Barron’s model—where symmetric oscillations and prior mass allocation make the prior predictive density exactly uniform—with the cosine-based parametric model, where oscillatory densities in the tail weakly target the uniform distribution but do not obviously average to it under natural priors.

They conjecture that typical priors (e.g., continuous and eventually decreasing in θ) cannot average tail oscillations to recover the uniform density, suggesting a structural barrier to reproducing Barron-type inconsistency in this parametric setting.