Constructing a KL-support prior that yields posterior inconsistency for the cosine-based parametric model

Construct a prior distribution with full Kullback–Leibler support on the one-parameter family of densities f_θ(x) ∝ (1 + cos(θ x)) 1_[0,1](x) for θ ≥ 0 that yields an inconsistent Bayesian posterior when the data are generated from the uniform density f_0 on [0,1].

Background

The paper analyzes mechanisms behind Hellinger inconsistency and shows that known counterexamples (e.g., Barron’s) rely on contrived priors and pathological oscillatory densities. It then discusses a parametric oscillatory family f_θ(x) ∝ (1 + cos(θ x)) 1_0,1, θ ≥ 0, recently studied for consistency properties.

Within this cosine-based model, prior work establishes consistency at any θ > 0 under full-support priors, but the uniform case (θ = 0 or θ → ∞) raises potential inconsistency due to oscillations. The authors note that despite this, it remains unresolved how to design a full KL-support prior that actually induces posterior inconsistency in this model.