Characterize experiment-level conditions yielding a directed KL cost

Identify the precise additional condition under which the experiment-level information-cost framework of PST_2023 specializes to the single-realized-posterior cost \theta D_{\mathrm{KL}}(p\|q), plausibly by restricting attention to one-directional, perfectly informative binary experiments.

Background

The paper distinguishes the single-posterior KL cost postulated in Assumption E3 from the experiment-level cost characterized by PST_2023. The latter is a weighted sum of directed KL divergences between signal distributions and generally does not reduce algebraically to \theta D_{\mathrm{KL}}(p|q) for an individual posterior.

The unresolved issue is to determine what additional structural restriction on experiments or information structures would make the experiment-level framework produce exactly the directed KL divergence from a posterior to the prior. The paper suggests one-directional, perfectly informative binary experiments as a possible route but does not establish it.