Weaker-than-L1(p) conditions ensuring computability of the joint distribution

Determine a natural, weaker-than L1(p) computability condition on the likelihood map θ ↦ ν([σ]) (as a function in θ) that still guarantees, for c.e. open sets U ⊆ Θ arising from a p-computable basis and for basic cylinders [σ] in the sample space, that the integrals ∫_U ν([σ]) dp are left-c.e. (in particular, sufficiently to ensure that the induced joint distribution μ on Θ × X is a computable probability measure).

Background

In Section 3, the joint distribution μ on the product space S = Θ × X is defined from a prior p on the parameter space Θ and a likelihood mapping θ ↦ ν on the sample space X. Proposition 3.1 shows that μ is a computable probability measure when the likelihood map θ ↦ ν([σ]) is L1(p)-computable (uniformly in σ).

The proof relies on the computability of numbers of the form μ(U × [σ]) = ∫_U ν([σ]) dp for c.e. open U from a p-computable basis. The authors note that, in fact, μ would be computable even if these integrals were merely left-c.e., but they do not know an interesting sufficient condition weaker than L1(p) computability that ensures this left-c.e. property. Identifying such a condition would broaden the applicability of their computable measure construction beyond the current L1(p)-computable likelihood assumption.