A PT-suitable reference family if: 1. (Full support): supp(π). 2. (Regularity): The log-likelihood ratio between π

B.1 Conditional convergence in distribution Suppose (X, d The proof of this Lemma is identical to the portmanteau lemma for weak convergence by replacing probabilities/expectations with conditional probabilities/expectations (for example, see [38, Section 2.1]). Lemma B.2. Suppose X, X X as m, and X is a constant a.s., then X A, where A is a constant. We can exchange the expectation and limit by the dominated convergence theorem. The result follows by taking ϵ 0. 4. Since X is a.s. For any K > 0, we have x x K is a bounded and continuous function. R. Because f g: X is a bounded and A. We now show that (X The result follows by an application of the continuous mapping theorem with the function (x, A) Ax. B.2 Model assumptions The following sets of assumptions are only used to prove the large-data limit results of Proposition 3.1, Proposition 3.2, and Proposition 3.3. We will always use a subscript m to indicate that the quantity is dependent on the data. For the remainder of this section we will assume the following regularity conditions.