make strong assumptions about $p$ independent of the parameters, $\mathbf{w}$, and make weaker assumptions about $q$. In the context of our algorithm for ABC, this allows us to treat $p$ as the prior and $q$ as the approximate posterior distribution.

We will require the following results:

@poczos2011estimation\
Suppose that $k\ge 2$ and that $\mathcal{M} = \text{supp}(p)$. Assume that (a) $q$ is bounded above, (b) $p$ is bounded away from zero, (c) $p$ is uniformly Lebesgue approximable [^1], (d) $\exists \, \delta_0$ such that $\forall \delta \in (0,\delta_0) \, \int_{\mathcal{M}} H(x,p,\delta,1/2)p(x) \mathrm{d}x < \infty,$ (e) $\int_{\mathcal{M}}||x-y||^{\gamma} p(y) \mathrm{d}y < \infty$ for almost all $x \in \mathcal{M}$, $\int \int_{\mathcal{M}^2}||x-y||^{\gamma} p(y) p(x) \mathrm{d}y \mathrm{d} x < \infty$, where $H(x,p,\delta,\psi) = \\ \sum_{j=0}^{k-1} \left(\frac{1}{j!}\right)^{\psi} \Gamma(1 - \alpha + j\psi) \left( \frac{p(x)+\delta}{p(x)-\delta}\right)^{j\psi} (p(x)-\delta)^{-(1-\alpha)}\left((1-\delta)\psi \right)^{-(1-\alpha)-j\psi}.$ Then $$\label{asymptoticly_unbiased}
\lim_{n \rightarrow \infty} \operatorname{\mathbb{E}}\left[ \left(\hat{D}_h\left( X_{1:n} || Y_{1:n}\right) - D_h\left(p || q\right) \right)^2 \right] = 0.$$

The proof of this results relies on constructing an integrable function as a bound such that Lebesgue’s dominated convergence theorem can be applied. See @poczos2011estimation for details. Lemma 1 specifies $L_2$ consistency of the nearest neighbour estimator, which ensures that the estimates of the distance between $p$ and $q$ become more concentrated around 