broad prior with a posterior distribution where we have gained some knowledge of parameter space and can exclude certain regions.

### Nearest neighbour distance estimator {#knn_estimator}

To estimate the distance between two distributions, based on samples from these distributions, we use a $k$ nearest neighbour estimator [@poczos2011estimation; @poczos2012nonparametric; @sutherland2012kernels] developed to describe a family of distances between distributions known as $\alpha$ divergences, of which the Hellinger distance is a special case. Suppose we have two probability distributions, $P$ and $Q$ with densities $p(x)$, and $q(x)$, and are interested in the distance between these. We suppose that we have some samples, $X_{1:n}$ and $Y_{1:n}$ from $p$ and $q$. If we define $ D_{\alpha} (p||q) = \int p^{\alpha}(x) q^{1-\alpha}(x) \, \mathrm{d} x$ for $\alpha \in \mathbb{R}$, then the Hellinger distance is $$D_h (p||q) = 1 - D_{1/2}(p||q).$$

The $k$ nearest neighbour estimator that we use depends only on distances between observations in a sample. Let $\rho_k(i)$ be the Euclidean distance from the sample $X_i$ to its $k$th nearest neighbour in $X_{1:n}$. Similarly, let $\nu_k(i)$ be the distance from $X_i$ to its $k$th nearest neighbour in the samples $Y_{1:n}$. Then the estimator [@poczos2011estimation] is given by $$\label{d_hat}
 \hat{D_{\alpha}}\left(X_{1:n} || Y_{1:n}\right) = \frac{1}{n} \sum_{i=1}^n 