Tree based credible set estimation

Abstract

Estimating a joint Highest Posterior Density credible set for a multivariate posterior density is challenging as dimension gets larger. Credible intervals for univariate marginals are usually presented for ease of computation and visualisation. There are often two layers of approximation, as we may need to compute a credible set for a target density which is itself only an approximation to the true posterior density. We obtain joint Highest Posterior Density credible sets for density estimation trees given by Li et al. (in: Lee, Sugiyama, Luxburg, Guyon, Garnett (eds) Advances in neural information processing systems, Curran Associates Inc, Red Hook, 2016) approximating a density truncated to a compact subset of $\mathbb {R}^d$ as this is preferred to a copula construction. These trees approximate a joint posterior distribution from posterior samples using a piecewise constant function defined by sequential binary splits. We use a consistent estimator to measure of the symmetric difference between our credible set estimate and the true HPD set of the target density samples. This quality measure can be computed without the need to know the true set. We show how the true-posterior-coverage of an approximate credible set estimated for an approximate target density may be estimated in doubly intractable cases where posterior samples are not available. We illustrate our methods with simulation studies and find that our estimator is competitive with existing methods.

Appendix

The density estimation tree algorithm of Li et al. (2016) and the maximum gap calculation it uses are given in this “Appendix”.

1.1 Maximum gap calculation

In order to find a good split for leaf $\Delta _k=[\pmb a^{(k)},\pmb b^{(k)}]$, defined in Step 5 of Algorithm 3, and given a set of points $\{{\tilde{\pmb s}}^{(k,j)}\}_{j=1}^{n_k}$ with ${\tilde{\pmb s}}^{(k,j)}=({\tilde{s}}^{(k,j)}_{1},\ldots ,{\tilde{s}}^{(k,j)}_{d})$, we divide the i-th dimension into $m_g$ equal-sized bins, $[a_{k,i}+(l-1)\delta _{k,i},a_{k,i}+l\delta _{k,i}]$, $i=1,\ldots ,d$ and $l=1,\ldots ,m_g$ where $\delta _{k,i}=(b_{k,i}-a_{k,i})/m_g$. There are in total $(m_g-1)d$ gaps. Each gap is defined by $h_{l,i}=| (1/n_k) \sum ^{n_k}_{j=1} \mathbbm {1}( {\tilde{s}}^{(k,j)}_i < a_{k,i}+l\delta _{k,i}) -l/m_g |$, $l=1,\ldots ,(m_g-1)$ and $i=1,\ldots ,d$. The splitting hyperplane is the gap with the maximum h-value.