  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. (2016)
approximating a density truncated to a compact subset of 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.
