Nested sampling for Bayesian computations: A discussion
Nicolas Chopin, Christian P. Robert · 2006
The approximation of marginal densities is central to the Bayesian approach to testing of hy-potheses since ratios m1(x)/m2(x) of those marginals are providing Bayes factors. It is thus of interest to see the emergence of a novel proposal for the approximative computation, although we are less confident than the author about the applicability of nested sampling in realistic Bayesian problems. Rewriting Z as an integral over [0, 1] A first difficulty stems from the convoluted presentation of the equivalence between Z in eqn (1) and z in eqn (4). We do not see why a discretisation and ordering would be necessary at this stage. Indeed, Z = Epi[L(θ)] = Ep̃i[L] = 0 X(λ) dλ where p̃i denotes the distribution of L(θ), associated with the cdf (1 −X(λ)). So it is only under the minimal restriction that X is strictly decreasing that we have the representation of eqn (4), since