Solve the following problem:
In a missing-variable setting where the sampling density can be written as
f(x|θ) = ∫z g(x,z|θ)dz,
we assume the prior π(θ) is such that a two-stage Gibbs sampler based on the simulation of g(z|x, θ) and π(θ|x, z) can be implemented. Using a Bayes' Theorem representation of the marginal density,
m(x) = f(x|θ)π(θ)/ π(θ|x)
deduce a converging estimator of m(x) based on the Rao-Blackwellized estimate of the posterior density π(θ|x) above.