  Monte Carlo methods are essential tools for Bayesian inference. Gibbs
sampling is a well-known Markov chain Monte Carlo (MCMC) algorithm, extensively
used in signal processing, machine learning, and statistics, employed to draw
samples from complicated high-dimensional posterior distributions. The key
point for the successful application of the Gibbs sampler is the ability to
draw efficiently samples from the full-conditional probability density
functions. Since in the general case this is not possible, in order to speed up
the convergence of the chain, it is required to generate auxiliary samples
whose information is eventually disregarded. In this work, we show that these
auxiliary samples can be recycled within the Gibbs estimators, improving their
efficiency with no extra cost. This novel scheme arises naturally after
pointing out the relationship between the standard Gibbs sampler and the chain
rule used for sampling purposes. Numerical simulations involving simple and
real inference problems confirm the excellent performance of the proposed
scheme in terms of accuracy and computational efficiency. In particular we give
empirical evidence of performance in a toy example, inference of Gaussian
processes hyperparameters, and learning dependence graphs through regression.
