  In the design of efficient simulation algorithms, one is often beset with a
poor choice of proposal distributions. Although the performance of a given
simulation kernel can clarify a posteriori how adequate this kernel is for the
problem at hand, a permanent on-line modification of kernels causes concerns
about the validity of the resulting algorithm. While the issue is most often
intractable for MCMC algorithms, the equivalent version for importance sampling
algorithms can be validated quite precisely. We derive sufficient convergence
conditions for adaptive mixtures of population Monte Carlo algorithms and show
that Rao--Blackwellized versions asymptotically achieve an optimum in terms of
a Kullback divergence criterion, while more rudimentary versions do not benefit
from repeated updating.
