  Probabilistic models in physics often require from the evaluation of
normalized Boltzmann factors, which in turn implies the computation of the
partition function Z. Getting the exact value of Z, though, becomes a
forbiddingly expensive task as the system size increases. This problem is also
present in probabilistic learning models such as the Restricted Boltzmann
Machine (RBM), where the situation is even worse as the exact learning rules
implies the computation of Z at each iteration. A possible way to tackle this
problem is to use the Annealed Importance Sampling (AIS) algorithm, which
provides a tool to stochastically estimate the partition function of the
system. So far, the standard application of the AIS algorithm starts from the
uniform probability distribution and uses a large number of Monte Carlo steps
to obtain reliable estimations of Z following an annealing process. In this
work we show that both the quality of the estimation and the cost of the
computation can be significantly improved by using a properly selected
mean-field starting probability distribution. We perform a systematic analysis
of AIS in both small- and large-sized problems, and compare the results to
exact values in problems where these are known. As a result of our systematic
analysis, we propose two successful strategies that work well in all the
problems analyzed. We conclude that these are good starting points to estimate
the partition function with AIS with a relatively low computational cost.
