  Adaptive importance sampling (AIS) algorithms are widely used to approximate
expectations with respect to complicated target probability distributions. When
the target has heavy tails, existing AIS algorithms can provide inconsistent
estimators or exhibit slow convergence, as they often neglect the target's tail
behaviour. To avoid this pitfall, we propose an AIS algorithm that approximates
the target by Student-t proposal distributions. We adapt location and scale
parameters by matching the escort moments - which are defined even for
heavy-tailed distributions - of the target and the proposal. These updates
minimize the $\alpha$-divergence between the target and the proposal, thereby
connecting with variational inference. We then show that the
$\alpha$-divergence can be approximated by a generalized notion of effective
sample size and leverage this new perspective to adapt the tail parameter with
Bayesian optimization. We demonstrate the efficacy of our approach through
applications to synthetic targets and a Bayesian Student-t regression task on a
real example with clinical trial data.
