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.

翻译：自适应重要性采样（AIS）算法广泛应用于对复杂目标概率分布的期望进行近似计算。当目标分布具有重尾特征时，现有AIS算法常忽略其尾部行为，导致估计量不一致或收敛缓慢。为避免此问题，我们提出了一种采用Student-t提议分布近似目标分布的AIS算法。通过匹配目标与提议分布的护航矩（即使对重尾分布仍可定义），对位置和尺度参数进行自适应调整。这些更新过程最小化目标分布与提议分布之间的α-散度，从而与变分推断建立联系。我们进一步证明，α-散度可通过广义有效样本量进行近似，并利用这一新视角采用贝叶斯优化方法自适应调整尾部参数。通过在合成目标分布及基于临床试验数据的真实贝叶斯Student-t回归任务中的应用，验证了本方法的有效性。