标题： 变分拉普拉斯的频率论渐近性 显示英文标题

标题： Frequentist Asymptotics of Variational Laplace

摘要： 变分推断是一种在贝叶斯框架中获得后验分布近似值的通用框架。 本质上，变分推断涉及在一个给定的概率分布族上进行优化，以选择最能逼近后验分布的成员。 变分拉普拉斯，一种受此目标启发的迭代更新方案，在认知神经科学界的不同背景下被广泛使用。 然而，到目前为止，该方案的理论性质尚未得到系统研究。 在这里，我们从频率论渐近统计的角度来研究变分拉普拉斯。 频率论渐近理论使我们能够通过其极限行为来判断点估计的质量。 我们将这个框架应用于两个玩具示例，发现由变分拉普拉斯生成的点估计具有渐近一致性和效率的优良性质。 此外，我们推导了在一般情况下建立这些性质的充分条件。 除了点估计外，我们还研究了在总变差距离意义上的分布频率论收敛性，这可能有助于将变分拉普拉斯与关于变分推断的最新发现以及对贝叶斯后验的经典频率论考虑联系起来。 最后，为了说明我们理论考虑的有效性，我们在研究示例中进行了模拟实验。 显示英文摘要

摘要： Variational inference is a general framework to obtain approximations to the posterior distribution in a Bayesian context. In essence, variational inference entails an optimization over a given family of probability distributions to choose the member of this family best approximating the posterior. Variational Laplace, an iterative update scheme motivated by this objective, is widely used in different contexts in the cognitive neuroscience community. However, until now, the theoretical properties of this scheme have not been systematically investigated. Here, we study variational Laplace in the light of frequentist asymptotic statistics. Asymptotical frequentist theory enables one to judge the quality of point estimates by their limit behaviour. We apply this framework to find that point estimates generated by variational Laplace enjoy the desirable properties of asymptotic consistency and efficiency in two toy examples. Furthermore, we derive conditions that are sufficient to establish these properties in a general setting. Besides of point estimates, we also study the frequentist convergence of distributions in the sense of total variation distance, which may be useful to relate variational Laplace both to recent findings regarding variational inference as well as to classical frequentist considerations on the Bayesian posterior. Finally, to illustrate the validity of our theoretical considerations, we conduct simulation experiments in our study examples.