论文标题
FMRIB变分贝叶斯推论教程II:随机变分贝叶斯
The FMRIB Variational Bayesian Inference Tutorial II: Stochastic Variational Bayes
论文作者
论文摘要
在许多应用程序中,贝叶斯方法已证明了数据的强大,用于从数据推断模型参数。这些方法基于贝叶斯定理,该定理本身是欺骗性的。但是,实际上,即使对于简单的情况,所需的计算也很棘手。因此,历史上,贝叶斯推断的方法要么显着近似,例如拉普拉斯近似值,要么以显着的计算费用(例如,马尔可夫链蒙特卡洛方法)从精确溶液中获得样品。自2000年左右以来,所谓的贝叶斯推论的变异方法已越来越多地部署。在其最通用的形式中,变异贝叶斯(VB)涉及通过另一个更“可管理的”分布近似真正的后验概率分布,目的是达到尽可能近的近似值。在原始的FMRIB变化贝叶斯教程中,我们记录了一种基于VB的方法,该方法采用了一种“平均场”方法来形成近似后部,需要先验和可能性的结合,并利用了变量的计算,以得出一系列迭代的更新方程,以最大程度地提高预期。在本教程中,我们重新访问VB,但是现在采用随机方法来解决该问题,该问题可能避免了早期方法施加的一些局限性。这种新方法与应用于机器学习算法的计算方法具有很多相似之处。虽然,我们在这里记录的仍然可以识别出经典意义上的贝叶斯推断,而不是试图将机器学习用作黑色框来解决推理问题。
Bayesian methods have proved powerful in many applications for the inference of model parameters from data. These methods are based on Bayes' theorem, which itself is deceptively simple. However, in practice the computations required are intractable even for simple cases. Hence methods for Bayesian inference have historically either been significantly approximate, e.g., the Laplace approximation, or achieve samples from the exact solution at significant computational expense, e.g., Markov Chain Monte Carlo methods. Since around the year 2000 so-called Variational approaches to Bayesian inference have been increasingly deployed. In its most general form Variational Bayes (VB) involves approximating the true posterior probability distribution via another more 'manageable' distribution, the aim being to achieve as good an approximation as possible. In the original FMRIB Variational Bayes tutorial we documented an approach to VB based that took a 'mean field' approach to forming the approximate posterior, required the conjugacy of prior and likelihood, and exploited the Calculus of Variations, to derive an iterative series of update equations, akin to Expectation Maximisation. In this tutorial we revisit VB, but now take a stochastic approach to the problem that potentially circumvents some of the limitations imposed by the earlier methodology. This new approach bears a lot of similarity to, and has benefited from, computational methods applied to machine learning algorithms. Although, what we document here is still recognisably Bayesian inference in the classic sense, and not an attempt to use machine learning as a black-box to solve the inference problem.