通过变分推断在贝叶斯分层模型中进行样本延续
Sample continuation in Bayesian hierarchical model via variational inference
AI总结：
研究通过变分推断跟踪先验参数变化对后验分布的影响，分析后验分布对先验假设的敏感性，并在先验假设显著改变时实现解的延续。
摘要：
在病态贝叶斯反问题中出现的后验分布往往在解析上不可行且对所选先验族的参数高度敏感。本文旨在通过跟踪后验样本表示在先验参数变化时的变化，理解不可行后验分布对先验假设变化的敏感性。这使得能够对先验的小扰动进行敏感性分析，提供后验估计在假设轻微变化下的鲁棒性见解。同时，它允许在先验信念发生显著改变时进行解的延续，从而全面理解假设大变化对后验分布的影响。我们专注于一类非共轭分层模型，该模型针对线性反问题鼓励稀疏性。所关注的特定分层模型因其参数化为少量形状参数而被选择，并且包含了大多数经典稀疏促进先验作为特殊情况。当形状参数变化时，后验可以连续地从可解析的单峰分布过渡到不可解析的多峰分布。为了跟踪后验的变化，我们采用基于粒子的变分推断方法，特别是Stein Variational Gradient Descent (SVGD)。SVGD通过迭代更新一组样本来最小化KL散度远离目标分布。我们通过Birth-Death采样增强SVGD，该方法可以高效地在分离的模式之间交换质量，同时优化用于推导SVGD更新的核带宽。此方法使能够通过追踪模式在更简单的单峰后验中分支出来而发现新模式。
英文摘要：
Posterior distributions arising in ill-posed Bayesian inverse problems are often both analytically intractable and highly sensitive to parameters of the chosen prior family. We aim to understand the sensitivity of intractable posterior distributions to changes in prior assumptions by tracking how a sample representation of the posterior changes as the prior parameters change. This enables sensitivity analysis for small perturbations in the prior, providing insights into the robustness of the posterior estimates under minor changes in assumptions. It also allows solution continuation when dealing with significant alterations in prior beliefs, facilitating a comprehensive understanding of how large shifts in assumptions affect the posterior distribution. We focus on a class of non-conjugate hierarchical models tailored to encourage sparsity in linear inverse problems. The specific hierarchical model of interest is chosen since it is parameterized by a small number of shape parameters, and includes most classical sparsity promoting priors as special cases. As the shape parameters change, the posterior can transition continuously from a tractable unimodal distribution to an intractable multimodal distribution. To track the change in the posterior, we adopt particle based variational inference methods, specifically Stein Variational Gradient Descent (SVGD). SVGD iteratively updates a set of samples to minimize the KL-divergence away from a desired target distribution. We augment SVGD by Birth-Death sampling, which can efficiently exchange mass between separated modes, while simultaneously optimizing the kernel bandwidth used to derive the SVGD update. This method enables the discovery of new modes by tracing the modes as they branch out of a simpler, unimodal posterior, derived within the same family of priors.
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