论文标题
贝叶斯逆问题的贝叶斯和rao-blackwellized SMC采样器的贝叶斯逆问题的无费用超参数选择/平均
Cost free hyper-parameter selection/averaging for Bayesian inverse problems with vanilla and Rao-Blackwellized SMC Samplers
论文作者
论文摘要
在贝叶斯逆问题中，一个旨在表征一组未知数的后验分布，给定间接测量。对于非线性/非高斯问题，很少有分析溶液可用：顺序蒙特卡洛采样器为近似复杂的后代提供了强大的工具，通过构造平稳到达后验的密度的辅助序列。通常，后部取决于标量超参数。在这项工作中，我们表明，正确设计的顺序蒙特卡洛（SMC）采样器自然可以免费提供与此超级参数相关的边际可能性的近似值，即以可忽略的额外计算成本。所提出的方法通过构造分布的辅助序列来进行，以使每个分布序列可以解释为与高参数的不同值相对应的后验分布。可以利用这来选择经验贝叶斯方法中的超参数，并根据完全贝叶斯方法中的某些高优点分布在超参数的值之间进行平均。对于FB方法，提出的方法具有进一步的好处，即以微不足道的计算成本允许先前的灵敏度分析。另外，该提出的方法利用了所有（相关）迭代的颗粒，从而减轻了SMC采样器的已知局限性之一，即通常丢弃中间迭代时的所有样品。我们在两个不同的情况下显示了数值结果，其中超参数仅影响可能性：一个玩具示例，其中SMC采样器用于近似后验分布；和大脑成像示例，其中rao-blackwellized SMC采样器用于近似条件线性高斯模型中参数子集的后验分布。
In Bayesian inverse problems, one aims at characterizing the posterior distribution of a set of unknowns, given indirect measurements. For non-linear/non-Gaussian problems, analytic solutions are seldom available: Sequential Monte Carlo samplers offer a powerful tool for approximating complex posteriors, by constructing an auxiliary sequence of densities that smoothly reaches the posterior. Often the posterior depends on a scalar hyper-parameter. In this work, we show that properly designed Sequential Monte Carlo (SMC) samplers naturally provide an approximation of the marginal likelihood associated with this hyper-parameter for free, i.e. at a negligible additional computational cost. The proposed method proceeds by constructing the auxiliary sequence of distributions in such a way that each of them can be interpreted as a posterior distribution corresponding to a different value of the hyper-parameter. This can be exploited to perform selection of the hyper-parameter in Empirical Bayes approaches, as well as averaging across values of the hyper-parameter according to some hyper-prior distribution in Fully Bayesian approaches. For FB approaches, the proposed method has the further benefit of allowing prior sensitivity analysis at a negligible computational cost. In addition, the proposed method exploits particles at all the (relevant) iterations, thus alleviating one of the known limitations of SMC samplers, i.e. the fact that all samples at intermediate iterations are typically discarded. We show numerical results for two distinct cases where the hyper-parameter affects only the likelihood: a toy example, where an SMC sampler is used to approximate the full posterior distribution; and a brain imaging example, where a Rao-Blackwellized SMC sampler is used to approximate the posterior distribution of a subset of parameters in a conditionally linear Gaussian model.
