Bayesian posterior contraction rates for linear severely ill-posed inverse problems
Bayesian posterior contraction rates for linear severely ill-posed inverse problems
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DOI：
10.1515/jip-2012-0071
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发表时间：
2012-10
影响因子：
1.1
通讯作者：
S. Agapiou;A. Stuart;Yuan-Xiang Zhang
中科院分区：
文献类型：
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作者：
S. Agapiou;A. Stuart;Yuan-Xiang Zhang
抽象的。考虑一类线性不适定反问题，该问题由具有指数衰减到零的奇异值的紧算子的逆引起。我们采用贝叶斯方法，假设高斯先验的未知函数。假设观测噪声为高斯分布，因此先验与似然共轭，使得后验分布也为高斯分布。研究了小观测噪声下的贝叶斯后验一致性。我们假设前向算子与先验和噪声协方差算子相互交换。我们展示了如何，对于给定的平滑性假设的真相，规模参数的先验，这是一个常数乘数的先验协方差算子，可以调整到优化后验收缩率的真理，我们明确计算的对数率。
Abstract. We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.