Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models
Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models
复制标题
重新审视对数凹模型中的最大后验估计
DOI：
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复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
M. Pereyra
中科院分区：
文献类型：
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作者：
M. Pereyra
最大后验概率(MAP)估计是成像科学中主要的贝叶斯估计方法，通常使用对数凹的贝叶斯模型来处理高维问题，其后验模式可以通过凸优化有效地计算出来。尽管MAP估计取得了成功并被广泛采用，但从理论上讲，它还没有得到很好的理解。社会上流行的观点是，在决策理论意义上，MAP估计不是适当的贝叶斯估计，因为它不会最小化有意义的预期损失函数(不像最小均方误差(MMSE)估计器将均方损失最小化)。本文通过给出对数凹的贝叶斯模型中MAP估计的决策理论推导来解决这一理论空白。一个主要的新奇之处是，我们的分析是基于微分几何的，并按如下方式进行。首先，我们利用贝叶斯模型的基本凸几何推导出参数空间上的黎曼几何。然后，我们使用微分几何来识别所谓的自然或正则损失函数，以执行该黎曼流形中的贝叶斯点估计。对于对数凹模型，这种典型损失是与负对数后验密度相关的Bregman发散。然后，我们证明了MAP估计是唯一能最小化期望正则损失的贝叶斯估计，后验均值估计或最小均方误差估计能最小化对偶正则损失。我们还研究了MAP和MSSE估计在大尺度下的性能问题，建立了期望标准误差作为维度的函数的普适界，为研究凸问题中观察到的良好性能提供了新的见解。这些结果为对数凹环境下的MAP和MMSE估计以及凸几何在成像问题中所扮演的多重角色提供了新的理解。
Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimisation. Despite its success and wide adoption, MAP estimation is not theoretically well understood yet. The prevalent view in the community is that MAP estimation is not proper Bayesian estimation in a decision-theoretic sense because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss). This paper addresses this theoretical gap by presenting a decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave. A main novelty is that our analysis is based on differential geometry, and proceeds as follows. First, we use the underlying convex geometry of the Bayesian model to induce a Riemannian geometry on the parameter space. We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For log-concave models, this canonical loss is the Bregman divergence associated with the negative log posterior density. We then show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss. We also study the question of MAP and MSSE estimation performance in large scales and establish a universal bound on the expected canonical error as a function of dimension, offering new insights into the good performance observed in convex problems. These results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple roles that convex geometry plays in imaging problems.
DOI：
10.1007/s10463-006-0099-8
发表时间：
2007-03-01
影响因子：
1
作者：
Dawid, A. P.
通讯作者：
Dawid, A. P.
影响因子：
4.8
作者：
Cai, Xiaohao;Pereyra, Marcelo;McEwen, Jason D.
通讯作者：
McEwen, Jason D.
影响因子：
2.5
作者：
Gribonval, Remi;Cevher, Volkan;Davies, Mike E.
通讯作者：
Davies, Mike E.