In this paper we review the concepts of Bayesian evidence and Bayes factors,
also known as log odds ratios, and their application to model selection. The
theory is presented along with a discussion of analytic, approximate and
numerical techniques. Specific attention is paid to the Laplace approximation,
variational Bayes, importance sampling, thermodynamic integration, and nested
sampling and its recent variants. Analogies to statistical physics, from which
many of these techniques originate, are discussed in order to provide readers
with deeper insights that may lead to new techniques. The utility of Bayesian
model testing in the domain sciences is demonstrated by presenting four
specific practical examples considered within the context of signal processing
in the areas of signal detection, sensor characterization, scientific model
selection and molecular force characterization.

本文介绍了贝叶斯证据和贝叶斯因子的概念及其在模型选择中的应用，包括解析、近似和数值技术，以及拉普拉斯近似、变分贝叶斯、重要性采样、热力学积分以及嵌套采样及其方法。该文基于统计物理学的相似性，阐述了这些技术，并通过四个具体实例展示了贝叶斯模型测试在信号处理、传感器表征、科学模型选择和分子力表征等领域的实用性。

贝叶斯证据和模型选择

Bayesian Evidence and Model Selection

We address the role of noise and the issue of efficient computation in
stochastic optimal control problems. We consider a class of non-linear control
problems that can be formulated as a path integral and where the noise plays
the role of temperature. The path integral displays symmetry breaking and there
exist a critical noise value that separates regimes where optimal control
yields qualitatively different solutions. The path integral can be computed
efficiently by Monte Carlo integration or by Laplace approximation, and can
therefore be used to solve high dimensional stochastic control problems.

本文研究噪音在随机最优控制问题中的作用以及高效计算的问题，通过将一类非线性控制问题转换成路径积分形式，探讨噪音在其中的作用，并针对这个问题提出蒙特卡罗积分或拉普拉斯逼近等高维随机控制问题的解法。