基于可解模型的中观贝叶斯推断
Mesoscopic Bayesian Inference by Solvable Models
AI总结：
本文提出一种解析理论处理线性回归中有限次测量情况，通过引入中观变量分析贝叶斯测量的参数估计、模型选择与测量整合，并证明自由能差异可由两个中观变量解析化简，为理解贝叶斯测量结果提供新方法。
摘要：
数据科学和人工智能的快速发展以多种方式影响了物理学，包括贝叶斯推断的应用，为研究方法的革命奠定了基础。我们的小组提出了贝叶斯测量，这是一个将贝叶斯推断应用于测量科学的框架，在各类自然科学中具有广泛的适用性。该框架能够确定系统参数的后验概率分布、进行模型选择以及整合多个测量数据集。然而，将贝叶斯测量应用于真实数据分析需要比传统统计方法（如赤池信息量准则（AIC）和贝叶斯信息量准则（BIC））更为复杂的方法，这些传统方法是针对无限次测量 $N$ 设计的。因此，在本文中，我们提出了一种解析理论，明确处理线性回归模型中 $N$ 为有限值的情况。我们为 $N$ 个观测噪声引入了 $O(1)$ 中观变量。利用这一中观理论，我们分析了贝叶斯测量的三个核心原则：参数估计、模型选择和测量整合。此外，通过引入这些中观变量，我们证明了对模型选择和测量整合都至关重要的自由能差异，可以通过 $N$ 个观测噪声的两个中观变量进行解析性化简。这为模型选择和测量整合提供了更深刻的定性理解，并进一步为非线性模型的实际测量提供了更深入的见解。我们的框架提出了一种理解贝叶斯测量结果的新颖方法。
英文摘要：
The rapid advancement of data science and artificial intelligence has affected physics in numerous ways, including the application of Bayesian inference, setting the stage for a revolution in research methodology. Our group has proposed Bayesian measurement, a framework that applies Bayesian inference to measurement science with broad applicability across various natural sciences. This framework enables the determination of posterior probability distributions of system parameters, model selection, and the integration of multiple measurement datasets. However, applying Bayesian measurement to real data analysis requires a more sophisticated approach than traditional statistical methods like Akaike information criterion (AIC) and Bayesian information criterion (BIC), which are designed for an infinite number of measurements $N$. Therefore, in this paper, we propose an analytical theory that explicitly addresses the case where $N$ is finite in the linear regression model. We introduce $O(1)$ mesoscopic variables for $N$ observation noises. Using this mesoscopic theory, we analyze the three core principles of Bayesian measurement: parameter estimation, model selection, and measurement integration. Furthermore, by introducing these mesoscopic variables, we demonstrate that the difference in free energies, critical for both model selection and measurement integration, can be analytically reduced by two mesoscopic variables of $N$ observation noises. This provides a deeper qualitative understanding of model selection and measurement integration and further provides deeper insights into actual measurements for nonlinear models. Our framework presents a novel approach to understanding Bayesian measurement results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
