AI 中文总结
本文针对似然为凸GLM型、先验源于局部有限差分正则化的贝叶斯逆问题，建立后验层面离散到连续理论，证明离散先验、后验、MAP估计等的收敛性，且两种取极限顺序结果一致。
AI 中文摘要
我们针对贝叶斯逆问题建立了后验层面的离散到连续理论，该问题的有限维先验源于局部有限差分正则化，似然为一般凸广义线性模型（GLM）型形式。与先采用连续方法的研究不同，本文未预先假设连续先验与后验，而是将其构造为数值计算中使用的有限维概率测度的极限。首先，在总信息参数τ和正则化强度κ固定的情况下，我们证明重构的离散高斯先验与后验测度在L²上弱收敛至明确定义的连续律。随后，我们考虑耦合网格细化与小噪声极限，其中N→∞且τ_N、κ_N→∞，同时κ_N/τ_N保持固定。在N与τ_N相关的显式增长条件下，我们证明重构的离散最大后验（MAP）估计收敛至极限连续代价泛函的唯一极小值u_*，重构的后验测度集中于u_*，且其中心化并重新标度的波动收敛至N(0,Q⁻¹)，其中Q为连续代价泛函在u_*处的海森矩阵。最后，我们证明先构造连续后验再取其小噪声、高正则化极限，可得到相同的确定性极限与高斯波动律。
英文摘要
We develop a posterior-level discrete-to-continuum theory for Bayesian inverse problems whose finite-dimensional priors arise from local finite-difference regularization and whose likelihoods are of a general convex GLM-type form. In contrast to continuum-first approaches, the continuum prior and posterior are not assumed at the outset but are constructed as limits of the finite-dimensional probability measures used in numerical computation. First, with the total information parameter $τ$ and the regularization strength $κ$ fixed, we prove weak convergence on $L^2$ of the reconstructed discrete Gaussian priors and posterior measures to well-defined continuum laws. We then consider a coupled grid-refinement and small-noise limit in which $N\to\infty$ and $τ_N,κ_N\to\infty$, while $κ_N/τ_N$ remains fixed. Under explicit growth conditions relating $N$ and $τ_N$, we prove that the reconstructed discrete MAP estimates converge to the unique minimizer $u_*$ of the limiting continuum cost functional, that the reconstructed posterior measures concentrate at $u_*$, and that their centered and rescaled fluctuations converge to $\mathcal N(\vec 0,Q^{-1})$, where $Q$ is the Hessian of the continuum cost functional at $u_*$. Finally, we show that the same deterministic limit and Gaussian fluctuation law are obtained by first constructing the continuum posterior and then taking its small-noise, high-regularization limit.
