用于贝叶斯曲线拟合的自适应回火可逆跳跃算法
Adaptive tempered reversible jump algorithm for Bayesian curve fitting
AI总结：
本文提出自适应回火可逆跳跃MCMC算法，通过自动调整建议分布并结合辅助回火策略，显著提升贝叶斯曲线拟合在高维复杂后验下的采样效率与收敛性能。
摘要：
贝叶斯曲线拟合在反问题中具有重要作用，通常使用可逆跳跃马尔可夫链蒙特卡洛（RJMCMC）算法来求解。然而，如果没有适当调优的建议分布，该算法在计算上可能效率低下。作为改进，我们通过将自适应Metropolis采样器从固定维情形扩展到跨维情形，提出了一种用于曲线拟合问题的自适应RJMCMC算法。在所提出的算法中，建议分布的大小和方向都可以在采样过程中自动调整。具体而言，曲线拟合的设置允许在代表性点网格上近似先验未知函数的后验协方差。这种近似有助于定义高效的建议分布。此外，我们通过非可逆并行回火引入该算法的辅助回火版本。为评估这些算法，我们进行了包含一系列受控实验的数值测试。结果表明，与传统算法相比，自适应算法表现出显著更高的效率。即使在后验分布高度复杂、导致辅助回火传统RJMCMC无法有效收敛的情况下，所提出的辅助回火自适应RJMCMC仍表现良好。此外，我们给出一个现实的反问题实例来测试这些算法。自适应算法的成功应用再次使其区别于传统算法，后者即使在数百万次迭代后仍无法有效收敛。
英文摘要：
Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the Adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the a priori unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.
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