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.

中文翻译：

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贝叶斯曲线拟合的自适应缓和可逆跳跃算法

贝叶斯曲线拟合在反问题中起着重要作用，通常使用可逆跳跃马尔可夫链蒙特卡罗 (RJMCMC) 算法来解决。然而，如果没有适当调整提案，该算法的计算效率可能会很低。作为补救措施，我们通过将自适应 Metropolis 采样器从固定维度扩展到跨维度情况，提出了一种针对曲线拟合问题的自适应 RJMCMC 算法。在该算法中，建议函数的大小和方向都可以在采样过程中自动调整。具体来说，曲线拟合设置允许近似后验协方差先验代表性点网格上的未知函数。这种近似有助于有效提案的定义。此外，我们通过不可逆并行调节引入了该算法的辅助调节版本。为了评估算法，我们进行了涉及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着更高的效率。即使在后验分布高度复杂，导致辅助调节传统 RJMCMC 无效收敛的情况下，所提出的辅助调节自适应 RJMCMC 也表现得令人满意。此外，我们提出了一个现实的逆示例来测试算法。自适应算法的成功应用使其再次区别于传统算法即使经过数百万次迭代也无法有效收敛的特点。