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首页> 外文期刊>Marine and Petroleum Geology >Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models, model resolution and model choice for Earth Science problems
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Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models, model resolution and model choice for Earth Science problems

机译：马尔可夫链蒙特卡洛（MCMC）采样方法，用于确定地球科学问题的最佳模型，模型分辨率和模型选择

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摘要

We present an overview of Markov chain Monte Carlo, a sampling method for model inference and uncertainty quantification. We focus on the Bayesian approach to MCMC, which allows us to estimate the posterior distribution of model parameters, without needing to know the normalising constant in Bayes' theorem. Given an estimate of the posterior, we can then determine representative models (such as the expected model, and the maximum posterior probability model), the probability distributions for individual parameters, and the uncertainty about the predictions from these models. We also consider variable dimensional problems in which the number of model parameters is unknown and needs to be inferred. Such problems can be addressed with reversible jump (RJ) MCMC. This leads us to model choice, where we may want to discriminate between models or theories of differing complexity. For problems where the models are hierarchical (e.g. similar structure but with a different number of parameters), the Bayesian approach naturally selects the simpler models. More complex problems require an estimate of the normalising constant in Bayes' theorem (also known as the evidence) and this is difficult to do reliably for high dimensional problems. We illustrate the applications of RJMCMC with 3 examples from our earlier working involving modelling distributions of geochronological age data, inference of sea-level and sediment supply histories from 2D stratigraphic cross-sections, and identification of spatially discontinuous thermal histories from a suite of apatite fission track samples distributed in 3D.
机译：我们介绍了马尔可夫链蒙特卡罗的概述，这是一种用于模型推断和不确定性量化的采样方法。我们专注于MCMC的贝叶斯方法，该方法使我们能够估计模型参数的后验分布，而无需知道贝叶斯定理中的归一化常数。给定后验的估计值，我们可以确定代表性模型（例如预期模型和最大后验概率模型），各个参数的概率分布以及这些模型的预测不确定性。我们还考虑了变量维问题，其中模型参数的数量未知，需要推断。此类问题可以通过可逆跳（RJ）MCMC解决。这导致我们进行模型选择，在这里我们可能要区分模型或具有不同复杂性的理论。对于模型是分层的问题（例如，相似的结构但参数数量不同），贝叶斯方法自然会选择较简单的模型。更复杂的问题需要估计贝叶斯定理中的归一化常数（也称为证据），而这对于高维问题很难可靠地完成。我们用之前的工作中的3个例子说明了RJMCMC的应用，这些例子涉及对年代年龄数据的分布进行建模，从2D地层剖面推断海平面和沉积物供应历史，以及从一组磷灰石裂变中识别空间不连续的热历史。跟踪3D分布的样本。

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