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Computationally efficient Bayesian estimation of high-dimensional Archimedean copulas with discrete and mixed margins

机译：具有离散和混合边距的高维阿基米德系算子的计算有效贝叶斯估计

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摘要

Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating terms, with J the number of discrete variables. Our article focuses on the estimation of Archimedean copulas, for example, Clayton and Gumbel copulas. Currently, data augmentation methods are used to carry out inference for discrete copulas and, in practice, the computation becomes infeasible when J is large. Our article proposes two new fast Bayesian approaches for estimating high-dimensional Archimedean copulas with discrete margins, or a combination of discrete and continuous margins. Both methods are based on recent advances in Bayesian methodology that work with an unbiased estimate of the likelihood rather than the likelihood itself, and our key observation is that we can estimate the likelihood of a discrete Archimedean copula unbiasedly with much less computation than evaluating the likelihood exactly or with current simulation methods that are based on augmenting the model with latent variables. The first approach builds on the pseudo-marginal method that allows Markov chain Monte Carlo simulation from the posterior distribution using only an unbiased estimate of the likelihood. The second approach is based on a variational Bayes approximation to the posterior and also uses an unbiased estimate of the likelihood. We show that the two new approaches enable us to carry out Bayesian inference for high values of J for the Archimedean copulas where the computation was previously too expensive. The methodology is illustrated through several real and simulated data examples.
机译：估计具有离散边际分布的copula具有挑战性，尤其是在高维中，因为计算每个观察值的似然贡献需要评估项，其中J为离散变量的数量。我们的文章集中在对阿基米德系势的估计，例如Clayton和Gumbel系势。当前，数据扩充方法被用于对离散的copula进行推理，并且在实践中，当J很大时，计算变得不可行。本文提出了两种新的快速贝叶斯方法来估计具有离散边距或离散边距和连续边距的组合的高维阿基米德系系。两种方法均基于贝叶斯方法的最新进展，该方法可用于可能性的无偏估计，而不是可能性本身，并且我们的主要观察结果是，与估计可能性相比，我们可以用更少的计算量来公正地估计离散阿基米德系势的可能性。精确地或使用基于以潜在变量扩充模型为基础的当前模拟方法。第一种方法建立在伪边际方法的基础上，该方法允许仅使用可能性的无偏估计从后验分布进行马尔可夫链蒙特卡罗模拟。第二种方法基于对后验的变分贝叶斯近似，并且还使用了可能性的无偏估计。我们证明了这两种新方法使我们能够对先前计算过于昂贵的阿基米德系势函数的高J值进行贝叶斯推断。通过几个真实的和模拟的数据示例说明了该方法。

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