Beyond Whittle's likelihood - new Bayesian semiparametric approaches to time series analysis

尽管非参数贝叶斯推理在过去十年中已经成为一个快速发展的主题，但只有很少的非参数贝叶斯方法被开发用于时间序列分析。主要的挑战在于必须为贝叶斯统计推断指定一个似然函数。一些作者通过使用惠特尔似然作为平稳时间序列的主要非参数特征的谱密度的贝叶斯建模的近似来解决这个问题。即使对于非高斯平稳时间序列，它不完全由其一阶和二阶结构指定，在许多情况下，惠特尔似然导致渐近正确的统计推断，但往往以效率损失为代价。另一方面，参数化模型更强大，但如果模型被错误指定就会失效。现代时间序列的非参数自举方法面临着类似的挑战，并且隐式地使用真似然的非参数或半参数近似。在这个项目中，我们将利用时间序列分析的自提领域的最新发展，并将时域的贝叶斯参数时间序列似然与基于非参数先验分布的频域校正相结合。这为贝叶斯时间序列分析提供了一种全新的半参数方法。

Even though nonparametric Bayesian inference has been a rapidly growing topic over the last decade, only very few nonparametric Bayesian approaches to time series analysis have been developed. The main challenge lies in the necessity to specify a likelihood function for Bayesian statistical inference.Several authors solved this problem by using Whittle's likelihood as an approximation for Bayesian modeling of the spectral density as the main nonparametric characteristic of stationary time series. Even for non-Gaussian stationary time series, which are not completely specified by their first and second-order structure, the Whittle likelihood results in asymptotically correct statistical inference in many situations but often at the cost of a loss of efficiency. Parametric models, on the other hand, are more powerful but fail if the model is misspecified. Modern nonparametric bootstrap methods for time series face similar challenges and implicitly use non- or semiparametric approximations of the true likelihood. In this project, we will take advantage of state-of-the-art developments in the bootstrap realm of time series analysis and combine Bayesian parametric time series likelihoods in the time domain with a frequency-domain correction based on nonparametric prior distributions.This yields an entirely new semiparametric approach to Bayesian time series analysis.