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

尽管非参数贝叶斯推断在过去十年中已经成为一个迅速增长的话题，但只有极少数非参数贝叶斯方法被开发出来用于时间序列分析。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，几位作者通过使用惠特尔的似然作为贝叶斯建模的近似来解决这个问题，谱密度是平稳时间序列的主要非参数特征。即使对于不完全由一阶和二阶结构描述的非高斯平稳时间序列，惠特尔似然在许多情况下也会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，参数模型功能更强大，但如果错误指定模型，则会失败。时间序列的现代非参数Bootstrap方法面临着类似的挑战，它们隐含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中Bootstrap领域的最新发展，将贝叶斯参数时间序列在时间域中的可能性与基于非参数先验分布的频域校正相结合，从而产生一种全新的贝叶斯时间序列分析的半参数方法。

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.