This paper considers a semiparametric approach within the general Bayesian linear model where the innovations consist of a stationary, mean zero Gaussian time series. While a parametric prior is specified for the linear model coefficients, the autocovariance structure of the time series is modeled nonparametrically using a Bernstein-Gamma process prior for the spectral density function, the Fourier transform of the autocovariance function. When updating this joint prior with Whittle's likelihood, a Bernstein-von-Mises result is established for the linear model coefficients showing the asymptotic equivalence of the corresponding estimators to those obtained from frequentist pseudo-maximum-likelihood estimation under the Whittle likelihood. Local asymptotic normality of the likelihood is shown, demonstrating that the marginal posterior distribution of the linear model coefficients shrinks at parametric rate towards the true value, and that the conditional posterior distribution of the spectral density contracts in the sup-norm, even in the case of a partially misspecified linear model.

翻译：本文在一般贝叶斯线性模型的框架下探讨了一种半参数方法，其中新息由平稳、均值为零的高斯时间序列构成。虽然对线性模型系数指定了参数先验，但时间序列的自协方差结构采用非参数建模方式，即通过Bernstein-Gamma过程先验对谱密度函数（自协方差函数的傅里叶变换）进行建模。当使用Whittle似然更新这一联合先验时，我们为线性模型系数建立了Bernstein-von-Mises结果，表明相应估计量与在Whittle似然下频率学派伪最大似然估计所获估计量具有渐近等价性。文中证明了似然的局部渐近正态性，表明线性模型系数的边缘后验分布以参数速率向真实值收缩，且谱密度的条件后验分布在sup-范数下具有收缩性，即使在部分误设线性模型的情形下该结论依然成立。