令$π$表示当来自多变量线性回归模型的可能性与标准非信息先验结合时产生的难以处理的后验密度，该模型具有来自正态的尺度混合的误差。有一种简单的数据增强算法（基于来自混合密度的潜在数据）可用于探索$π$。令$h(\cdot)$和$d$分别表示混合密度和回归模型的维数。Hobert等人（2016）[arXiv：1506.03113v2]最近表明，如果$h$以适当的速率在原点收敛到0，并且$\int_0^\infty u^{\frac{d}{2}} \, h(u) \, du < \infty$，则DA算法和替代Haar PX-DA算法的马尔可夫链都是几何遍历的。事实上，比几何遍历性更强的东西常常成立。本文证明了在$h$上的简单条件下，由DA和Haar PX-DA马尔可夫链定义的马氏算子是迹族，即，具有可和特征值的紧致的。许多混合密度，满足霍伯特等人。（2016）的条件也满足本文提出的新条件。因此，对于这组混合密度，新的结果提供了一个实质性的加强霍伯特等人。”（2016年）的结论，没有任何额外的假设。例如，Hobert et al.（2016）表明，只要混合密度是广义逆高斯、对数正态、Fréchet（形状参数大于$d/2$）或逆伽马（形状参数大于$d/2$），DA和Haar PX-DA马尔可夫链就是几何遍历的。本文的结果表明，在这些情况下，DA和Haar PX DA马尔可夫算子，事实上，迹族。
英文摘要：
Let $π$ denote the intractable posterior density that results when the likelihood from a multivariate linear regression model with errors from a scale mixture of normals is combined with the standard non-informative prior. There is a simple data augmentation algorithm (based on latent data from the mixing density) that can be used to explore $π$. Let $h(\cdot)$ and $d$ denote the mixing density and the dimension of the regression model, respectively. Hobert et al. (2016) [arXiv:1506.03113v2] have recently shown that, if $h$ converges to 0 at the origin at an appropriate rate, and $\int_0^\infty u^{\frac{d}{2}} \, h(u) \, du < \infty$, then the Markov chains underlying the DA algorithm and an alternative Haar PX-DA algorithm are both geometrically ergodic. In fact, something much stronger than geometric ergodicity often holds. Indeed, it is shown in this paper that, under simple conditions on $h$, the Markov operators defined by the DA and Haar PX-DA Markov chains are trace-class, i.e., compact with summable eigenvalues. Many of the mixing densities that satisfy Hobert et al.'s (2016) conditions also satisfy the new conditions developed in this paper. Thus, for this set of mixing densities, the new results provide a substantial strengthening of Hobert et al.'s (2016) conclusion without any additional assumptions. For example, Hobert et al. (2016) showed that the DA and Haar PX-DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log-normal, Fréchet (with shape parameter larger than $d/2$), or inverted gamma (with shape parameter larger than $d/2$). The results in this paper show that, in each of these cases, the DA and Haar PX-DA Markov operators are, in fact, trace-class.
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