Standard likelihood penalties to learn Gaussian graphical models are based on regularizing the off-diagonal entries of the precision matrix. Such methods, and their Bayesian counterparts, are not invariant to scalar multiplication of the variables, unless one standardizes the observed data to unit sample variances. We show that such standardization can have a strong effect on inference and introduce a new family of penalties based on partial correlations. We show that the latter, as well as the maximum likelihood, and logarithmic penalties are scale invariant. We illustrate the use of one such penalty, the partial correlation graphical LASSO, which sets an penalty on partial correlations. The associated optimization problem is no longer convex, but is conditionally convex. We show via simulated examples and in two real datasets that, besides being scale invariant, there can be important gains in terms of inference.

中文翻译：

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偏相关图形LASSO

学习高斯图模型的标准似然惩罚基于对精度矩阵的非对角线条目进行正则化。此类方法及其贝叶斯方法对于变量的标量乘法并不是不变的，除非将观察到的数据标准化为单位样本方差。我们表明，这种标准化可以对推理产生强大的影响，并引入基于偏相关的新惩罚系列。我们证明了后者以及最大可能性，对数惩罚是尺度不变的。我们演示了一种这样的惩罚的使用，即偏相关图形 LASSO，它设置了一个对部分相关性的惩罚。相关的优化问题不再是凸的，而是条件凸的。我们通过模拟示例和两个真实数据集表明，除了尺度不变之外，在推理方面还可以获得重要的收益。