We consider problems in model selection caused by the geometry of models close to their points of intersection. In some cases---including common classes of causal or graphical models, as well as time series models---distinct models may nevertheless have identical tangent spaces. This has two immediate consequences: first, in order to obtain constant power to reject one model in favour of another we need local alternative hypotheses that decrease to the null at a slower rate than the usual parametric $n^{-1/2}$ (typically we will require $n^{-1/4}$ or slower); in other words, to distinguish between the models we need large effect sizes or very large sample sizes. Second, we show that under even weaker conditions on their tangent cones, models in these classes cannot be made simultaneously convex by a reparameterization. This shows that Bayesian network models, amongst others, cannot be learned directly with a convex method similar to the graphical lasso. However, we are able to use our results to suggest methods for model selection that learn the tangent space directly, rather than the model itself. In particular, we give a generic algorithm for learning Bayesian network models.

中文翻译：

---

模型选择和局部几何

我们考虑由靠近交点的模型的几何形状引起的模型选择问题。在某些情况下——包括常见类别的因果或图形模型，以及时间序列模型——不同的模型可能仍然具有相同的切线空间。这有两个直接后果：首先，为了获得拒绝一个模型而支持另一个模型的恒定功率，我们需要局部替代假设，这些假设以比通常的参数 $n^{-1/2}$ 更慢的速度减少到零（通常我们需要 $n^{-1/4}$ 或更慢）；换句话说，为了区分模型，我们需要大的效应量或非常大的样本量。其次，我们表明，在它们的切锥上甚至更弱的条件下，这些类中的模型不能通过重新参数化同时成为凸的。这表明贝叶斯网络模型等不能通过类似于图形套索的凸方法直接学习。然而，我们能够使用我们的结果来建议直接学习切线空间的模型选择方法，而不是模型本身。特别是，我们给出了学习贝叶斯网络模型的通用算法。