This paper gives a self-contained introduction to the Hilbert projective metric $\mathcal{H}$ and its fundamental properties, with a particular focus on the space of probability measures. We start by defining the Hilbert pseudo-metric on convex cones, focusing mainly on dual formulations of $\mathcal{H}$ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the dual formulation of $\mathcal{H}$ in the general setting, and explore the geometry of the probability simplex under $\mathcal{H}$ in the special case of discrete probability measures. Throughout, we compare $\mathcal{H}$ with other distances between probability measures. In particular, we show how convergence in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound for the total variation between two probability measures in terms of their Hilbert distance.

翻译：本文对希尔伯特投影度量 $\mathcal{H}$ 及其基本性质进行了自包含的引介，特别关注概率测度空间。首先在凸锥上定义希尔伯特伪度量，重点探讨 $\mathcal{H}$ 的对偶形式。我们证明凸锥上的线性算子在 $\mathcal{H}$ 的双曲正切所定义的距离下具有压缩性，这一结果尤其蕴含了 Birkhoff 关于 $\mathcal{H}$ 的经典压缩定理。转向作为度量的 $\mathcal{H}$ 所在的概率测度空间时，我们在一般框架下分析 $\mathcal{H}$ 的对偶形式，并在离散概率测度的特例中探索概率单纯形在 $\mathcal{H}$ 下的几何结构。全文将 $\mathcal{H}$ 与其他概率测度距离进行比较，特别证明了 $\mathcal{H}$ 收敛性蕴含全变差距离、$p$-Wasserstein 距离以及任意 $f$-散度下的收敛性。此外，我们利用希尔伯特距离推导出两个概率测度之间全变差距离的一个新型紧界。