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：本文对希尔伯特射影度量 $\mathcal{H}$ 及其基本性质进行了独立完整的介绍，特别聚焦于概率测度空间。我们首先在凸锥上定义希尔伯特伪度量，重点探讨 $\mathcal{H}$ 的对偶表述。我们证明了凸锥上的线性算子在由 $\mathcal{H}$ 的双曲正切给出的距离下具有收缩性，这尤其蕴含了 Birkhoff 关于 $\mathcal{H}$ 的经典收缩定理。转向概率测度空间（此处 $\mathcal{H}$ 成为度量），我们在一般设定下分析了 $\mathcal{H}$ 的对偶表述，并探讨了在离散概率测度这一特殊情形下概率单纯形在 $\mathcal{H}$ 下的几何性质。自始至终，我们将 $\mathcal{H}$ 与概率测度间的其他距离进行比较。特别地，我们证明了 $\mathcal{H}$ 收敛如何蕴含全变差收敛、$p$-Wasserstein 距离收敛以及任意 $f$-散度收敛。此外，我们基于两个概率测度的希尔伯特距离，推导出了一个关于它们之间全变差的新颖尖锐上界。