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Spaces of probability distributions occur naturally in probability, statistics and machine learning. Optimal transport and information geometry are both mathematical frameworks for studying geometries on these spaces. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. After reviewing both frameworks, we explain a remarkable relationship – based on pseudo-Riemannian geometry – between the two fields. Finally, we discuss some statistical applications inspired by these results.