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Scoring rules and their approximations on manifolds

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RCLW02 - Calibrating prediction uncertainty : statistics and machine learning perspectives

On metric spaces of strong negative type an energy or kernel-based strictly proper scoring rule for probabilistic forecasts may be defined. However, the relationship between the strong negative type property and the curvature of a metric space that is a manifold is not well understood. I will comment on this issue while drawing parallels to conditions on the curvature that determine efficient sampling on manifolds using intrinsic stochastic differential equations (SDEs). I will then discuss error bounds for SDE -based sampling from forecasts distributions on manifolds, and their application to computing the corresponding scoring rules. 

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