Probabilistic forecast reconciliation: Properties, evaluation and score optimisation

•Forecast reconciliation in the probabilistic setting is rigorously developed. Point forecast reconciliation is extended in a novel way to the probabilistic setting.•Results are derived for the Gaussian and non-Gaussian case.•Theorems on scoring rules are derived with recommendations for forecast evaluation.•A new reconciliation method based on score optimisation and stochastic gradient descent is proposed.•The new methods are shown to improve forecast accuracy in a simulated and empirical setting.