Log score: Difference between revisions

I. J. Good, 1954</ref> (also sometimes referred to ascalled [https://en.wikipedia.org/wiki/Information_content surprisal]) is a [[proper scoring rule|strictly proper scoring rule]] used to evaluate ahow forecastgood in light of the actually observedforecasts outcomewere. A forecaster scored by the log score will, in expectation, obtain the best score by providing a predictive distribution that is equal to the data-generating distribution. The log score therefore incentivizes forecasters to report their true belief about the future. It was first proposed by [https://en.wikipedia.org/wiki/I._J._Good I. J. Good] in 1952 to evaluate binary forecasts. The score is commonly to score predictions and evaluate how good a distribution captures the observed data, for example in Bayesian statistics. For non-binary questions and up to affine transformations, the log score is the only<ref>[https://en.wikipedia.org/wiki/Scoring_rule#Locality]</ref> strictly proper local scoring rule, meaning that the score depends only on the probability (or [https://en.wikipedia.org/wiki/Probability_density_function probability density]) assigned to the actually observed outcome, rather than on the entire predictive distribution.

The log score is usually computed as the negative logarithm of the predictive density evaluated at the observed value <math>y</math>, i.e. <math>\text{log score}(y) = -\log f(y)</math>, where <math>f()</math> is the predictive probability density function. Usually the natural logarithm is used, but the log score remains strictly proper for any base <math>> 1</math> used for the logarithm.