While I don’t have my notes in front of me, I do recall from the decision analysis class I recently took that log score is related to the weight one would give to one forecaster among several when combining forecasts. Unfortunately it does not appear that the professor uploaded the slides on ensemble forecasting, so I can’t provide any more right now. I am emailing the professor. Thought this would help in the meantime.
I can see that I misremembered the lecture. Seems to be an application of Bayes as Lumifer suggested for the basic approach. Other more complex approaches were also discussed.