I have multiple weather forecasters who each use their own unique, independent calculation for prediction of the weather for the next day. We are only concerned with rain predictions to know if we may need an umbrella (i.e. we don't care about temperature or other factors).
Forecaster A, who has a track record of 34% accuracy of predictions, predicts it will rain tomorrow.
Forecaster B, who has a track record of 75% accuracy of predictions, also predicts that it will rain tomorrow.
Forecaster A and B agree on their forecasts 42% of the time (although this may not always be known).
Empirically (using rainfall average data) it would otherwise be 27% likely for rain tomorrow.
How would I practically calculate with Bayes Theorem a revised likelihood that it will rain tomorrow and what would the answer be? How would it change if we introduce n additional forecasters? I should add that I can work out a single-forecaster scenario as a straight-forward Bayes Theorem, but I've had trouble working out how to include/chain multiple forecasters of varying accuracy.
Many thanks in advance!