Is it possible to add up the accuracy rates of 2 predictors?

Question

Weather channel 1 has a 65% accuracy rate of predicting tomorrows weather

Weather channel 2 has a 59% accuracy rate of predicting tomorrows weather

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Is it somehow possible to take into account their respective accuracy rates and predictions to somehow make a more accurate predictor?

What would the rates be if the both channels had the same prediction?

What would the rates be if the both channels had different predictions?

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What specific topics/subjects cover this particular problem (that of combining different predictors)?

• accuracy_rate will be defined in this question to avoid confusion as the

  (# of correct predictions / # of total predictions) the channel has made

Answer 1

To do this well, you really need to understand the covariance between the two channels. For example, in the 59% of cases where ch 2 is correct, the probability of channel 1 could be anywhere from 24% (always right when 2 is wrong) to 100% (always right when 2 is right), but I would imagine somewhere in between.

It is tempting to assume there is no correlation, in which case you can simply calculate the odds by multiplying the probabilities of the independent events and dividing by the sum of the probabilities of those which could have been. For example, $$P(both right) = \frac{P(ch 1 right) * P(ch 2 right)}{ P(channels agree)}$$ $$=\frac{P(ch 1 right) * P(ch 2 right)}{ P(ch 1 right) P(ch 2 right) + P( ch 1 wrong) * P(ch 2 wrong)}$$ $$=\frac{0.65 \cdot 0.59}{0.65 \cdot 0.59 + 0.35 \cdot 0.41}\approx0.728$$ This stands to reason in the sense that it is better (more likely) than either alone when both are better than 50%.

However, considering they are both likely looking at the same radar maps, were instructed on the same techniques during school/training, etc, I find it very hard to believe that these would be independent.