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I have a question regarding how to solve this problem: There are two predictors A and B. If A is positive, there is 60% chance of raining. If B is positive, there is 60% chance of raining. A and B is independent decision maker. What's the chance of raining when both A and B are positive.

How to think about this problem in the bayesian framework? I guess there is not enough condition to solve this problem, but can we build some sort of confidence interval?

thanks

Jack2019
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    This question has been asked at [math](http://math.stackexchange.com/questions/457866/a-question-on-raining-probability-using-conditional-probability) with an answer. – TooTone Aug 09 '13 at 11:11
  • This kind of problem is easy to solve visually with diagrams like that in http://stats.stackexchange.com/a/30850/919, which depicts three binary events (AX3 or not, short or not, female or not). Here the three binary events are A predicts rain/A does not predict rain, B predicts rain/B does not predict rain, and rains/does not rain. With this translation I believe the present question is a duplicate (although I concede it would be next to impossible to dig up the duplicate through any reasonable search!). – whuber Aug 09 '13 at 14:39

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A simplistic way of looking at it is from the probabilistic point of view like so: if $P(R|A) = 0.6$ and $P(R|B) = 0.6$, then $P(R|A \cap B) = 1 - 0.4^2 = 0.84$.

But, it is not that simple. We can't really consider predictors as some sort of random variable producing machines which for some reason correlate with the real outcome... This 60% is an average number of times when it did rain. It doesn't mean that it is always 60% of raining probability tomorrow if A is positive. It can capture more of some raining events than other (say, if previous day it was raining, A is always positive and it did rain next day in 90% cases as well).

If the two predictors are identical, then adding B will merely reproduce the same result and give 60% confidence.

On the other hand, if B is able to filter out all those events which A wrongfully predicted it will produce 100% confidence. Imagine that B predicted no rain in all those events when A wrongfully predicted rain. So, 40% of all events when A predicted rain and it was no rain, those B didn't predict.

Yet again, it is possible that when A and B predict rain at the same time, there is always no rain at all! Really, we have no information about how often they do predict rain and about the error of "no rain" event.

Moreover, it is quite possible that A and B never give positive prediction at the same time at all. I.e., imagine A is suited only for summer and by default <0 for other seasons, while B is for spring. In such case the answer is undefined since they simply will never predict rain at the same time at all.

All in all, in reality the union of the two could range from 0% to 100% or even be undefined.

sashkello
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  • hi sashikello. Thank you very much for your reply. I agree with what you said. However, I wonder how to put what you said in a math framework. Can I use any probability term to describe the relationship between A and B, e.g "if B is able to filter out all those events which A wrongfully predicted", so that the solution can be described as a formula with variables ? – Jack2019 Aug 09 '13 at 19:22
  • Your first line is great, but it seems to go astray after that. The problem specifically notes that A and B are independent though, which seems to rule out the rest of your answer. – Matt Krause Mar 06 '14 at 00:13