Combining two probability scores [duplicate]

Question

This question is rather mathematic, and I am quite sure some theory should already exist but I can not find it...
Say I have some data records, and for a certain record X two independent statistical tests give two different probabilities of X being something (for example an outlier).

Lets say these probabilities are 0.9 and 0.87. Since each test is independent and also calculated using a different part of X, I would like to see a combination method that would make the final outcome actually higher than 0.9.

Since each test gives a very high probability, the overall probability should be even higher in my opinion (they strengthen each other). Is there some standard way to combine these probabilities?

Perhaps (something that comes to mind to me now) is this the same as Fishers Method (for p-values)?

Please let me know if I am not clear.

Answer 1

The easiest and most common way to do this is to take the "conflation" of the probabilities given by the different tests. That is, you can multiply all the probabilities of the different tests together and then renormalize the result in the end. For your case this is:

\begin{align} P(X \in C) = \frac{0.9 \cdot 0.87}{0.9 \cdot 0.87 + (1 - 0.9) (1- 0.87)} = \frac{0.783}{0.783 + 0.013} \approx 0.984 \end{align}

Or, in more general terms:

\begin{align} P(X \in C) &= P(X = C)\\ &= \frac{P_{test_1}(X = C) \cdot ... \cdot P_{test_N}(X = C)} {P_{test_1}(X = C) \cdot ... \cdot P_{test_N}(X = C) + P_{test_1}(X \neq C) \cdot ... \cdot P_{test_N}(X \neq C)} \\ &= \frac{\prod_i P_{test_i}(X=C)}{\prod_i P_{test_i}(X = C) + \prod_i P_{test_i}(X \neq C)} \end{align}

This minimizes the loss of information when combining the different tests and is often the way to go for simple cases. However, there are also many other possible methods to combine probability distributions and depending on your context or knowledge about the tests you might need something else. For a more in-depth discussion of the topic see the discussion on Combining probabilities/information from different sources as pointed out in the comments by Tim, or, alternatively read a full paper on the topic (e.g. Genest, C. and Zidek, J. (1986) Combining probability distributions: a critique and an annotated bibliography).