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Let's say I have 2 probabilistic regression models that each output a predictive mean and variance:

$$f_1(x) = \mathcal{N}(\mu_1, \sigma_1^2)$$ $$f_2(x) = \mathcal{N}(\mu_2, \sigma_2^2)$$

And I want to combine them into one predictive distribution. The ensemble predicted mean is the (possibly weighted) average of the individual means. How would I combine the variances? If the distributions are independent, I believe I can once again take the weighted average, but what if they're not?

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