I have N measurements, each reports a Gaussian distribution of the scalar value (i.e. each reports mean and variance of it: $(\mu_i, \sigma_i^2)$).
What is the probability that these measurements are "correct"? Or, to phrase it differently, what is the probability that there exists a true value that all the N measurements detected?
This is a practical problem: I have ensemble of N neural networks, they get the same input and output $(\mu_i, \sigma_i^2)$ each. I want to measure how much they "agree" in a principled way. I.e. if they output exactly the same distribution the agreement should be high, if they output low sigma and means that are far away from each other - the agreement should be low.