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This is a softmax probability distribution:
$$P(i\mid w_1, w_2, \ldots, w_n) = \frac{\exp(w_i)}{\sum_{i=1}^n \exp(w_i)}.$$

It known also as Boltzmann distribution. It is used in generalized Bradley-Terry model and in multinomial logistic regression. There are efficient minorization-maximization algorithms for infering $\vec{w}$ from data through Maximal Likelihood principle.

I'm looking for similar distribution but extended with "variance" parameter vector. A parameter that would represent the confidence in $\vec{w}$. Then I would like to infer both $\vec{w}$ and its confidence.

Does anybody of you happen to know such distribution or a research on the topic?

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