This question is a bit generic.
Let's suppose we have some quantities measured $X_i$ and we want to estimate some quantity, for example $\mu$, the mean of the distribution.
For every quantity we also have a way to estimate the probability of $X_i$ being an outlier, $p_i$ dependent on an other parallel measurement not involving the quantity if interest, in this case the mean of the distribution (e.g. measurements involving higher moments...). Is there a standard way to incorporate this information in a rigorous way in the analysis?
For example if we want to estimate the mean of the distribution one could try something like: $\sum_i p_i X_i$ (and normalize). Is there a standard, Bayesian maybe, way to make such an estimate rigorous? Do you have references of examples where in a particular setting such an analysis is performed? At the end such things are done every time someone identifies and removes by hand outliers.