Loss function for conditional variance? Minimizing square loss results in predicting conditional means.
Minimizing absolute loss results in predicting conditional medians.
What loss function results in predicting conditional variances?
I have seen this answer, which works if we can assume a conditional mean of zero (which might be a safe assumption for much of my particular application), but what if we can't assume a conditional mean of zero?

In econometrics, people have been using GARCH to model the conditional variances of time series with various degrees of success.  The basic idea there is to impose a model for the conditional variances --- i.e. trying to "predict" them with known information.  The parameters involved in the model for the conditional variance can then be estimated through various techniques using the log-likelihood as the objective.
I guess one can probably extend this line of thought and apply it to non-time-series data.
