How to accurately quantify forecast uncertainty in a special case of robust linear regression? If I'm using OLS linear regression, and I want to know the uncertainty of my forecasts I can quantify it using residuals (MSE, median absolute deviation, etc). But if I'm using robust linear regression that down-weighs some observations as outliers and I don't have access to those weights, how can I accurately quantity the forecast uncertainty? 
 A: I don't think you will get reasonable results with the model you use.
Why?  There is no real way to distinguish between the uncertainty of the model and the error of the model, because it's not explicitly trained. Unless if you are absolutely sure that you are using the right model, you cannot tell whether your model is wrong or if the error corresponds to uncertainty.
(If you know that your model is right, that is of course different...)
Maybe you should resort to some method that models uncertainty. E.g. Gaussian processes or mixture density networks.
Another options is the following. 
Robust linear regression (as I know it) gives you the maximum likelihood solution under the assumption that your targets are distributed according to a Laplace distribution with constant scale for each sample $(x_i, z_i)$:
$$argmin_W \prod_i \frac{1}{2}\exp (|z_i - Wx_i|)$$
Why don't you attempt to regress on that scale as well?
$$argmin_{W,W'} \prod_i \frac{1}{2W'x_i}\exp (\frac{|z_i - Wx_i|}{W'x_i})$$
Procedure should be straightforward: take the log, take the derivative, use an optimizer to find a minimu. Make sure that the scale has to be positive, so maybe you want to use $\exp(W'x_i)$ as a model instead.
A: Why not just take the loss function used in robust linear regression, and plug in your forecasts with the corresponding actuals for a hold-out sample? (where "forecasts" are calculated using backtesting for that hold-out sample).
For example, if Huber loss function is used in the regression then just use that functional form to calculate average value of the loss function on a hold-out sample.
