If they combine, you need to specify the recipe. It just doesn't magically happen. Again, what exact model are you trying to fit? That model would be specific about the weights.

I don't think my data has the appropriate structure for this, but thanks

You wrote "I normally use the means of the bins as predictor (independent) variables, but I want to use the variances of the bins too". Maybe it would be interesting for you to use quantile regression to obtain such a predictor?

See :

https://community.wolfram.com/groups/-/m/t/1395719

Admittedly "error" is a loaded word. The standard requirement for regression is that you know the predictor variable exactly. Even though your predictors might be measured without error, the fact that your data set has "binned" the predictors essentially constitutes something very akin to "error": the predictor variable is not known exactly.

It is not clear how you expect to use the variability within a bin. Is that a new predictor? How specifically is that variability to be used? To know if Mathematica or R or SAS will do what you want it to do, a specific model describing the function of the predictors and how the predictors affect the error structure is required. Then someone could tell you whether a particular piece of software can estimate the coefficients.

You have to supply the regularization parameter. There is no automatic selection process. (But I certainly could be wrong about that.) And the predicted responses depend on how one scales the predictors (subtract the mean or subtract mean and divide by standard deviation, etc.) given the same regularization parameter.

It's great that FitRegularization has been added but it currently seems both under-documented and lacking necessary regularization parameter selection features that are found in other more complete functions such as those in packages found in R.

OK, thanks. I read it cursorily, because I don't have the time to become an expert in statistics, and hope to use Mathematica for this purpose. It looks like what I need is total least squares. Is there a Mathematica function which can do Total Least Squares analysis? Ideally I could just use ANOVA or LinearModelFit and specify all the errors, not just for dependent variable. Is there a way to do this? There doesn't seem to be as far as I can tell. Why not?

Also, even though I could write code to find out the estimate on the regression coefficient from the formulas given, I also want all the other statistical "goodies" such as p values, confidence intervals, etc. Do I need to write code for all of these myself?

https://reference.wolfram.com/language/ref/ParameterMixtureDistribution.html

I'm going to try to revive this post. If there are known variances in dependent variable, their inverse can be used as a weight in computing the chi square sum in the regression analysis. But what if we also have information about the independent variables' variances? Can we use that information somehow? Naively, I would say (for one independent variable xi and one dependent variable yi) to find the regression coefficient r that minimizes chisquare=Sumi[(r xi -yi)^2/(r^2 sigmaxi^2+sigmay_i^2)]. Is this correct? And if so, how can the Mathematica stat package implement it (I suppose I could write my own code to do it, if not available in stat package)