If I perform a linear transform on an input and output variable of a regression, and then want to "untransform" the coefficients, how do I do that?

Question

I am running a regression, with X predicting Y. I have performed linear transforms on X and Y to put them on the same scale (did this to be able to fit priors for Bayesian parameter estimation).

If I want the resulting coefficients to be on the original scale, how do I untransform them?

Is it:

Coefficient * (slope of the transformation for X / slope of the transformation for Y)

?

Answer 1

Transform as $y=a+bY$ and $x=c+dX$ then the regression on x-y is $y=Ax+B$. Putting the untransformed variables back in gives $a+bY=Ac+AdX+B$. It is then easy to see what the untransformed coefficients would be. It looks like $a=c=0$ in your case so what you suggest is correct.