# Projection of multivariate distribution to lower dimensional subspace

Say that $$X \in \mathbb{R}^n$$ is a vector of $$n$$ r.v.'s with pdf $$p(x_1,\ldots,x_n)$$. Let's consider now the linear map $$Y = A X$$ where $$Y \in \mathbb{R}^m$$ with $$m < n$$. I am seeking $$p(y_1,\ldots,y_m)$$. The case where $$n=m$$ is treated in all elementary probability textbooks (change of variables formula) however I could not find much about what happens when $$m < n$$ since in this case the Jacobian is not invertible. I am interested both in the case where $$X$$ is normally distributed and the case where is $$p(x_1,\ldots,x_n)$$ is a generic continuous density. In my specific problem the map $$A$$ is an orthogonal projection to a lower dimensional subspace ($$m = n-1$$).

• By far the easiest way (IMO) to deal with this uses differential algebra, as I describe (very generally) at stats.stackexchange.com/a/154298/919. However, this isn't needed in your case, because orthogonal projection is equivalent to computing the marginal distribution when you choose appropriately adapted coordinates for the calculation: In short, you simply integrate. – whuber Dec 31 '18 at 20:58
• @whuber thanks for your answer. Do you know any "modern" textbook on differential algebra treating these kind of applications? – Arrigo Benedetti Dec 31 '18 at 22:44
• The most elementary and rigorous one I know, and reference often, is Michael Spivak's Calculus on Manifolds. Although it was written 50 years ago, it is still fairly modern. I don't know of any textbook devoted to illustrating applications of differential algebra to statistical problems, though. – whuber Dec 31 '18 at 22:51