What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?

Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.

Let $$p(x) \propto \exp(-\|x\|_1/\sigma)$$ be the pdf of the Laplace distribution, where $$x = (x_1, \dots, x_d)\in \mathbb{R}^d$$. Consider the conditional distribution $$p(x\,|\,\sum^d_{i = 1}x_i = 0)$$, which is the Laplace distribution $$p(x)$$ supported on the linear subspace $$\sum^d_{i = 1}x_i = 0$$.

How does $$p(x\,|\,\sum^d_{i = 1}x_i = 0)$$ look like? Also, how do we sample from $$p(x\,|\,\sum^d_{i=1}x_i)$$?

• Isn’t the event being conditioned on of measure zero? – Nawaf Bou-Rabee Dec 22 '18 at 20:18
• @NawafBou-Rabee I think the op means that conditional density restricted to the linear subspace. Since the domain is also of measure zero in $R^d$, you can still define density. – Minkov Dec 23 '18 at 6:30

Let $$n:=d$$. By rescalling, without loss of generality $$\sigma=1$$. Let $$X_1,\dots,X_n$$ be iid random variables (r.v.'s) with the standard Laplace distribution, so that the joint pdf of $$X:=(X_1,\dots,X_n)$$ is $$$$f_X(x)=\frac1{2^n}\,\exp\Big\{-\sum_1^n|x_i|\Big\}$$$$ for $$x=(x_1,\dots,x_n)$$. Let $$Y_1:=X_1+\dots+X_n$$, $$Y_2:=X_2,\dots,Y_n:=X_n$$, so that $$X_1=Y_1-Y_2-\dots-Y_n$$, $$X_2=Y_2,\dots,X_n=Y_n$$. So, $$X=(X_1,\dots,X_n)$$ and $$Y:=(Y_1,\dots,Y_n)$$ are related by an invertible linear transformation of determinant $$1$$. Therefore, using transformation technique for systems of r.v.'s/change of variables in multi-fold integrals, we see that the joint pdf of $$Y=(Y_1,\dots,Y_n)$$ is $$\begin{multline} f_Y(y)=f_X(y_1-y_2-\dots-y_n, y_2,\dots,y_n) \\ =\frac1{2^n}\,\exp\Big\{-|y_1-y_2-\dots-y_n|-\sum_2^n|y_i|\Big\} \end{multline}$$ for $$y=(y_1,\dots,y_n)$$. Therefore, the joint distribution of $$X=(X_1,\dots,X_n)$$ given that $$X_1+\dots+X_n=0$$ is determined by the conditional joint pdf of $$(Y_2,\dots,Y_n)$$ given $$Y_1=0$$, which is given by the expression $$$$f_{Y_2,\dots,Y_n|Y_1=0}(y_2,\dots,y_n)=\frac{f_Y(0,y_2,\dots,y_n)}{f_{Y_1}(0)},$$$$ where $$f_{Y_1}$$ is the pdf of $$Y_1$$.
So, to complete the calculation of $$f_{Y_2,\dots,Y_n|Y_1=0}(y_2,\dots,y_n)$$, it remains to compute $$f_{Y_1}(0)$$. This can be done quickly by using characteristic functions (c.f.'s). Indeed, the c.f. of $$X_1$$ is given by $$$$\phi(t)=\int_{-\infty}^\infty e^{itx}\frac1{2}\,e^{-|x|}\,dx=\frac1{1+t^2}$$$$ for real $$t$$. Hence, the c.f. of $$Y_1=X_1+\dots+X_n$$ is $$t\mapsto\frac1{(1+t^2)^n}$$ and $$$$f_{Y_1}(0)=\frac1{2\pi}\int_{-\infty}^\infty e^{-it0}\frac{dt}{(1+t^2)^n} =2^{-2 n-1} \binom{2 n}{n}.$$$$
Thus, the conditional joint pdf of $$(Y_2,\dots,Y_n)$$ given $$Y_1=0$$ is $$$$2^{n+1}\exp\Big\{-|y_2+\dots+y_n|-\sum_2^n|y_i|\Big\}\Big/\binom{2 n}{n}.$$$$