5
$\begingroup$

Let us consider a probability distribution $(g_n)_{n \in \mathbb{N}}$ which we want to approximate by a mixture of $(f_n(\lambda))_{n \in \mathbb{N}}$ where $\lambda \in \mathbb{R}$ is a parameter.

Are there known techniques that allow one to find the mixture minimizing the $L^1$ norm: \begin{equation} \min_{p} \sum_{n=0}^{\infty} \left|g_n - \int \rm{d} \lambda \; p(\lambda) f_n(\lambda) \right| \end{equation} where $p(\lambda)$ is a normalized probability distribution?

The motivation of this problem is linked to experimental physics: ideally one would like to generate an experimental process characterized by the probability distribution $g$ but this is really not practical. What is really easy, however, is to generate an experimental process with the distribution $f(\lambda)$ where $\lambda$ is a tunable parameter. Therefore, the goal is to approximate $g$ as closely as possible with such a mixture of $f(\lambda)$, where the distance between the two distribution is computed with the $L^1$ norm, that is, I want to minimize the variation distance between the two distributions.

In the specific problem I consider, $f(\lambda)$ is a Poisson distribution with parameter $\lambda \geq 0$, but I really am interested in a general method to approach this problem-

Any pointer to the relevant literature would be greatly appreciated. Thanks a lot!

$\endgroup$
7
  • $\begingroup$ Can you motivate a bit more your problem (why do you need that in two lines, is it some sort of least favorable prior for simultaneous testing)? it is probability over $\mathbb{R}$ ? The norm you use in your sum is the $L^1$ norm between distribution right ? note that $p$ should have integral = 1. $\endgroup$ Sep 1, 2010 at 11:02
  • $\begingroup$ I edited my question according to your remarks. $\endgroup$ Sep 1, 2010 at 12:08
  • 1
    $\begingroup$ Maybe it's better to bound $L^1$ distance by $L^2$, then Fourier analytic techniques can be used. That is also the strategy used in length minimization via energy minimization, well known to differential geometers. $\endgroup$
    – John Jiang
    Sep 1, 2010 at 20:58
  • $\begingroup$ Unfortunately, it is really the $L^1$ distance which is relevant in my problem so I cannot switch from the $L^1$ to the $L^2$ distance. Furthermore, as the distributions are defined over $\mathbb{N}$, I cannot see how a bound on the $L^2$ distance could give any information concerning the $L^1$ distance? $\endgroup$ Sep 2, 2010 at 16:05
  • $\begingroup$ The iteration algorithm called Expectation-maximization is often suitable for approximations with mixitures, though it might not actually converge to the minimum you asked for. $\endgroup$ Sep 16, 2010 at 14:39

1 Answer 1

2
$\begingroup$

perhaps for starters you could take $p$ to be supported on a finite number of points. then the constraints on $p$ become simple linear inequalities and you have a [convex - perhaps even linear] programming problem.

there has been attention in the statistical literature to fitting models using least absolute deviations, rather than least squares. [in the simplest case, the minimizer of

$$\sum_{i=1}^n |x_i - a|$$

is the sample median - rather than the sample mean one gets for $a$ using least squares.]

you could see if references in the monograph by yadolah dodge [L$_1$ statistical procedures and related topics, ims lecture notes - monograph series vol 31 1997] give anything useful.

$\endgroup$
1
  • $\begingroup$ Thanks for the reference! Yes, I indeed started by taking $p$ supported on a finite number of points, and it works pretty well in practice. Still, it's a bit frustrating as a continuous density would be much more natural. $\endgroup$ Sep 16, 2010 at 16:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.