2

In his Introduction to this paper, Ferguson says that we can model an arbitrary density f(x) on the real line as the mixture of a countable number of normal distributions in the form:

$f(x) = \sum_1^{\infty} p_i h(x|\mu_i,\sigma_i)$

with $h(x|\mu,\sigma)$ is the density of normal distribution, $N(\mu,\sigma^2)$. There are countably infinite number of parameters of the model $(p_1, p_2, ..., \mu_1,\mu_2,...,\sigma_1,\sigma_2,...)$. Using such mixtures any f(x) can be approximated to within any preassigned accuracy in the levy metric or in the L1 norm. Thus the model can be considered nonparametric.

Question: Where is it given, that any f(x) can be written as this infinite mixture model? Can you provide a proof or reference please.

Thanks

  • It should be related with Fourier series... Also, there is an x^2 in the definition of the Gaussian density, so maybe we just need to have some L2 convergence... Maybe use that the density has a finite measure... – ahstat Jun 22 '17 at 10:01
  • For intuition, consider approximating the graph of $f$ by a countable sequence of equally (horizontally)-spaced points, then replace each point by a Gaussian with extremely small standard deviation. That ought to guide you to a straightforward proof. – whuber Jun 22 '17 at 16:31

0 Answers0