reaching transcendence for Gaussian mixtures

September 2, 2015
By

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

Nested sampling sample on top of a mixture log-likelihood“…likelihood inference is in a fundamental way more complicated than the classical method of moments.”

Carlos Amendola, Mathias Drton, and Bernd Sturmfels arXived a paper this Friday on “maximum likelihood estimates for Gaussian mixtures are transcendental”. By which they mean that trying to solve the five likelihood equations for a two-component Gaussian mixture does not lead to an algebraic function of the data. (When excluding the trivial global maxima spiking at any observation.) This is not highly surprising when considering two observations, 0 and x, from a mixture of N(0,1/2) and N(μ,1/2) because the likelihood equation

(x-\mu)\exp\{\mu^2\}-x+\mu\exp\{-\mu(2x-\mu)\}=0

involves both exponential and algebraic terms. While this is not directly impacting (statistical) inference, this result has the computational consequence that the number of critical points ‘and also the maximum number of local maxima, depends on the sample size and increases beyond any bound’, which means that EM faces increasing difficulties in finding a global finite maximum as the sample size increases…

Filed under: Books, R, Statistics Tagged: algebraic geometry, computational statistics, EM, mixtures of distributions, transcendental equations

To leave a comment for the author, please follow the link and comment on their blog: Xi'an's Og » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Search R-bloggers


Sponsors

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)