Tài liệu Independent component analysis P9 docx

9

ICA by Maximum

Likelihood Estimation

A very popular approach for estimating the independent component analysis (ICA)

model is maximum likelihood (ML) estimation. Maximum likelihood estimation is

a fundamental method of statistical estimation; a short introduction was provided in

Section 4.5. One interpretation of ML estimation is that we take those parameter

values as estimates that give the highest probability for the observations. In this

section, we show how to apply ML estimation to ICA estimation. We also show its

close connection to the neural network principle of maximization of information flow

(infomax).

9.1 THE LIKELIHOOD OF THE ICA MODEL

9.1.1 Deriving the likelihood

It is not difficult to derive the likelihood in the noise-free ICA model. This is based

on using the well-known result on the density of a linear transform, given in (2.82).

According to this result, the density px of the mixture vector x
As (9.1)

can be formulated as

pxx
j det Bjpss
j det BjY

i

pisi (9.2)

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Independent Component Analysis. Aapo Hyvarinen, Juha Karhunen, Erkki Oja ¨

Copyright  2001 John Wiley & Sons, Inc.

ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic)