Pattern Recognition

$\iff p\left(b|a\right)=p\left(b\right).\left(2.205\right)$

Definition 2.12 (Induced uncorrelatedness). If two random variables are independent, then they are also uncorrelated. The converse is not necessarily true.

As an example, let a ∼ N(0,1) and b:= a2. Certainly, a and b are not stochastically independent, since if b is observed, one knows the magnitude |a| of a. If a is observed, one knows b. Nevertheless, a and b are uncorrelated, because Cov {a, b ∝ E {a3} = 0 since the third central moment of a Gaussian distribution is zero.

As already mentioned, the ICA is composed of two steps (see Figure 2.39). Instead of directly looking for a matrix U such that

$\underset{\_}{m}\prime \prime =U\left(\underset{\_}{m}-\text{E}\left\{\underset{\_}{m}\right\}\right)$