# Maximum Likelihood notation

My notes for determining the Maximum Likelihood decision rule for classifying two fish given two features (lightness and width) have a different decision rule if the features are correlated to when they're uncorrelated.

ML Decision rule for 2 independant features:

if p(lightness | sea bass)p(width | sea bass) >= p(lightness | salmon)p(width | salmon)

no problem with that.

For 2 correlated features, the ML Decision Rule is:

if p(lightness, width | sea bass) >= p(lightness, width | salmon)

The notation is a bit confusing for me. what does p(width, lightness) actually mean. I've seen things like p(x , $\omega$) before but what does that mean? I know that p(x | $\omega$) means the probability of x given $\omega$, but not what p(x , $\omega$).

Also, my notes say that when two features are correlated, their covariance matrices will be different. Why is this? What does having different covariance matrices mean in terms of understanding the data?

You seem to use $p(a,b|c)$ to denote $p(d|c)$ where $d$ denotes "$a$ and $b$" or, equivalently $d=a\cap b$.