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mml-book / mml-book.github.io

Companion webpage to the book "Mathematics For Machine Learning"
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Page 196: Why inner product requires zero mean for random variables X and Y (further explaination request) #645

Open Luo-Chang opened 3 years ago

Luo-Chang commented 3 years ago

In Page 196, equation (6.59), where the definition of inner product of random variables are given, the text states that 'if we define

<X, Y>:= Cov [x,y]

for zero mean random variables X and Y, we obtain an inner product'.

I was so confused that why zero mean is needed here? Since we define inner product as Covariance, it already removes the effects of mean.

Could you please add some extra explanations like one or two sentences for this?

I did a lot Google search but seems there is no related websites found.

Thanks!

AlbertoGuastalla commented 3 years ago

For me, the authors required that the means of both variables be 0 since the inner product is a bilinear, symmetric and positive definite form (as well as the covariance) that takes exactly two arguments, and therefore setting the means to 0 makes that there is no other two arguments (and therefore the arguments are exactly two: X, Y).