Hi everybody, I apologize if this question is too basic but I did 1 hour of solid Google searching and couldn't find an answer and I'm stuck.(adsbygoogle = window.adsbygoogle || []).push({});

I'm reading Bishop'sPattern Recognition and Machine Learningand in the second chapter he introduces partitioned vectors. Say, if X is a D-dimensional vector, it can be partitioned like:

X = [Xa, Xb] where Xa is the first M components of X and Xb is the remaining D-M components of X.

I have no problem with this simple concept. Later in the same chapter he talks about conditional and marginal multivariate Gaussian distributions and he uses the notation p(Xa, Xb). I'm trying to understand how certain integrals involving this notation are expanded but I'm actually struggling to understand even this expression. It seems to suggest that we're denoting the joint probability of the components of Xa and the components of Xb. But those are just the components of X anyway!

What is the difference between P(Xa, Xb) and P(X)?

It will be more helpful for me if we considered a more concrete example. Say, X = [X1, X2, X3, X4] and Xa = [X1, X2] while Xb = [X3, X4]. Now, the joint probability P(X) would simply be P(X1, X2, X3, X4), right? What is P(Xa, Xb) in this case?

Thanks in advance!

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# Joint probability of partitioned vectors

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