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I am studying Gaussian mixtures from the book Pattern Recognition and Machine Learning by Chris Bishop. Figure (a) are contours of the mixture components (Gaussians) with the corresponding weights. Figure (c) is the surface graph of the same.

Figure (b) is the 'combination' of the individual components of the mixture. The book labels it as the contour of the marginal probability density of the mixture.

I do not understand why it is 'marginal'. Is it correct to say that 'marginal' is 'combination' of the individual mixtures because that is how is see it. Please provide some insights. Thanks


Quick recap:

Conditional distribution: $f(x|i)$
Marginal distribution: $f(x) = \sum_{i} f(x|i) \times \pi(i) $
where $\pi$ is the prior of $i$.

In this case the conditional distributions are: $f(x|red),f(x|green),f(x|blue)$. So if we know our object is a "red" then we know the distribution of $x$, it is $f(x|red)$. Otherwise if we don't know what type of object it is, we have to 'marginalise' over the groups, or as we stats people like to say 'integrate out' the groups. I assume the numbers below the ellipses represent the prior probabilities for each group so the marginal probability is $f(x) = \sum_{red,green,blue} f(x|colour) \times \pi(colour)$.


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