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Hello everyone, In Bishop's ML&PR, Page 490, 10.4 Exponential Family Distributions, 1st Paragraph He mentions that in a Mixture of Gaussian, the Joint distributions of X and Z is a member of the exponential family, while the marginal of X isn't. Is this because the Joint Distribution corresponds to a single Gaussian Distribution for a single observation; while the Marginal consists of the mix of all the Normal's over the space. Thanks a lot |
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The joint distribution is the product of a multinomial and a Gaussian distribution, both of which are in the exponential family. The product of two exponential family distributions is always in the exponential family itself (basically, because you can simply sum the terms inside the exp), so the joint distribution is exponential family. The marginal distribution, on the other hand, is a weighted sum of Gaussian distributions. There is no way to express such a sum in terms of an exponential distribution (note that the sum of two distributions cannot be written as a sum inside the exp). |
