If I have a piecewise distribution with density f on [0,1) so that f=f1 on [0,0.5) and f=f2 on [0.5,1), with both f1 and f2 densities, is it necessarily a mixture distribution? My reasoning is that we can always introduce a latent variable z taking values in [0,1) uniformly and then choose either f1 or f2 depending on whether z>0.5 or not. What is the difference between a piecewise distribution and a mixture distribution here?
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1you can write any piecewise distribution as a mixture distribution, yes. Actually, there's no need for it to be a piecewise distribution: a standard normal distribution is a 50-50 mixture between two folded normal distributions, one positive and one negative.– John MaddenCommented May 12, 2023 at 14:06
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1Every distribution, without exception, can be expressed as a mixture in many ways. Just arbitrarily partition R into up to a countable union of measurable disjoint subsets R=E1∪E2∪⋯, let pi=Prf(Ei) for each i, and for all nonzero pi set fi(x)=f(x)piIEi(x) to be the truncation of f supported on Ei.. It is immediate that f is the mixture of these fi with weights pi.– whuber ♦Commented May 12, 2023 at 14:47
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1It's more clear now that mixture distributions constitute every distribution, in particular the piecewise defined distributions, thanks.– giorgi nguyenCommented May 13, 2023 at 6:33
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