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I understand the EM algorithm and know there are other optimisation algorithms.

However, I have seen that with a mixture copula, EM algorithm always seems to be used. Is there any reason why EM algorithm is such a commonly used method for mixture models in general and for mixture copula in particular?

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There are many varieties of EM algorithms. In some of the problems I have worked on, fancy EM algorithms are absolutely state of the art.

That being said, in regards to mixture models, the "vanilla EM" algorithm, which I am pretty sure you are referring to, has two major advantages; extremely easy to implement and extremely numerically stable. As such, if you are exploring new models, this makes the vanilla EM algorithm extremely attractive. If using this vanilla EM algorithm allows you to fit your model, and you've shown that your new model is much better than previous models, then it makes sense to start designing a fancier algorithm that computes your estimate faster if the basic algorithm is prohibitively slow.

Basically, the simplest EM algorithm is often used when fitting new models because "pre-mature optimization is the root of all evil". My guess is that in the copula mixture model papers you've looked at, the authors are trying to demonstrate the superiority of their model, not the computational efficiency of their algorithm. As such, they provide the simplest algorithm to do the job.

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