“The paper defines a new solution to the problem of defining a suitable parameter space for mixture models.”
When I received the table of contents of the incoming Statistics & Computing and saw a paper by V. Maroufy and P. Marriott about the above, I was quite excited about a new approach to mixture parameterisation. Especially after our recent reposting of the weakly informative reparameterisation paper. Alas, after reading the paper, I fail to see the (statistical) point of the whole exercise.
Starting from the basic fact that mixtures face many identifiability issues, not only invariance by component permutation, but the possibility to add spurious components as well, the authors move to an entirely different galaxy by defining mixtures of so-called local mixtures. Developed by one of the authors. The notion is just incomprehensible for me: the object is a weighted sum of the basic component of the original mixture, e.g., a Normal density, and of k of its derivatives wrt its mean, a sort of parameterised Taylor expansion. Which implies the parameter is unidimensional, incidentally. The weights of this strange mixture are furthermore constrained by the positivity of the resulting mixture, a constraint that seems impossible to satisfy in the Normal case when the number of derivatives is odd. And hard to analyse in any case since possibly negative components do not enjoy an interpretation as a probability density. In exponential families, the local mixture is the original exponential family density multiplied by a polynomial. The current paper moves one step further [from the reasonable] by considering mixtures [in the standard sense] of such objects. Which components are parameterised by their mean parameter and a collection of weights. The authors then restrict the mean parameters to belong to a finite and fixed set, which elements are coerced by a maximum error rate on any compound distribution derived from this exponential family structure. The remainder of the paper discusses of the choice of the mean parameters and of an EM algorithm to estimate the parameters, with a confusing lower bound on the mixture weights that impacts the estimation of the weights. And no mention made of the positivity constraint. I remain completely bemused by the paper and its purpose: I do not even fathom how this qualifies as a mixture.