Practical Bayesian inference using mixtures of mixtures

Discrete mixtures of normal distributions are widely used in modeling amplitude fluctuations of electrical potentials at synapses of human and other animal nervous systems. The usual framework has independent data values yj arising as yj = mu j + xn0 + j, where the means mu j come from some discrete prior G(mu) and the unknown xno + j's and observed xj, j = 1,...,n0, are Gaussian noise terms. A practically important development of the associated statistical methods is the issue of nonnormality of the noise terms, often the norm rather than the exception in the neurological context. We have recently developed models, based on convolutions of Dirichlet process mixtures, for such problems. Explicitly, we model the noise data values xj as arising from a Dirichlet process mixture of normals, in addition to modeling the location prior G(mu) as a Dirichlet process itself. This induces a Dirichlet mixture of mixtures of normals, whose analysis may be developed using Gibbs sampling techniques. We discuss these models and their analysis, and illustrate them in the context of neurological response analysis.