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There has been great interest recently in applying nonparametric kernelmixtures in a hierarchical manner to model multiple related data samplesjointly. In such settings several data features are commonly present: (i) therelated samples often share some, if not all, of the mixture components butwith differing weights, (ii) only some, not all, of the mixture components varyacross the samples, and (iii) often the shared mixture components acrosssamples are not aligned perfectly in terms of their location and spread, butrather display small misalignments either due to systematic cross-sampledifference or more often due to uncontrolled, extraneous causes. Properlyincorporating these features in mixture modeling will enhance the efficiency ofinference, whereas ignoring them not only reduces efficiency but can jeopardizethe validity of the inference due to issues such as confounding. We introducetwo techniques for incorporating these features in modeling related datasamples using kernel mixtures. The first technique, called $\psi$-stickbreaking, is a joint generative process for the mixing weights through thebreaking of both a stick shared by all the samples for the components that donot vary in size across samples and an idiosyncratic stick for each sample forthose components that do vary in size. The second technique is to imbue randomperturbation into the kernels, thereby accounting for cross-samplemisalignment. These techniques can be used either separately or together inboth parametric and nonparametric kernel mixtures. We derive efficient Bayesianinference recipes based on MCMC sampling for models featuring these techniques,and illustrate their work through both simulated data and a real flow cytometrydata set in prediction/estimation, cross-sample calibration, and testingmulti-sample differences.

Mixture modeling on related samples by $\psi$-stick breaking and kernel perturbation