A note on the identifiability of nonparametric and semiparametric mixtures of GLMs

In this article, we first propose a semiparametric mixture of generalized linear models (GLMs) and a nonparametric mixture of GLMs, and then establish identifiability results under mild conditions.     

On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.     

Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach

There is ample evidence that in applications of self-exciting point-process models, the intensity of background events is often far from constant. If a constant background is imposed that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modeling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this article, we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam’s razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy, in cases where there is not sufficient empirical evidence to suggest that the background intensity function is more complex than those models. This approach is seldom, if ever, used for nonparametric function estimation in other settings, not least because in those cases more data are typically available. However, our “ontological parsimony” argument is appropriate in the context of self-exciting point-process models. Supplementary materials are available online.     

A Pivotal Allocation-Based Algorithm for Solving the Label-Switching Problem in Bayesian Mixture Models

In Bayesian analysis of mixture models, the label-switching problem occurs as a result of the posterior distribution being invariant to any permutation of cluster indices under symmetric priors. To solve this problem, we propose a novel relabeling algorithm and its variants by investigating an approximate posterior distribution of the latent allocation variables instead of dealing with the component parameters directly. We demonstrate that our relabeling algorithm can be formulated in a rigorous framework based on information theory. Under some circumstances, it is shown to resemble the classical Kullback-Leibler relabeling algorithm and include the recently proposed equivalence classes representatives relabeling algorithm as a special case. Using simulation studies and real data examples, we illustrate the efficiency of our algorithm in dealing with various label-switching phenomena. Supplemental materials for this article are available online.