Bayesian mixture labeling and clustering

Label switching is one of the fundamental issues for Bayesian mixture modeling. It occurs due to the nonidentifiability of the components under symmetric priors. Without solving the label switching, the ergodic averages of component specific quantities will be identical and thus useless for inference relating to individual components, such as the posterior means, predictive component densities, and marginal classification probabilities. In this article, we establish the equivalence between the labeling and clustering and propose two simple clustering criteria to solve the label switching. The first method can be considered as an extension of K-means clustering. The second method is to find the labels by minimizing the volume of labeled samples and this method is invariant to the scale transformation of the parameters. Using a simulation example and two real data sets application, we demonstrate the success of our new methods in dealing with the label switching problem

Statistical inference with anchored Bayesian mixture of regressions models: A case study analysis of allometric data

We present a case study in which we use a mixture of regressions model to improve on an ill-fitting simple linear regression model relating log brain mass to log body mass for 100 placental mammalian species. The slope of this regression model is of particular scientific interest because it corresponds to a constant that governs a hypothesized allometric power law relating brain mass to body mass. A specific line of investigation is to determine whether the regression parameters vary across subgroups of related species. We model these data using an anchored Bayesian mixture of regressions model, which modifies the standard Bayesian Gaussian mixture by pre-assigning small subsets of observations to given mixture components with probability one. These observations (called anchor points) break the relabeling invariance typical of exchangeable model specifications (the so-called label-switching problem). A careful choice of which observations to pre-classify to which mixture components is key to the specification of a well-fitting anchor model. In the article we compare three strategies for the selection of anchor points. The first assumes that the underlying mixture of regressions model holds and assigns anchor points to different components to maximize the information about their labeling. The second makes no assumption about the relationship between x and y and instead identifies anchor points using a bivariate Gaussian mixture model. The third strategy begins with the assumption that there is only one mixture regression component and identifies anchor points that are representative of a clustering structure based on case-deletion importance sampling weights. We compare the performance of the three strategies on the allometric data set and use auxiliary taxonomic information about the species to evaluate the model-based classifications estimated from these models

Better Optimism By Bayes: Adaptive Planning with Rich Models

The computational costs of inference and planning have confined Bayesian model-based reinforcement learning to one of two dismal fates: powerful Bayes-adaptive planning but only for simplistic models, or powerful, Bayesian non-parametric models but using simple, myopic planning strategies such as Thompson sampling. We ask whether it is feasible and truly beneficial to combine rich probabilistic models with a closer approximation to fully Bayesian planning. First, we use a collection of counterexamples to show formal problems with the over-optimism inherent in Thompson sampling. Then we leverage state-of-the-art techniques in efficient Bayes-adaptive planning and non-parametric Bayesian methods to perform qualitatively better than both existing conventional algorithms and Thompson sampling on two contextual bandit-like problems.Comment: 11 pages, 11 figure