Estimating the number of endmembers in hyperspectral images using the normal compositional model and a hierarchical Bayesian algorithm.

This paper studies a semi-supervised Bayesian unmixing algorithm for hyperspectral images. This algorithm is based on the normal compositional model recently introduced by Eismann and Stein. The normal compositional model assumes that each pixel of the image is modeled as a linear combination of an unknown number of pure materials, called endmembers. However, contrary to the classical linear mixing model, these endmembers are supposed to be random in order to model uncertainties regarding their knowledge. This paper proposes to estimate the mixture coefficients of the Normal Compositional Model (referred to as abundances) as well as their number using a reversible jump Bayesian algorithm. The performance of the proposed methodology is evaluated thanks to simulations conducted on synthetic and real AVIRIS images

A Bayesian information criterion for singular models

We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher-information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion (BIC) and the penalty structure in BIC generally does not reflect the frequentist large-sample behavior of their marginal likelihood. While large-sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components of mixture models, the number of factors in latent factor models or the rank in reduced-rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems

Bayesian variable selection for high dimensional generalized linear models: convergence rates of the fitted densities

Bayesian variable selection has gained much empirical success recently in a variety of applications when the number $K$ of explanatory variables $(x_1,...,x_K)$ is possibly much larger than the sample size $n$. For generalized linear models, if most of the $x_j$'s have very small effects on the response $y$, we show that it is possible to use Bayesian variable selection to reduce overfitting caused by the curse of dimensionality $K\gg n$. In this approach a suitable prior can be used to choose a few out of the many $x_j$'s to model $y$, so that the posterior will propose probability densities $p$ that are ``often close'' to the true density $p^*$ in some sense. The closeness can be described by a Hellinger distance between $p$ and $p^*$ that scales at a power very close to $n^{-1/2}$, which is the ``finite-dimensional rate'' corresponding to a low-dimensional situation. These findings extend some recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics, Northwestern Univ.] on consistency of Bayesian variable selection for binary classification.Comment: Published in at http://dx.doi.org/10.1214/009053607000000019 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org