Spatially-dependent Indian Buffet Processes

We develop a new stochastic process called spatially-dependent Indian buffet processes (SIBP) for spatially correlated binary matrices and propose general spatial factor models for various multivariate response variables. We introduce spatial dependency through the stick-breaking representation of the original Indian buffet process (IBP) and latent Gaussian process for the logit-transformed breaking proportion to capture underlying spatial correlation. We show that the marginal limiting properties of the number of non-zero entries under SIBP are the same as those in the original IBP, while the joint probability is affected by the spatial correlation. Using binomial expansion and Polya-gamma data augmentation, we provide a novel Gibbs sampling algorithm for posterior computation. The usefulness of the SIBP is demonstrated through simulation studies and two applications for large-dimensional multinomial data of areal dialects and geographical distribution of multiple tree species.