Inconsistency of Pitman-Yor process mixtures for the number of components

In many applications, a finite mixture is a natural model, but it can be difficult to choose an appropriate number of components. To circumvent this choice, investigators are increasingly turning to Dirichlet process mixtures (DPMs), and Pitman-Yor process mixtures (PYMs), more generally. While these models may be well-suited for Bayesian density estimation, many investigators are using them for inferences about the number of components, by considering the posterior on the number of components represented in the observed data. We show that this posterior is not consistent --- that is, on data from a finite mixture, it does not concentrate at the true number of components. This result applies to a large class of nonparametric mixtures, including DPMs and PYMs, over a wide variety of families of component distributions, including essentially all discrete families, as well as continuous exponential families satisfying mild regularity conditions (such as multivariate Gaussians).Comment: This is a general treatment of the problem discussed in our related article, "A simple example of Dirichlet process mixture inconsistency for the number of components", Miller and Harrison (2013) arXiv:1301.270

Contribution of Land Conservation and Freshwater Resources to Residential Property Values in the Matanuska-Susitna Borough

Growing interest in quantifying values of ecosystem services has generated numerous studies attempting to measure the contribution of neighborhood environmental amenities to urban and suburban property values. Proximity to freshwater resources -- lakes and streams -- has also figured prominently in many of these studies. Alaska’s Matanuska-Susitna (Mat-Su) Borough, analogous to a county under state law, is a large and rapidly urbanizing local government jurisdiction adjacent to Anchorage, the state’s largest metropolitan area. As the population of the borough grows, and more land becomes subdivided and developed, an important question arises regarding the contribution of remaining undeveloped land and natural amenities to the economy of the borough. Visitors who are attracted to the scenery and recreation opportunities of the borough capture some of that value, and contribute to the borough economy through local purchases of goods and services. Private owners of borough real estate, who are willing to pay more for property located close to natural areas and recreation sites, also appropriate a portion of the value, however. This study focuses on this latter component of value of ecosystem services. It provides estimates of the enhanced value of private residential property and undeveloped land in the Mat-Su borough created by local protected open space and outdoor recreation opportunities. After briefly describing the Mat-Su Borough region, we summarize the valuation methods and the data available for the study. Then we present statistical results, followed by a discussion of the implications of the findings for valuing ecosystem services in the Borough. We conclude with suggestions for future research to improve the estimates.The Nature Conservancy of Alaska. The Bullitt Foundation. The U.S. Fish and Wildlife Servic

Exact Enumeration and Sampling of Matrices with Specified Margins

We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or non-negative integer matrices are handled. The method is distinguished by applicability to non-regular margins, tractability on large matrices, and the capacity for exact sampling

Point/Counterpoint: The Gettysburg Battlefield Marathon

Jeff: On November 6, the small town of Gettysburg will be swarmed by runners during the first ever Gettysburg Battlefield Marathon. The event has provoked heated discussion from many in the Civil War community, bringing up many questions regarding the use of our most hallowed grounds for recreational use. In this post, Matt and I will engage in a back and forth conversation about the concerns and advantages of the race. I’d like to begin by noting that the views that we each express in this piece may not necessarily be our own and that we may merely be bringing them up to contribute to the conversation surrounding the marathon. [excerpt

Exact sampling and counting for fixed-margin matrices

The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in many nontrivial real-world cases. We illustrate with real datasets including ecological co-occurrence matrices and contingency tables.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1131 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1104.032