arXiv subjects
Publications and source records attributed to Pierre Pudlo.
This report is a collection of comments on the Read Paper of Fearnhead and Prangle (2011), to appear in the Journal of the Royal Statistical Society Series B, along with a reply from the authors.
The development of statistical methods and numerical algorithms for model choice is vital to many real-world applications. In practice, the ABC approach can be instrumental for sequential model design; however, the theoretical basis of its use has been questioned. We present a measure-theoretic framework for using the ABC error towards model choice and describe how easily existing rejection, Metropolis-Hastings and sequential importance sampling ABC algorithms are extended for the purpose of model checking. Considering a panel of applications from evolutionary biology to dynamic systems, we discuss the choice of summaries which differs from standard ABC approaches. The methods and algorithms presented here may provide the workhorse machinery for an exploratory approach to ABC model choice, particularly as the application of standard Bayesian tools can prove impossible.
Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.
Also known as likelihood-free methods, approximate Bayesian computational (ABC) methods have appeared in the past ten years as the most satisfactory approach to untractable likelihood problems, first in genetics then in a broader spectrum of applications. However, these methods suffer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey, we study the various improvements and extensions made to the original ABC algorithm over the recent years.
Following Hartigan, a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two steps. First, a nonparametric density estimate is used to extract the data points for which the estimated density takes a value greater than t. Next, the extracted points are clustered based on the eigenvectors of a graph Laplacian matrix. Under mild assumptions, we prove the almost sure convergence in operator norm of the empirical graph Laplacian operator associated with the algorithm. Furthermore, we give the typical behavior of the representation of the dataset into the feature space, which establishes the strong consistency of our proposed algorithm.