this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining to convergence. Such an effort can see wide applications in many other problems related to dynamic systems where the SIS is useful (Berzuini et al. 1996; Liu and Chen 1996). Section 2 describes the specific model that we consider. For illustration we focus discussion on the beta-binomial model, although the methods are applicable to other conjugate families. In Section 3, we describe the first generation of the SIS and Gibbs sampler in this context, and present the necessary conditional distributions upon which the techniques rely. Section 4 describes the alterations that create the second generation techniques, and provides specific algorithms for the model we consider. Section 5 presents a comparison of the techniques on a large set of data. Section 6 provides theory that ensures the proposed methods work and that is generally applicable to many other problems using importance sampling approaches. The final section presents discussion. 2 The Model

. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...

This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the b...

Abstract. A class of probability models is introduced with the objective of representing certain properties of the geometric optics of the human eye. Astigmatic probability laws are those in which the extreme curvature values in the anterior corneal surface, measured at circularly arranged and equally spaced locations, are displaced by an approximate 90 deg angular separation. The relationship between the symmetry invariance of these probability laws for curvature data and probability laws for the ranking permutations associated with the ordering of these data is obtained. A distinction is made between the condition in which the components of the curvature ensemble are represented as real numbers from that in which these curvatures are color-coded and take value on a finite totally ordered set. A constructive principle for astigmatic laws is outlined based on algebraic arguments for the analysis of structured data. 1.

This paper proposes a new efficient merge-split sampler for both conjugate and nonconjugate Dirichlet process mixture (DPM) models. These Bayesian nonparametric models are usually fit using Markov chain Monte Carlo (MCMC) or sequential importance sampling (SIS). The latest generation of Gibbs and Gibbs-like samplers for both conjugate and nonconjugate DPM models effectively update the model parameters, but can have difficulty in updating the clustering of the data. To overcome this deficiency, merge-split samplers have been developed, but until now these have been limited to conjugate or conditionally-conjugate DPM models. This paper proposes a new MCMC sampler, called the sequentially-allocated merge-split (SAMS) sampler. The sampler borrows ideas from sequential importance sampling. Splits are proposed by sequentially allocating observations to one of two split components using allocation probabilities that condition on previously allocated data. The SAMS sampler is applicable to general nonconjugate DPM models as well as conjugate models. Further, the proposed sampler is substantially more efficient than existing conjugate and nonconjugate samplers.

. Spectral analysis on the symmetric group Sn calls for computing projections of functions defined on Sn and its homogeneous spaces, onto invariant subspaces. In particular, for the analysis of partially ranked data, the appropriate homogeneous spaces are given as quotients by Young subgroups. Here the naive character theoretic approach to computing projections requires O(n \Delta n!) operations. In this paper two types of polynomial time algorithms (quadratic in the size of the homogeneous space) are presented for partially ranked data. The first approachmakes use of a more careful organization of the character theoretic computation and is applicable to arbitrary finite groups and their homogeneous spaces. The second approach makes use of the techniques of the combinatorial Radon transform. 1. Introduction Let G be a finite group acting transitively on a set X. Often X is called a homogeneous space for G. Let L(X) denote the vector space of complex-valued functions on X. Then L(X) na...