Results 1  10
of
35
A bayesian interpretation of interpolated kneserney
, 2006
"... Interpolated KneserNey is one of the best smoothing methods for ngram language models. Previous explanations for its superiority have been based on intuitive and empirical justifications of specific properties of the method. We propose a novel interpretation of interpolated KneserNey as approxima ..."
Abstract

Cited by 33 (3 self)
 Add to MetaCart
Interpolated KneserNey is one of the best smoothing methods for ngram language models. Previous explanations for its superiority have been based on intuitive and empirical justifications of specific properties of the method. We propose a novel interpretation of interpolated KneserNey as approximate inference in a hierarchical Bayesian model consisting of PitmanYor processes. As opposed to past explanations, our interpretation can recover exactly the formulation of interpolated KneserNey, and performs better than interpolated KneserNey when a better inference procedure is used. 1
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a sub ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.
Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times
 ELECTRON. J. PROBAB
, 1999
"... For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variabl ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X . The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae a...
Rook theory, generalized Stirling numbers and (p,q)analogues
, 2004
"... In this paper, we define two natural (p, q)analogues of the generalized Stirling numbers of the first and second kind S 1 (α, β, r) andS 2 (α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β =0andα and r are nonnegative integers both of our (p, q)analogues have natural ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
(Show Context)
In this paper, we define two natural (p, q)analogues of the generalized Stirling numbers of the first and second kind S 1 (α, β, r) andS 2 (α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β =0andα and r are nonnegative integers both of our (p, q)analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our (p, q)analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our (p, q)analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.
A Bayesian Review of the PoissonDirichlet Process
, 2010
"... The two parameter PoissonDirichlet process is also known as the PitmanYor Process and related to the ChineseRestaurant Process, is a generalisation of the Dirichlet Process, and is increasingly being used for probabilistic modelling in discrete areas such as language and images. This article revie ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
(Show Context)
The two parameter PoissonDirichlet process is also known as the PitmanYor Process and related to the ChineseRestaurant Process, is a generalisation of the Dirichlet Process, and is increasingly being used for probabilistic modelling in discrete areas such as language and images. This article reviews the theory of the PoissonDirichlet process in terms of its consistency for estimation, the convergence rates and the posteriors of data. This theory has been well developed for continuous distributions (more generally referred to as nonatomic distributions). This article then presents a Bayesian interpretation of the PoissonDirichlet process: it is a mixture using an improper and infinite dimensional Dirichlet distribution. This interpretation requires technicalities of priors, posteriors and Hilbert spaces, but conceptually, this means we can understand the process as just another Dirichlet and thus all its sampling properties fit naturally. Finally, this article also presents results for the discrete case which is the case seeing widespread use now in computer science, but which has received less attention in the literature.
Topic Segmentation with a Structured Topic Model
"... We present a new hierarchical Bayesian model for unsupervised topic segmentation. This new model integrates a pointwise boundary sampling algorithm used in Bayesian segmentation into a structured topic model that can capture a simple hierarchical topic structure latent in documents. We develop an M ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
We present a new hierarchical Bayesian model for unsupervised topic segmentation. This new model integrates a pointwise boundary sampling algorithm used in Bayesian segmentation into a structured topic model that can capture a simple hierarchical topic structure latent in documents. We develop an MCMC inference algorithm to split/merge segment(s). Experimental results show that our model outperforms previous unsupervised segmentation methods using only lexical information on Choi’s datasets and two meeting transcripts and has performance comparable to those previous methods on two written datasets. 1
Congruences for Degenerate Number Sequences
"... The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as padic integrals of generalized factorials. As an application of this ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as padic integrals of generalized factorials. As an application of this identification we deduce systems of congruences which are analogues and generalizations of the Kummer congruences for the ordinary Bernoulli numbers. Keywords: Degenerate weighted Stirling numbers; Degenerate Eulerian polynomials; Partial Stirling numbers; Kummer congruences; padic integration 1.
A lattice path model for the Bessel polynomials
, 1999
"... The (n \Gamma 1)th Bessel polynomial is represented by an exponential generating function derived from the number of returns to 0 of a sequence with 2n increments of \Sigma1 which starts and ends at 0. AMS 1991 subject classification. Primary: 05A15. Secondary: 33C10, 33C45. It is well known [21, x3 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
The (n \Gamma 1)th Bessel polynomial is represented by an exponential generating function derived from the number of returns to 0 of a sequence with 2n increments of \Sigma1 which starts and ends at 0. AMS 1991 subject classification. Primary: 05A15. Secondary: 33C10, 33C45. It is well known [21, x3.71 (12)],[6, (7.2(40)] that the McDonald function or Bessel function of imaginary argument K (x) := 1 2 ` x 2 ' \Gamma Z 1 0 t \Gamma1 e \Gammat\Gamma(x=2) 2 =t dt (1) admits the evaluation K n+1=2 (x) = r ß 2x e \Gammax ` n (x)x \Gamman (n = 0; 1; 2; : : :) (2) Research supported in part by N.S.F. Grant 9703961 where ` n (x) := n X m=0 fi n;n\Gammam x m with fi n;k := (n + k)! 2 k (n \Gamma k)!k! : (3) The Bessel polynomials ` n (x) and y n (x) := n X k=0 fi n;k x k = x n ` n (x \Gamma1 ) (4) have been extensively studied and applied: see the book of Grosswald [9] for a review. Dulucq and Favreau [4, 5] gave a combinatorial model for the Bes...
The generalized Stirling and Bell numbers revisited
 J. Integer Seq
"... The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs; ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs;h(n) and Bell polynomials B s;hn(x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel 1 polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a qanalogue Ss;h(n,kq) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the qdeformed numbers Ss;h(n,kq) are special cases of the typeII p,qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the qmeromorphic Weyl algebra considered by Diaz and Pariguan. 1