# On Random Stable Partitions

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### CS 6240: Parallel Data Processing in MapReduce

a random assignment, we can rely on the default hash Partitioner. If the data is very skewed, i.e., the amount of work varies significantly between different keys, one should use a more fine-grained partitioning or carefully design a custom Partitioner to balance load. Let us look at two MapReduce examples next. 15

### Partition Structures Derived from Brownian Motion and Stable

motion and stable subordinators JIM PITMAN Department of Statistics, University of California, Berkeley CA 94720, USA Explicit formulae are obtained for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following

### 2 Random Fields - Stanford University

Random Fields 2.1 Stochastic Processes and Random Fields As you read in the Preface, for us a random eld is simply a stochastic pro-cess, taking values in a Euclidean space, and de ned over a parameter space of dimensionality at least one. Actually, we shall be rather loose about exchang-ing the terms random eld and stochastic process

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### Stick-breaking priors: The Pitman-Yor process and randomized

Basic Deﬁnitions: mass Partitions, random cdf s, Pitman-Yor process, -stable Poisson Kingman laws Stick-breaking reps derived from species sampling models (size biased sampling) What is explicitly known? Surprisingly not much Lancelot F. James ([email protected]) HKUST

### RANDOM PARTITIONS APPROXIMATING THE COALESCENCE

RANDOM PARTITIONS APPROXIMATING THE COALESCENCE OF LINEAGES DURING A SELECTIVE SWEEP BY JASON SCHWEINSBERG1 AND RICK DURRETT2 University of California, San Diego and Cornell University When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep.

### INTERNATIONAL CONFERENCE ON RANDOM MAPPINGS, PARTITIONS, AND

xex = n for random partitions of the set 1,2, , n}, x = exp (-n/&) for random partitions of the integer n and x = exp (-n/ fi 12n) for partitions with no repeated parts. For most examples, with an appropriate choice of x, for large n and individually each i = 1 to n, Z, is a good approximation for C,(n).

### BayesianAnalysis(2020) ,Number3,pp.683 710 Gibbs

able sequence sampled from a random probability measure, and the laws of the par-tition and measure are one-to-one. The measures inducing the Gibbs-type partitions are called Gibbs-type random measures. (The reader should consult Kingman (1975) for background on random probability measures.) We will focus on subclasses of the

### Limit shape of a random integer partition with a bounded max

random from A∩Λn,orfromB ∩Λn, the random function yn(x):=m−n1λ xm n converges to y =f(x); here mn =an1/2. In terminology of [5], Andrews s partitions were thus conjectured to be stable. Igor Pak also anticipated that the uniformly random partition from {λ λ1 2λ}∩Λn should have the same limit shape y =f(x)[6].

### Consensus clustering in complex networks

and their results typically depend on the specific random seeds, initial conditions and tie-break rules proven to lead quickly to consistent and stable partitions in real

### Probab. Theory Relat. Fields 92, 21-39 (1992) Probability Theory

arcsine laws and interval partitions associated with a stable subordinator. In settings for both the gamma and stable cases mentioned above, another random sequence which turns up naturally is the sequence (P~, P2 ) derived from (P(1), P~2), 9 - - ) by size-biased random permutation.

### Random Tessellation Forests - NeurIPS

for the random tessellations associated with such (Here, for a hyperplane h, h+ xrefers to the set fy+ x: y2hg, and for a tessellation Y and a domain W0, Y W0is the tessellation fa W0: a2Yg.) In [27], such tessellations are referred to as stable iterated tessellations. If we assume that is the product measure d d

### CONTINUUM-SITES STEPPING-STONE MODELS, COALESCING

TheAnnalsofProbability 2000,Vol.28,No.3,1063 1110 CONTINUUM-SITES STEPPING-STONE MODELS, COALESCING EXCHANGEABLE PARTITIONS AND RANDOM TREES ByPeterDonnelly,1 StevenN.Evans,2 KlausFleischmann,

### Are Stable Instances Easy? - huji.ac.il

be solved eﬃciently on (locally) stable graphs. In Section 5.1 we show how to deduce an improved approximation bound for the Goemans Williamson algorithm on stable instances, and that Max-Cut is easy in a certain random model for such instances. Finally, while we claim that all interesting clustering instances are stable (in an intuitive

### Limit shape of partitions via bijections

Theorem 4.1. Let A C P be an asymptotically stable set of partitions, let B C P and let : A -+ B be a size preserving one-to-one correspondence. Suppose a natural geometric Injection ç defines ð. Then ç is asymptotically stable. Moreover, this holds ifç is an AG-bijection with asymptotically stable parameters.

### Design and Estimation for Domains

(c)The pair of subclasses may be exhaustive partitions (xc +Xb= x for the entire sample), but often they are only two of k partitions, hence the pair represents two out of k(k-1)/2possible pairs. (d) Usually they are random variables, especially when they are cross classes, even if the total sample size were fixed. This feature has conse

### Entropy-based Generating Markov Partitions for Complex Systems

lating the unstable and stable manifolds, previous meth-ods have obtained approximate Markov partitions for dy-namical systems by searching alternative approaches14. For example, Ref.15 shows how to calculate partitions by the primary homoclinic tangencies of dissipative systems, later extended to conservative systems16. In Ref.17, au-

### On a class of random probability measures with general

on completely random measures and Bell polynomials as well as the proofs of our results. 2 Gibbs-type r.p.m.s A random partition of the set of natural numbers N is de ned as a consistent sequence := f n;n 1gof random elements, with ntaking values in the set of all partitions of f1;:::;ng into some number of disjoint blocks.

### Beta-coalescents and continuous stable random trees

Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning

### A Bayesian Nonparametric View on Count-Min Sketch

on the stable beta process, which enables the modeling of power law behavior in the stream. 2 The Count-Min Sketch The count-min (CM) sketch [8] is a randomized data structure that uses random hashing to approxi-

### Package mri

random and independent partitions. The maximum value of the index is 1. Higher value indicates more similar (stable) partitions. Both splitting of clusters and merging of clusters lower the value of the indices. Note The special cases of the modiﬁed indices (when only outgoers or only newcomers are present) are

### Beta-coalescents and continuous stable random trees

to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly Kurtz lookdown process using continuous random trees, which is of independent in-terest.

### Poisson-Kingman Partitions

shown in [2, 3], it is this stable(1 2) model which governs the asymptotic distribution of partitions derived in various ways from random forests, random mappings, and the additive coalescent. See also [5, 9] for further developments in terms of Brownian paths, and [10, 25] for applications to hashing and parking algorithms. This paper

### Ensembles of Partitions via Data Resampling

partitions as well as considering object-distributed and feature-distributed formulations of the problem [4]. They also examined the combination of partitions with a deterministic overlap of points between data subsets (non-random). Resampling methods have been traditionally used to obtain more accurate estimates of data statistics. Efron

### An upper bound for the solvability probability of a random

RANDOM STABLE ROOMMATES 469 Theorem 2.3. roommates instance, then C is a cycle in all stable partitions for that instance. If C is an odd-length cycle in a stable partition for a given It follows from Theorem 2.3 that an instance that admits a stable partition with

### Hypergraph Partitioning and Clustering

nonempty partitions. The k-way partitioning problem seeks to minimize a given cost function of such an assignment. A standard cost function is net cut, which is the number of hyperedges that span more than one partition, or, more generally, the sum of weights of such edges. Con-

### THE NATURE OF PARTITION BIJECTIONS II. ASYMPTOTIC STABILITY

of Euler s theorem only Sylvester s bijection is asymptotically stable. The basic tool in our study is the limit shape for sets of partitions. This idea originated in the study of random Young tableaux [24], and has recently appeared in the context of random partitions [23]. Roughly, it says that the diagram of a random partition scaled by

### A Mixture Model for Clustering Ensembles

most stable partitions from an evolving ensemble (population) of clustering algorithms along with a special objective function. The objective function evaluates multiple partitions according to changes caused by data perturbations and prefers those clusterings that are least susceptible to those perturbations.

### Stable behavior and generalized partition

that the behavior of a Bayesian expected utility maximizer is stable for this family of contingent preferences, if and only if it is generated by a generalized partition. Importantly, the class of generalized partitions includes information structures that support all events,3 so that it may be possible to elicit I for all I ⊂ S. In other

### Combinatorial Stochastic Processes

random partitions and random trees, and the asymptotics of these models re-lated to continuous parameter stochastic processes. A basic feature of models for random partitions is that the sum of the parts is usually constant. So the sizes of the parts cannot be independent. But the structure of many natural mod-

### 92-27593

On random partitions of the integer n 3:50 - 4:10 B. FRISTEDT, University of Minnesota Random permutations and stable matchings 12:00 - 1:30 Lunch

### Spectral Graph Theory and its Applications

Random walks Graph Partitioning and clustering Refine partitions using other partitions. Use vertex classes to split eigenspaces. Eigenvectors stable too, if

### -coalescents and stable Galton-Watson trees

of partitions of the integers using the pruning of a L evy continuum random tree. More precisely, we consider the weighted stable L evy tree (T;d;mT) associated with the branching mechanism ( ) = for 2(1;2) (the case = 2 is studied inABRAHAM and DELMAS(2013a) and requires a di erent pruning). We recall

### artition structures deriv

, is a random v ariable n with v alues in the set of partitions of n. Kingman (1978) in tro duced the concept of a p artition structur e, that is a sequence (P n, n =1; 2;:::)of dis-tributions for random partitions n of n,whic his c onsistent in the follo wing sense: if n ob jects are partitioned in to subsets with sizes giv en b y n,and

### Learning Ensembles from Bites: A Scalable and Accurate Approach

random disjoint partitions of data, and combine predictions from all those classiﬁers to achieve high classiﬁcation accuracies; in some cases similar to or better than boosting or distributed boosting (Lazarevic and Obradovic, 2002). We utilize Breiman s algorithms for pasting small votes: Ivote and Rvote (Breiman, 1999).

### Maximally Stable Gaussian Partitions With Discrete Applications

1.1 Maximally Stable Gaussian Partitions We will be concerned with ﬁnding partitions of Rn that maximize the probability that correlated Gaussian vectors remain within the same part. More speciﬁcally we would like to partition Rn into q ≥ 2 disjoint sets of ﬁxed measure.

### Exchangeable random partitions and random discrete

Exchangeable random partitions and random discrete probability measures: a brief tour guided by the Dirichlet Process Notes for Oxford Statistics Grad Lecture Benjamin Bloem-Reddy [email protected] 15 February, 2018 These notes are a work in progress; if you ﬁnd any mistakes, please let me know. 1 Overview and preliminaries

### Theory of the Stability of Punishment, A.

to speak of a stable level of crime in this strict sense. There are indications in Durkheim's and Erik-son's works that both men, when they speak of 7 K. EanxsoN, supra note 5, at 163-81. stable levels of crime, are referring only to those crimes which come to public attention through the punishment of the wrongdoer.

### Beta-Negative Binomial Process and Exchangeable Random

BNBP to describe the joint distribution of the random group sizes and their random partitions over a random, potentially unbounded number of exchangeable clusters shared across all the groups. Later we will show how to derive the EPPF from the ECPF using Bayes rule. Lemma 1. Denote k(z ji) as a unit point mass at z ji = k, n jk = P m j P i=1

### Random Tessellation Forests

partitions of(v 1,z 1), ,(v n,z n)induced by tessellations. 2.1 The Random Tessellation Process A tessellationY of a bounded domainW #Rd is a Þnite collection of closed polytopes such that the union of the polytopes is all ofW, and such that the polytopes have pairwise disjoint interiors [6]. We denote tessellations ofW byY(W)or the symbol