Combinatorial Stochastic Processes Research Articles

A reinforced urn process modeling of recovery rates and recovery times

Abstract Answering a major demand in modern credit risk management, we propose a nonparametric survival approach for the modeling of the recovery rate and the recovery time of a defaulted counterparty, by introducing what we call the Recovery Reinforced Urn Process, a special type of combinatorial stochastic process. The new model allows for the elicitation and exploitation of prior knowledge and experts’ judgements, and for the constant update of this information over time, as soon as new data become available. We show how to use it to perform Bayesian nonparametric prediction about the recovered amounts and the (total) recovery time of a series of defaulted exposures. An application to real data is provided using the Single Family Loan-Level Dataset by Freddie Mac.

A Conversation with Jim Pitman

Jim Pitman was born in June 1949, received a Ph.D. in 1974 from the University of Sheffield with advisor Terry Speed, and since 1979 has been in the U.C. Berkeley Statistics department. He is known for research on many topics within probability, in particular for a long-running collaboration with Marc Yor on distributional properties of Brownian motion, and for his influential lecture notes Combinatorial Stochastic Processes. The following conversation took place at his home in December 2017 and February 2018.

A Reinforced Urn Process Modeling of Recovery Rates and Recovery Times

Answering a major demand in modern credit risk management, we propose a nonparametric survival approach for the modeling of the recovery rate and the recovery time of a defaulted counterparty, by introducing what we call the Recovery Reinforced Urn Process, a special type of combinatorial stochastic process. The new model allows for the elicitation and exploitation of prior knowledge and experts’ judgements, and for the constant update of this information over time, as soon as new data become available. We show how to use it to perform Bayesian nonparametric prediction about the recovered amounts and the (total) recovery time of a series of defaulted exposures. An application to real data is provided using the Single Family Loan-Level Dataset by Freddie Mac.

The transmission process: A combinatorial stochastic process for the evolution of transmission trees over networks

We derive a combinatorial stochastic process for the evolution of the transmission tree over the infected vertices of a host contact network in a susceptible-infected (SI) model of an epidemic. Models of transmission trees are crucial to understanding the evolution of pathogen populations. We provide an explicit description of the transmission process on the product state space of (rooted planar ranked labelled) binary transmission trees and labelled host contact networks with SI-tags as a discrete-state continuous-time Markov chain. We give the exact probability of any transmission tree when the host contact network is a complete, star or path network – three illustrative examples. We then develop a biparametric Beta-splitting model that directly generates transmission trees with exact probabilities as a function of the model parameters, but without explicitly modelling the underlying contact network, and show that for specific values of the parameters we can recover the exact probabilities for our three example networks through the Markov chain construction that explicitly models the underlying contact network. We use the maximum likelihood estimator (MLE) to consistently infer the two parameters driving the transmission process based on observations of the transmission trees and use the exact MLE to characterize equivalence classes over the space of contact networks with a single initial infection. An exploratory simulation study of the MLEs from transmission trees sampled from three other deterministic and four random families of classical contact networks is conducted to shed light on the relation between the MLEs of these families with some implications for statistical inference along with pointers to further extensions of our models. The insights developed here are also applicable to the simplest models of “meme” evolution in online social media networks through transmission events that can be distilled from observable actions such as “likes”, “mentions”, “retweets” and “+1s” along with any concomitant comments.

The Ubiquitous Ewens Sampling Formula

Ewens’s sampling formula exemplifies the harmony of mathematical theory, statistical application, and scientific discovery. The formula not only contributes to the foundations of evolutionary molecular genetics, the neutral theory of biodiversity, Bayesian nonparametrics, combinatorial stochastic processes, and inductive inference but also emerges from fundamental concepts in probability theory, algebra, and number theory. With an emphasis on its far-reaching influence throughout statistics and probability, we highlight these and many other consequences of Ewens’s seminal discovery.

Clustering from Categorical Data Sequences

The three-parameter cluster model is a combinatorial stochastic process that generates categorical response sequences by randomly perturbing a fixed clustering parameter. This clear relationship between the observed data and the underlying clustering is particularly attractive in cluster analysis, in which supervised learning is a common goal and missing data is a familiar issue. The model is well equipped for this task, as it can handle missing data, perform out-of-sample inference, and accommodate both independent and dependent data sequences. Moreover, its clustering parameter lies in the unrestricted space of partitions, so that the number of clusters need not be specified beforehand. We establish these and other theoretical properties and also demonstrate the model on datasets from epidemiology, genetics, political science, and legal studies.

Combinatorial Clustering and the Beta Negative Binomial Process

We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process ( NBP ) as an infinite-dimensional prior appropriate for such problems. We show that the NBP is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process ( BNBP) and hierarchical models based on the BNBP (the HBNBP). We study the asymptotic properties of the BNBP and develop a three-parameter extension of the BNBP that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the HBNBP , and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis.

Analysis of the Strategy “Hiring Above the $$m$$ m -th Best Candidate”

The hiring problem is a simple model for on-line decision-making, inspired by the well-known and time-honored secretary problem (see, for instance, the surveys of Freeman Int Stat 51:189---206, 1983 or Samuels Handbook of sequential analysis, 1991). In the combinatorial model of the hiring problem (Archibald and Martinez Proc. of the 21st Int. Col. on Formal Power Series and Algebraic Combinatorics (FPSAC), 2009), there is a sequence of candidates of unknown length, and we can rank all candidates from best to worst without ties; all orders are equally likely. Thus any finite prefix of the sequence of candidates can be modeled as a random permutation. Candidates come one after another, and a decision must be taken immediately either to hire or to discard the current candidate, based on her relative rank among all candidates seen so far. In this paper we analyze in depth the strategy hiring above the $$m$$ m -th best candidate (or hiring above the $$m$$ m -th best, for short), formally introduced by Archibald and Martinez (Proc. of the 21st Int. Col. on Formal Power Series and Algebraic Combinatorics (FPSAC), 2009). This hiring strategy hires the first $$m$$ m candidates in the sequence whatever their relative ranks, then any further candidate is hired if her relative rank is better than the $$m$$ m -th best candidate seen so far. The close connection of this hiring strategy to the notion of records in sequences and permutations is quite evident; records and their generalizations (namely, $$m$$ m -records) have been throughly studied in the literature (see, for instance, the book of Arnold et al. Records, Wiley Series in Probability and Mathematical Statistics, 1998), and we explore here that relationship. We also discuss the relationship between this hiring strategy and the seating plan $$(0,m)$$ ( 0 , m ) for the well-known Chinese restaurant process (Pitman Combinatorial stochastic processes, 2006). We analyze in this paper several random variables (we also call them hiring parameters) that give us an accurate description of the main probabilistic features of "hiring above the $$m$$ m -th best", from the point of view of the hiring rate (number of hired candidates, waiting time to hire a certain number of candidates, time between consecutive hirings,...) and also of the quality of the hired staff (rank of the last hired candidate, best discarded candidate,...). We are able to obtain the exact and asymptotic probability distributions for the most fundamental parameter, namely, the number of hired candidates, and also for most of the other parameters. Another novel quantity that we analyze here is the number of replacements, a quantity naturally arising when we consider "hiring above the $$m$$ m -th best" endowed with a replacement mechanism: that is, an incoming candidate can be: (1) hired anew, because the candidate is among the $$m$$ m best seen so far, (2) hired to replace some previously hired worse candidate, or (3) discarded.

Cluster and Feature Modeling from Combinatorial Stochastic Processes

One of the focal points of the modern literature on Bayesian nonparametrics has been the problem of clustering, or partitioning, where each data point is modeled as being associated with one and only one of some collection of groups called clusters or partition blocks. Underlying these Bayesian nonparametric models are a set of interrelated stochastic processes, most notably the Dirichlet process and the Chinese restaurant process. In this paper we provide a formal development of an analogous problem, called feature modeling, for associating data points with arbitrary nonnegative integer numbers of groups, now called features or topics. We review the existing combinatorial stochastic process representations for the clustering problem and develop analogous representations for the feature modeling problem. These representations include the beta process and the Indian buffet process as well as new representations that provide insight into the connections between these processes. We thereby bring the same level of completeness to the treatment of Bayesian nonparametric feature modeling that has previously been achieved for Bayesian nonparametric clustering.

Reconstructing Macroeconomics, A Perspective from Statistical Physics and Combinatorial Stochastic Processes: Book Review

Professors Aoki and Yoshikawa adopted a variety of concepts from statistical physics and combinatorial stochastic processes to various problems in economics (such as labor markets, real growth, consumption, unemployment, financial market, productivity difference, etc.). In order to analyze these phenomena, they depart from the standard methods of model construction and analysis in mainstream economics and use methods that generally fall into two broad categories. One deals with stochastic dynamics, and the other with the formation of clusters and random combinatorial analysis. The authors build their new macroeconomics on the observation that the huge number of heterogeneous agents act stochastically different based on individual insights, tastes, goals, etc. In this approach the properties of economic systems are described by macroscopic dynamical equations of motion that are nonlinear partial differential equations, such as backward and forward Kolmogorov equations (known as a Chapman-Kolmogorov equation and Fokker-Planck equations).

Analysis of the “Hiring Above the Median” Selection Strategy for the Hiring Problem

This paper gives a precise mathematical analysis of the behaviour of “hiring above the median” strategies for a problem in the context of “on-line selection under uncertainty” that is known (at least in computer science related literature) as the “hiring problem”. Here a sequence of candidates is interviewed sequentially and based on the “score” of the current candidate an immediate decision whether to hire him or not has to be made. For “hiring above the median” selection rules, a new candidate will be hired if he has a score better than the median score of the already recruited candidates. Under the natural probabilistic model assuming that the ranks of the first n candidates are forming a random permutation, we show exact and asymptotic results for various quantities of interest to describe the dynamics of the hiring process and the quality of the hired staff. In particular we can characterize the limiting distribution of the number of hired candidates of a sequence of n candidates, which reveals the somewhat surprising effect, that slight changes in the selection rule, i.e., assuming the “lower” or the “upper” median as the threshold, have a strong influence on the asymptotic behaviour. Thus we could extend considerably previous analyses (Krieger et al., Ann. Appl. Probab., 17:360–385, 2007; Broder et al., Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1184–1193, ACM/SIAM, New York/Philadelphia, 2008 and Archibald and Martinez, Proceedings of the 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Mathematics and Theoretical Computer Science, pp. 63–76, 2009) of such selection rules. Furthermore, we discuss connections between the hiring process and the Chinese restaurant process introduced by Pitman (Combinatorial Stochastic Processes, Springer, Berlin, 2006).

A Statistical Mechanic View of Macro-dynamics in Economics

In this study, based on a statistical mechanic perspective, some technical insights for macroeconomic analysis are presented with regard to (1) "modeling the behaviour of a many-interacting-heterogeneous agents system through the analysis of fractions' stochastic dynamics over a state space" (Delli Gatti and Gallegati, Eterogeneita degli agenti economici ed interazione sociale: teorie e verifiche empiriche, 2005) and (2) suggesting a new microfoundation method based on jump Markov processes to mimic fluctuation dynamics of macro variables (Aoki, New approaches to macroeconomic modeling, 1996; Aoki, Modeling aggregate behaviour and fluctuations in economics, 2002; Aoki and Yoshikawa, Reconstructuring macroeconomics. A perspective from statistical physics and combinatorial stochastic processes, 2006). The main aim is to review and to describe how the statistical mechanic methodology provides systematic methods to solve master equation problems to be applied in macroscopic dynamics analysis in economics. The results are achieved in terms of a mean field system of equations to model average and volatility of a macroscopic quantity. By means of a macroscopic ordinary differential equation, the drift component is detached and, by means of a Fokker---Planck equation, the spread component around such a deterministic path is derived.

M. Aoki and H. Yoshikawa: Reconstructing macroeconomics, a perspective from statistical physics and combinatorial stochastic processes

M. Aoki and H. Yoshikawa: Reconstructing macroeconomics, a perspective from statistical physics and combinatorial stochastic processes

Combinatorial stochastic processes

Well-known results for sums of independent stochastic processes are extended to processes S = Σ i = 1 n φ iΠ( i) , where φ = ( φ i ) 1 ≤ i, j ≤ n is a collection of independent stochastic processes φ ij on some set T , and Π is a random permutation of {1, 2,…, n} such that Π, φ are independent. The general results, a uniform Law of Large Numbers and a functional Central Limit Theorem, are applied to permutation processes and randomized trials.