Statistics, probability, and game theory : papers in honor of David Blackwell
01 Dec 1999-Journal of the American Statistical Association (Institute of Mathematical Statistics)-Vol. 94, Iss: 448, pp 1385
About: This article is published in Journal of the American Statistical Association.The article was published on 1999-12-01. It has received 48 citations till now. The article focuses on the topics: Game theory & Honor.
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TL;DR: In this paper, it was shown that the complexity of the payoff function for the Blackwell game is approximately the same as that of the perfect information game with Borel measurable payoff functions.
Abstract: Games of infinite length and perfect information have been studied for many years. There are numerous determinacy results for these games, and there is a wide body of work on consequences of their determinacy. Except for games with very special payoff functions, games of infinite length and imperfect information have been little studied. In 1969, David Blackwell [1] introduced a class of such games and proved a determinacy theorem for a subclass. During the intervening time, there has not been much progress in proving the determinacy of Blackwell's games. Orkin [17] extended Blackwell's result to a slightly wider class. Blackwell [2] found a new proof of his own result. Maitra and Sudderth [9, 10] improved Blackwell's result in a different direction from that of Orkin and also generalized to the case of stochastic games. Recently Vervoort [18] has obtained a substantial improvement. Nevertheless, almost all the basic questions have remained open. In this paper we associate with each Blackwell game a family of perfect information games, and we show that the (mixed strategy) determinacy of the former follows from the (pure strategy) determinacy of the latter. The complexity of the payoff function for the Blackwell game is approximately the same as the complexity of the payoff sets for the perfect information games. In particular, this means that the determinacy of Blackwell games with Borel measurable payoff functions follows from the known determinacy of perfect information games with Borel payoff sets.
298 citations
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TL;DR: In this paper, the authors presented 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.
Abstract: 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 two-parameter family of Poisson-Dirichlet 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.
216 citations
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TL;DR: Inference under models with nonparametric Bayesian (BNP) priors is reviewed for density estimation, clustering, regression and for mixed effects models with random effects distributions.
Abstract: We review inference under models with nonparametric Bayesian (BNP) priors The discussion follows a set of examples for some common inference problems The examples are chosen to highlight problems that are challenging for standard parametric inference We discuss inference for density estimation, clustering, regression and for mixed effects models with random effects distributions While we focus on arguing for the need for the flexibility of BNP models, we also review some of the more commonly used BNP models, thus hopefully answering a bit of both questions, why and how to use BNP
158 citations
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TL;DR: A Bayesian framework for modeling individual differences, in which subjects are assumed to belong to one of a potentially infinite number of groups, is introduced, allowing us to learn flexible parameter distributions without overfitting the data, or requiring the complex computations typically required for determining the dimensionality of a model.
147 citations
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TL;DR: A model-based clustering algorithm that exploits available covariates is developed that is suitable for any combination of continuous, categorical, count, and ordinal covariates and formalizes Posterior predictive inference in this model.
Abstract: We propose a probability model for random partitions in the presence of covariates. In other words, we develop a model-based clustering algorithm that exploits available covariates. The motivating application is predicting time to progression for patients in a breast cancer trial. We proceed by reporting a weighted average of the responses of clusters of earlier patients. The weights should be determined by the similarity of the new patient's covariate with the covariates of patients in each cluster. We achieve the desired inference by defining a random partition model that includes a regression on covariates. Patients with similar covariates are a priori more likely to be clustered together. Posterior predictive inference in this model formalizes the desired prediction.We build on product partition models (PPM). We define an extension of the PPM to include a regression on covariates by including in the cohesion function a new factor that increases the probability of experimental units with similar covariates to be included in the same cluster. We discuss implementations suitable for any combination of continuous, categorical, count, and ordinal covariates.An implementation of the proposed model as R-package is available for download.
140 citations
Cites background from "Statistics, probability, and game t..."
...Alternatively, the species sampling model (SSM) (Pitman 1996; Ishwaran and James 2003) defines an exchangeable probability model p(ρn) that depends on ρn only indirectly through the cardinality of the partitioning subsets, p(ρn) = p(|S1|, . . . , |Skn |)....
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...The popular Dirichlet process (DP) model (Ferguson 1973) is a special case of a SSM....
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...The SSM can be alternatively characterized by a sequence of predictive probability functions (PPFs) that describe how individuals are sequentially assigned to either already formed clusters or to start new ones....
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