• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.823-855
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• 1996

We study Dirichlet forms and the associated diffusion processes for the Gibbs measures related to the quantum unbounded spin systems (lattice boson systems) interacting via superstable and regular potentials. This work is a continuation of the author's previous study on the classical systems [LPY] to the quantum cases. In [LPY], we constructed Dirichlet forms and the associated diffusion processes for the Gibbs measures of classical unbounded spin systems. Furthermore, we also showed the essential self-adjointness of the Dirichlet operator and the log-Sobolev inequality for any Gibbs measure under appropriate conditions on the potentials. In this atudy we try to extend the results of the classical systems to the quantum cases. Because of some technical difficulties, we are only able to construct a Dirichlet form and the associated diffusion process for any Gibbs measure of the quantum systems. We utilize the general scheme of the previous work on the theory in infinite dimensional spaces [AH-K1-2, AKR, AR1-2, Kus, MR, Ro, Sch] and the ideas we employed in our study of the calssical systems ]LPY].

• Journal of the Korean Data and Information Science Society
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• v.15 no.3
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• pp.575-584
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• 2004

We propose a Bayesian model-based approach using a mixture of Dirichlet processes model with discrete wavelet transform, for curve clustering in the microarray data with time-course gene expressions.

• Journal of the Korean Statistical Society
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• v.30 no.1
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• pp.127-138
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• 2001

In order to combine parametric and nonparametric approaches together for survival analysis with censored observations, a new class of priors called mixtures of the beta processes is introduced. It is shown that mixtures of beta processes priors generalized the well known priors - mixtures of Dirichlet processes, and they are conjugate with right censored observations. Formulas for computing the posterior distribution are derived. Finally, a real data set is analyzed for illustrational purpose.

• 한국데이터정보과학회:학술대회논문집
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• 2006.04a
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• pp.39-42
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• 2006

We propose a Bayesian model-based approach using a mixture of Dirichlet processes model with discrete wavelet transform, for curve clustering in the microarray data with time-course gene expressions.

• Proceedings of the Korean Statistical Society Conference
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• 2003.10a
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• pp.39-45
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• 2003

Ghosh and Ramamoorthi (1996) studied the posterior consistency for survival models and showed that the posterior was consistent, when the prior on the distribution of survival times was the Dirichlet process prior. In this paper, we study the posterior consistency of survival models with neutral to the right process priors which include Dirichlet process priors. A set of sufficient conditions for the posterior consistency with neutral to the right process priors are given. Interestingly, not all the neutral to the right process priors have consistent posteriors, but most of the popular priors such as Dirichlet processes, beta processes and gamma processes have consistent posteriors. For extended beta processes, a necessary and sufficient condition for the consistency is also established.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.731-770
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• 1997

We study Diriclet forms and related subjects for the Gibbs measures of classical unbounded sping systems interacting via potentials which are superstable and regular. For any Gibbs measure $\mu$, we construct a Dirichlet form and the associated diffusion process on $L^2(\Omega, d\mu), where \Omega = (R^d)^Z^\nu$. Under appropriate conditions on the potential we show that the Dirichlet operator associated to a Gibbs measure $\mu$ is essentially self-adjoint on the space of smooth bounded cylinder functions. Under the condition of uniform log-concavity, the Gibbs measure exists uniquely and there exists a mass gap in the lower end of the spectrum of the Dirichlet operator. We also show that under the condition of uniform log-concavity, the unique Gibbs measure satisfies the log-Sobolev inequality. We utilize the general scheme of the previous works on the theory in infinite dimensional spaces developed by e.g., Albeverio, Antonjuk, Hoegh-Krohn, Kondratiev, Rockner, and Kusuoka, etc, and also use the equilibrium condition and the regularity of Gibbs measures extensively.

• The Pure and Applied Mathematics
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• v.5 no.2
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• pp.123-132
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• 1998

Let F and G denote the distribution functions of the failure times and the censoring variables in a random censorship model. Susarla and Van Ryzin(1978) verified consistency of $F_{\alpha}$, he NPBE of F with respect to the Dirichlet process prior D($\alpha$), in which they assumed F and G are continuous. Assuming that A, the cumulative hazard function, is distributed according to a beta process with parameters c, $\alpha$, Hjort(1990) obtained the Bayes estimator $A_{c,\alpha}$ of A under a squared error loss function. By the theory of product-integral developed by Gill and Johansen(1990), the Bayes estimator $F_{c,\alpha}$ is recovered from $A_{c,\alpha}$. Continuity assumption on F and G is removed in our proof of the consistency of $A_{c,\alpha}$ and $F_{c,\alpha}$. Our result extends Susarla and Van Ryzin(1978) since a particular transform of a beta process is a Dirichlet process and the class of beta processes forms a much larger class than the class of Dirichlet processes.

• Journal of the Korean Data and Information Science Society
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• v.18 no.2
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• pp.553-560
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• 2007

In this paper, we consider two components system which lifetimes have Freund's bivariate exponential model with equal failure rates. We propose Bayesian multiple comparisons procedure for the failure rates of I Freund's bivariate exponential populations based on Dirichlet process priors(DPP). The family of DPP is applied in the form of baseline prior and likelihood combination to provide the comparisons. Computation of the posterior probabilities of all possible hypotheses are carried out through Markov Chain Monte Carlo(MCMC) method, namely, Gibbs sampling, due to the intractability of analytic evaluation. The whole process of multiple comparisons problem for the failure rates of bivariate exponential populations is illustrated through a numerical example.

• 한국데이터정보과학회:학술대회논문집
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• 2006.11a
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• pp.71-80
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• 2006

A nonparametric Bayesian multiple comparisons problem (MCP) for dependence parameters in I bivariate exponential populations is studied here. A simple method for pairwise comparisons of these parameters is also suggested. Here we extend the methodology studied by Gopalan and Berry (1998) using Dirichlet process priors. The family of Dirichlet process priors is applied in the form of baseline prior and likelihood combination to provide the comparisons. Computation of the posterior probabilities of all possible hypotheses are carried out through Markov Chain Monte Carlo method, namely, Gibbs sampling, due to the intractability of analytic evaluation. The whole process of MCP for the dependent parameters of bivariate exponential populations is illustrated through a numerical example.

• Communications for Statistical Applications and Methods
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• v.23 no.6
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• pp.445-466
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• 2016

Nonparametric Bayesian methods have seen rapid and sustained growth over the past 25 years. We present a gentle introduction to the methods, motivating the methods through the twin perspectives of consistency and false consistency. We then step through the various constructions of the Dirichlet process, outline a number of the basic properties of this process and move on to the mixture of Dirichlet processes model, including a quick discussion of the computational methods used to fit the model. We touch on the main philosophies for nonparametric Bayesian data analysis and then reanalyze a famous data set. The reanalysis illustrates the concept of admissibility through a novel perturbation of the problem and data, showing the benefit of shrinkage estimation and the much greater benefit of nonparametric Bayesian modelling. We conclude with a too-brief survey of fancier nonparametric Bayesian methods.