On the Convergence of Dirichlet Processes

Zipf’s, Heaps’ and Taylor’s laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the scaling of fluctuations for systems as diverse as written texts and cities, ecological systems and stock markets. Many modeling schemes have been proposed in literature to explain those laws, but only recently a modeling framework has been introduced that accounts for the emergence of those laws without deducing the emergence of one of the laws from the others or without ad hoc assumptions. This modeling framework is based on the concept of adjacent possible space and its key feature of being dynamically restructured while its boundaries get explored, i.e., conditional to the occurrence of novel events. Here, we illustrate this approach and show how this simple modeling framework, instantiated through a modified Pólya’s urn model, is able to reproduce Zipf’s, Heaps’ and Taylor’s laws within a unique self-consistent scheme. In addition, the same modeling scheme embraces other less common evolutionary laws (Hoppe’s model and Dirichlet processes) as particular cases.

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Strong Markov Local Dirichlet Processes and Stochastic Differential Equations

Theory of Probability and Its Applications ◽

10.1137/s0040585x97976829 ◽

1999 ◽

Vol 43 (2) ◽

pp. 189-202

Author(s):

H. J. Engelbert ◽

J. Wolf

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Dirichlet Processes ◽

Strong Markov

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Computations of Mixtures of Dirichlet Processes

SIAM Journal on Scientific and Statistical Computing ◽

10.1137/0907004 ◽

1986 ◽

Vol 7 (1) ◽

pp. 60-71 ◽

Cited By ~ 41

Author(s):

Lynn Kuo

Keyword(s):

Dirichlet Processes

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Kernel Analysis Based on Dirichlet Processes Mixture Models

Entropy ◽

10.3390/e21090857 ◽

2019 ◽

Vol 21 (9) ◽

pp. 857

Author(s):

Jinkai Tian ◽

Peifeng Yan ◽

Da Huang

Keyword(s):

Mixture Models ◽

Crucial Role ◽

Gaussian Mixture Models ◽

Gaussian Process Regression ◽

Gaussian Mixture ◽

Spectral Domain ◽

Dirichlet Processes ◽

Number Of Components ◽

Kernel Analysis ◽

Nonparametric Bayesian Models

Kernels play a crucial role in Gaussian process regression. Analyzing kernels from their spectral domain has attracted extensive attention in recent years. Gaussian mixture models (GMM) are used to model the spectrum of kernels. However, the number of components in a GMM is fixed. Thus, this model suffers from overfitting or underfitting. In this paper, we try to combine the spectral domain of kernels with nonparametric Bayesian models. Dirichlet processes mixture models are used to resolve this problem by changing the number of components according to the data size. Multiple experiments have been conducted on this model and it shows competitive performance.

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