Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems

By C. Antoniak. 1974

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A Neural Dirichlet Process Mixture Model for Task-Free Continual Learning (2020, methodology): “We briefly review the Dirichlet process mixture (DPM) model (Antoniak, 1974; Ferguson, 1983), and a variational method to approximate the posterior of DPM models in an online setting: Sequential Variational Approximation (SVA) (Lin, 2013).”
Variational inference for Dirichlet process mixtures (2006, methodology): “In the Dirichlet process mixture model, the DP is used as a nonparametric prior in a hierarchical Bayesian specification (Antoniak 1974): G | {α, G 0 } ∼ DP(α, G 0 ) η n | G ∼ G X n | η n ∼ p(x n | η n ).”
Search for patterns of functional specificity in the brain: A nonparametric hierarchical Bayesian model for group fMRI data (2011, methodology): “To overcome this challenge, we note that for 0 ≤ r ≤ n, we have are unsigned Stirling numbers of the first kind (Antoniak, 1974).”
Bayesian Semiparametric Regression for Median Residual Life (2003, unknown): “In particular, Antoniak (1974) noted that this Bayesian model can be marginalized over G; a result that forms the basis of several MCMC algorithms (Escobar, 1994; West et al., 1994; Escobar & West, 1995; Bush & MacEachern, 1996; MacEachern & Müller, 1998; Neal, 2000) which can be implemented to…”
A Bayesian Nonparametric Approach to Inference for Quantile Regression (2010, methodology): “Based on the work of Antoniak (1974), the full posterior can be expressed as p(G, ,α,ψ | data) = p(G | ,α,ψ)p( ,α,ψ | data).”
Semi-Supervised Deep Learning Using Pseudo Labels for Hyperspectral Image Classification (2018, methodology): “The DPMM [33]–[35] is a non-parametric model for clustering, which has the ability to infer the number of clusters from the data.”
Variational Russian Roulette for Deep Bayesian Nonparametrics (2019, methodology): “Within machine learning, Bayesian nonparametric methods have been applied within models as diverse as clustering (Antoniak, 1974; Görür & Rasmussen, 2010; Teh et al., 2005), topic modeling (Teh et al., 2006), and infinite deep neural networks (Adams et al., 2010).”
Mixture Modeling for Marked Poisson Processes (2010, methodology): “A.2 Posterior Simulation Using results from Antoniak (1974), the posterior distribution for the DP mixture model is partitioned as p(G,θ?, s, α, ψ | data) = p(G | θ?, s, α, ψ)p(θ?, s, α, ψ | data), where G, given θ?, s, α, ψ, is distributed as a DP with precision parameter α + N and base…”
Correcting the mathematical structure of a hydrological model via Bayesian data assimilation (2011, unknown): “…js1:Tð Þ / P j¼1 T f sij ið Þ aG0 ið Þ þ P j<i i jð Þ aþ i 1 ; ðA3Þ where f (si∣ i) is the likelihood component for i derived from the density of the Gaussian distribution of si∣ i and a and G0 characterize the Dirichlet process [Antoniak, 1974] used to specify a prior for the mixture parameters.”
Discovering linguistic structures in speech: models and applications (2014, methodology): “More specifically, we formulate a Dirichlet Process mixture model [3, 42, 120, 35] where each mixture is a Hidden Markov Model (DPHMM) that is used to model an acoustic unit and to generate observed segments of that unit.”
Nonparametric Bayesian Multiple Imputation for Incomplete Categorical Variables in Large-Scale Assessment Surveys (2013, methodology): “We note that Dirichlet process mixture models (Antoniak, 1974) are used in many fields including, for example, econometrics (Chib & Hamilton, 2002; Hirano, 2002), social science (Kyung, Gill, & Casella, 2010), and finance (Rodrı́guez & Dunson, 2011).”
Mixtures of products of Dirichlet processes for variable selection in survival analysis (2003, methodology): “This can be done following the procedure illustrated in Antoniak (1973); see also Petrone and Raftery (1997). Suppose that, among the ni observations in the ith partition group, the number of distinct observations is equal to ri, organized in nondecreasing order, as follows: x∗ i(1); x ∗ i(2); : : :”
Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages (2005, methodology): “One may note, for instance, the nontrivial arguments used by Antoniak [ 3 ] to derive (8).”
Bayesian mixture modeling for spatial Poisson process intensities, with applications to extreme value analysis (2007, background): “For example, a useful approximation (for moderately large n) to the marginal prior mean for n∗ is given by E(n∗) ≈ (aα/bα) log{1+(nbα/aα)} (see, e.g., Antoniak, 1974; Escobar and West, 1995; Liu, 1996).”
Introduction to the Dirichlet Distribution and Related Processes (2010, background): “When α = [1, 1, 1], the Dirichlet distribution reduces to the uniform distribution over the simplex (as a quick exercise, check this using the density of the Dirichlet in (1).”
A Latent Dirichlet Model for Unsupervised Entity Resolution (2005, methodology): “2 Relation to the Dirichlet Process The Dirichlet process was introduced by Ferguson [14] and Antoniak [2] as a non-parametric statistical approach that allows the complexity of the model to grow with increasing size of the data.”