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Construction of Dependent Dirichlet Processes based on Poisson Processes

Construction of Dependent Dirichlet Processes based on Poisson Processes. Dahua Lin, Eric Grimson , John Fisher Presented by Yingjian Wang Jun. 03, 2011. Outline. Measure space; Stochastic processes; Three operations on measure; Build Markov chain of Dirichlet processes;

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Construction of Dependent Dirichlet Processes based on Poisson Processes

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  1. Construction of Dependent Dirichlet Processesbased on Poisson Processes Dahua Lin, Eric Grimson, John Fisher Presented by Yingjian Wang Jun. 03, 2011

  2. Outline • Measure space; • Stochastic processes; • Three operations on measure; • Build Markov chain of Dirichlet processes; • The sampling algorithm; • Experiments;

  3. Measure space • The triple (X, Σ, μ) is called a measure space, with Σ is the σ-field of X; μ is a measure. • Σ is closed for complementation and countable unions of subsets of X (measurable). • If μ(X)=1, it is a probability space. • For a probability space (X, Σ, μ), X is the ‘sample space’; Σ is the ‘event space’. • Non-measurable set is a non-trivial result of the axiom of choice.

  4. Stochastic processes • Well, we are interested in stochastic processes (maybe for nonparametric Bayesian methods), why you talk the measure space? • Stochastic processes are defined/live in measure spaces. • Gamma process G on (X, Σ, μ): • Dirichlet process G on (X, Σ, μ):

  5. Operations on measure • Superposition (innovation): • Subsampling (removal): • Transition (move):

  6. Building Markov chain of DPs • Markov chain of base measures: • DPs with Markov base measures:

  7. The sampling algorithm • Previous phase DP: • Sample the next phase DP - D’:

  8. Experiments • Synthetic data: Gaussian mixtures with birth & death of components.

  9. Experiments • People flows at New York Grand Central Station: observation is location-velocity pair. • Infers a much smaller 20 flows with the average likelihood -3.34, compared with D-FMM’s -3.34 of 50 flows.

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