Hilbert space embeddings of conditional distributions with applications to dynamical systems

ABSTRACT

In this paper, we extend the Hilbert space embedding approach to handle conditional distributions. We derive a kernel estimate for the conditional embedding, and show its connection to ordinary embeddings. Conditional embeddings largely extend our ability to manipulate distributions in Hilbert spaces, and as an example, we derive a nonparametric method for modeling dynamical systems where the belief state of the system is maintained as a conditional embedding. Our method is very general in terms of both the domains and the types of distributions that it can handle, and we demonstrate the effectiveness of our method in various dynamical systems. We expect that conditional embeddings will have wider applications beyond modeling dynamical systems.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.6 Learning

Additional Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Nonparametric statistics; Distribution functions

  I. Computing Methodologies
  I.6 SIMULATION AND MODELING
      I.6.4 Model Validation and Analysis
      I.6.5 Model Development
          Subjects: Modeling methodologies

General Terms:
Algorithms, Design, Performance, Theory