Characterizing the Function Space for Bayesian Kernel Models

Natesh S. Pillai, Qiang Wu, Feng LiangSayan Mukherjee and Robert L. Wolpert

JMLR 2007

Presented by: Mingyuan ZhouDuke UniversityJanuary 20, 2012

Outline

• Reproducing kernel Hilbert space (RKHS)• Bayesian kernel model

– Gaussian processes– Levy processes

• Gamma process• Dirichlet process• Stable process

– Computational and modeling considerations• Posterior inference• Discussion

RKHS

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels.

http://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space

A finite kernel based solution

The direct adoption of the finite representation is not a fully Bayesian model since it depends on the (arbitrary) training data sample size . In addition, this prior distribution is supported on a finite-dimensional subspace of the RKHS. Our coherent fully Bayesian approach requires the specification of a prior distribution over the entire space H.

Mercer kernel

Bayesian kernel model

Properties of the RKHS

Properties of the RKHS

Bayesian kernel models and integral operators

Two concrete examples

Two concrete examples

Bayesian kernel models

Gaussian processes

Levy processes

Levy processes

Poisson random fields

Poisson random fields

Dirichlet Process

Symmetric alpha-stable processes

Symmetric alpha-stable processes

Computational and modeling considerations

• Finite approximation for Gaussian processes

• Discretization for pure jump processes

Posterior inference

• Levy process model

– Transition probability proposal– The MCMC algorithm

Classification of gene expression data

Classification of gene expression data

Discussion• This paper formulates a coherent Bayesian perspective for

regression using a RHKS model.• The paper stated an equivalence under certain conditions of

the function class G and the RKHS induced by the kernel. This implies: – (a) a theoretical foundation for the use of Gaussian processes, Dirichlet

processes, and other jump processes for non-parametric Bayesian kernel models.

– (b) an equivalence between regularization approaches and the Bayesian kernel approach.

– (c) an illustration of why placing a prior on the distribution is natural approach in Bayesian non-parametric modelling.

• A better understanding of this interface may lead to a better understanding of the following research problems:– Posterior consistency– Priors on function spaces– Comparison of process priors for modeling– Numerical stability and robust estimation