Bayesian Analysis

Laplace Approximation for Logistic Gaussian Process Density Estimation and Regression

Jaakko Riihimäki and Aki Vehtari

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Abstract

Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximate Bayesian inference for LGP density estimation in a grid using Laplace’s method to integrate over the non-Gaussian posterior distribution of latent function values and to determine the covariance function parameters with type-II maximum a posteriori (MAP) estimation. We demonstrate that Laplace’s method with MAP is sufficiently fast for practical interactive visualisation of 1D and 2D densities. Our experiments with simulated and real 1D data sets show that the estimation accuracy is close to a Markov chain Monte Carlo approximation and state-of-the-art hierarchical infinite Gaussian mixture models. We also construct a reduced-rank approximation to speed up the computations for dense 2D grids, and demonstrate density regression with the proposed Laplace approach.

Article information

Source
Bayesian Anal. Volume 9, Number 2 (2014), 425-448.

Dates
First available in Project Euclid: 26 May 2014

Permanent link to this document
http://projecteuclid.org/euclid.ba/1401148315

Digital Object Identifier
doi:10.1214/14-BA872

Mathematical Reviews number (MathSciNet)
MR3217002

Zentralblatt MATH identifier
1327.62248

Keywords
Gaussian process logistic transformation density estimation density regression approximate inference Laplace’s method

Citation

Riihimäki, Jaakko; Vehtari, Aki. Laplace Approximation for Logistic Gaussian Process Density Estimation and Regression. Bayesian Anal. 9 (2014), no. 2, 425--448. doi:10.1214/14-BA872. http://projecteuclid.org/euclid.ba/1401148315.


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