Bayesian inverse regression for supervised dimension reduction with small datasets

ABSTRACT

We consider supervised dimension reduction problems, namely to identify a low dimensional projection of the predictors $x$ which can retain the statistical relationship between $x$ and the response variable y. We follow the idea of the sliced inverse regression (SIR) and the sliced average variance estimation (SAVE) type of methods, which is to use the statistical information of the conditional distribution $\pi (x|y)$ to identify the dimension reduction (DR) space. In particular we focus on the task of computing this conditional distribution without slicing the data. We propose a Bayesian framework to compute the conditional distribution where the likelihood function is constructed using the Gaussian process regression model. The conditional distribution $\pi (x|y)$ can then be computed directly via Monte Carlo sampling. We then can perform DR by considering certain moment functions (e.g. the first or the second moment) of the samples of the posterior distribution. With numerical examples, we demonstrate that the proposed method is especially effective for small data problems.

Keywords:

• Dimension reduction
• Gaussian process
• Monte Carlo simulation
• sliced inverse regression
• supervised learning

Acknowledgments

XC and JL were partially supported by the National Natural Science Foundation of China under grant number 11301337

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

XC and JL were partially supported by the National Natural Science Foundation of China [grant number 11301337].