Variational Bayesian inference for interval regression with an asymmetric Laplace distribution

Abstract

Abstract This paper proposes a Bayesian nonparametric interval regression model assuming the noise on the lower and upper bounds of interval data follows an asymmetric Laplace distribution. In order to address various uncertainties in real applications and make model training more convenient and efficient, the asymmetric Laplace distribution is represented as a scale mixture of Gaussian distribution, which is amenable to variational inference (VI for short) (Blei et al., 2016). Inspired by multi-task learning, the marginal quantile regression functions for lower and upper bounds relate each other by the correlation parameter of the coefficients in infinite feature space. This model can give point estimate of interval bounds or predicted occurrence probability for some events, which provides new potential for the problems which cannot be solved by previous interval regression models. In addition, the probability mass waste problem is analyzed and alleviated by a local correction strategy. Moreover, we modify the model to be fit for the interval data represented as center and radius. We verify above analysis results by numerical experiment on artificial and real dataset. Finally, the approximation ability of this model and that of the other existing methods are compared by applications on public data sets.