Online Bayesian learning for mixtures of spatial spline regressions with mixed effects

Abstract

Classification and clustering methods based on univariate functions have been well developed. Recent work has extended the techniques to the domain of bivariate functions by incorporating the techniques based on mixtures of spatial spline regression with mixed-effects models. An Expectation Maximization (EM) algorithm is implemented to facilitate model inference. In this paper, we further extend the mixtures of spatial spline regression with mixed-effects model under the Bayesian framework to accommodate streaming image data. First, we derive a Markov chain Monte Carlo (MCMC) algorithm as an alternative approach to the EM algorithm to make inference on the model. However, MCMC is not scalable to streaming image data since it requires all observed information to update the posterior distribution of the parameters. To tackle this issue, we propose a sequential Monte Carlo (SMC) algorithm to analyse online fashion image data. The existence of model sufficient statistics improves the efficiency of the proposed online SMC algorithm. Instead of saving all batch data for inference, we only require storage of the model sufficient statistics and every data point is only used once, which is well suited for large-scale stream type data. In addition, the proposed algorithm provides an unbiased estimator of the marginal likelihood as a by-product of the approach, which can be used for model selection. Numerical experiments are used to demonstrate the effectiveness of our method. Our implementation is available at https://github.com/ShufeiGe/Online-Bayesian-learning-for-MMSRm.