**1. Introduction**

Bayesian inference derives from Bayesian theory, which depicts the probability of occurrence of an event given some prior information. Due to the huge advances in computational and modeling techniques, Bayesian inference has been increasingly become an important tool for data analysis in the Bayesian framework and has widely been applied to various fields, including social science, engineering, philosophy, medicine, sport, law and psychology, for parameter/nonparameter estimation, hypothesis test, and prediction. Various Bayesian methods including Markov chain Monte Carlo, objective Bayesian method, subjective Bayesian method, approximate Bayesian computation, and variational Bayesian methods have been developed to make Bayesian inference on various problems such as large-scale image classification and cluster analysis of microarray, and models including parametric, nonparametric, semiparametric models, and other complicated models such as joint models of survival data and longitudinal data, graphical models, computer models, neural network models, and spatial econometric models. In particular, in the big data era, various Bayesian fronts including theories, methods, and computational algorithms have been developed for accommodating the applications of AI and data science in recent years [1], for example, the prior learning, Bayes factor evaluation, Bayesian variable selection, robust Bayesian inference, variational Bayesian inference, resampling, approximation of posterior distribution, approximate Bayesian computation, and debias methods for high/ultrahigh-dimensional data, multisource heterogeneous data, imbalanced data, missing data, and data stream. But there are some challenging problems, for example, how to balance the computational times and statistical efficiency, design efficient Bayesian computational algorithm and robust sampling schemes for big/massive data, distributed data and streaming data in the privacy protection and the defense of malicious attacks framework, and modeling so that they can adapt to the development of AI and the requirement of data mining, to be addressed and solved for Bayesian inference. In what follows, we will introduce the recent development and some topics of interest on Bayesian inference.
