To ensure process safety and product quality, multivariate statistical process monitoring (MSPM) has been widely used in industry for decades. Among the steps of MSPM, fault isolation is an important link between the fault detection and the root-cause diagnosis, which identifies the variables closely related to the detected process abnormality. The existing methods for fault isolation often suffer from the smearing effect or rely on the impractical requirement of a sufficient amount of historical fault data. To solve these problems, a reconstruction method based on the Bayesian Lasso (short for least absolute shrinkage and selection operator) is proposed in this work, which transforms the problem of statistical fault isolation to the variable selection in regression analysis and solves it in a Bayesian framework. If the posterior distribution of a Lasso coefficient changes significantly before and after the occurrence of the fault, the corresponding process variable has a large probability to be faulty. Furthermore, the Bayesian framework permits the tracking of fault propagation, thereby facilitating the subsequent root-cause diagnosis. The feasibility of the proposed method is illustrated by case studies.