Fault diagnosis is an effective way to address the dimensional quality challenges by identifying failing process characteristics
in multistage manufacturing processes. A significant number of literature is found addressing the problem with an implicit  assumption of number of measurement points exceeds the number of process error sources. In reality due to economical constraints this assumption is frequently violated, resulting into an ill-conditioned process that provides less than full column rank models to describe the relationship between the product quality and process error sources. Under such circumstances, no mathematically unique solution do exist which is a hindrance in accurately diagnosing the faults arising. In this study, the fault diagnosis problem in such processes is presented as a variable selection problem and assuming sparsity of faults, an amalgamation of state space model and Bayesian Hierarchical Model for LASSO based regularized regression is adopted. To estimate the failing process elements through the Bayesian LASSO problem the popular stochastic simulation methods, viz., Gibbs Sampling is employed. A floor pan assembly problem is studied to demonstrate the applicability of the proposed method. Results obtained are promising and effective in successfully identifying process faults.