  Fog manufacturing can greatly enhance traditional manufacturing systems
through distributed Fog computation units, which are governed by predictive
computational workload offloading methods under different Industrial Internet
architectures. It is known that the predictive offloading methods highly depend
on accurate prediction and uncertainty quantification of runtime performance
metrics, containing multivariate mixed-type responses (i.e., continuous,
counting, binary). In this work, we propose a Bayesian sparse regression for
multivariate mixed responses to enhance the prediction of runtime performance
metrics and to enable the statistical inferences. The proposed method considers
both group and individual variable selection to jointly model the mixed types
of runtime performance metrics. The conditional dependency among multiple
responses is described by a graphical model using the precision matrix, where a
spike-and-slab prior is used to enable the sparse estimation of the graph. The
proposed method not only achieves accurate prediction, but also makes the
predictive model more interpretable with statistical inferences on model
parameters and prediction in the Fog manufacturing. A simulation study and a
real case example in a Fog manufacturing are conducted to demonstrate the
merits of the proposed model.
