Our current understanding of galaxy formation is still at a stage a wide range of physical processes may play a role but exactly how they operate to shape the observed galaxy population is not clear. The problem of galaxy formation is therefore best tackled with a Bayesian-inference based approach, which allows one to constrain theory with data in a statistically rigorous way. The semi-analytic model (SAM) of galaxy formation, which uses multi-parameter parameterizations to describe the underlying physical processes of galaxy formation, provides an important avenue to develop a Bayesian-inference based SAM of galaxy formation.We develop a generalized SAM and a scheme that incorporates such a SAM into the framework of Bayesian inference.With an advanced Markov-Chain Monte-Carlo algorithm and current parallel computations, it is now possible to rigorously sample the posterior distribution of a large number of model parameters. We find that the posterior probability distribution is in general very complex, showing strong degeneracy among some important parameters.Because of the strong covariance of model parameters, some conclusions obtained from early SAMs are not reliable.Our Bayesian inferences from from a number of observational data sets indicate that some new physics is still required in the current model of galaxy formation.