Linear and Gaussian models with regime switching are popular in signal processing. In this letter, we revisit Bayesian inference in such models under the variational Bayesian framework. We propose a structured but implicit variational distribution which can be seen as the posterior distribution of an alternative statistical model in which the first and second order moments of the filtering distribution can be computed exactly. A critical property of this alternative model is that the Kullback-Leibler Divergence between its associated posterior distribution and that of linear and Gaussian models with regime switching can also be computed exactly, at a cost linear in the number of observations; hence, we propose a parameter estimation method of our variational model but also of the original one from a sequence of observations. Once the parameters are estimated, our inference method is efficient as compared to the Rao-Blackwell Particle Filter.