The MAPK/ERK pathway is a major signal transduction system which regulates many fundamental cellular processes including the growth control and the cell death. As a result of these roles, it has a crucial importance in cancer as well as normal developmental processes. Therefore, it has been intensively studied resulting in a wealth of knowledge about its activation. It is also well documented that the activation kinetics of the pathway is crucial to determine the nature of the biological response. However, while individual biochemical steps are well characterized, it is still difficult to predict or even understand how the activation kinetics works. The aim of this paper is to estimate the stochastic rate constants of the MAPK/ERK network dynamics. Accordingly, taking a Bayesian approach, we combined underlying qualitative biological knowledge in several competing dynamic models via sets of quasireactions and estimated the stochastic rate constants of these reactions. Comparing the resulting estimates via the BIC and DIC criteria, we chose a biological model which includes EGFR degradation-Raf-MEK-ERK cascade without the involvement of RKIPs.