The Bayesian model selection and parameter estimation framework developed by Khalil et al. (J Sound Vib 332(15):3670–3691, 2013, Bayesian inference for complex and large-scale engineering systems. Ph.D. thesis, Carleton University, 2013) and Sandhu et al. (J Comput Methods Appl Mech Eng 282:161–183, 2014, Bayesian model selection and parameter estimation for a nonlinear fluid–structure interaction problem. Master’s thesis, Carleton University, 2012, 54th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, 2013) is extended to handle strongly nonlinear systems. The evidence required to estimate the posterior probability of each proposed model is computed using the Chib–Jeliazkov method. The posterior parameter samples generated using Metropolis–Hastings (M–H) Markov Chain Monte Carlo (MCMC) simulations are required by this method. The M–H MCMC-based parameter estimation procedure is complemented by an efficient particle filter and an ensemble Kalman filter for the strongly non-Gaussian state estimation problem. A strongly nonlinear system having multiple fixed (equilibrium) points is analyzed to demonstrate the efficacy of the algorithm.