This paper derives two new optimization-driven Monte Carlo algorithms inspired from variable splitting and data augmentation. In particular, the formulation of one of the proposed approaches is closely related to the alternating direction method of multipliers (ADMM) main steps. The proposed framework enables to derive faster and more efficient sampling schemes than the current state-of-the-art methods and can embed the latter. By sampling efficiently the parameter to infer as well as the hyperparameters of the problem, the generated samples can be used to approximate Bayesian estimators of the parameters to infer. Additionally, the proposed approach brings confidence intervals at a low cost contrary to optimization methods. Simulations on two often-studied signal processing problems illustrate the performance of the two proposed samplers. All results are compared to those obtained by recent state-of-the-art optimization and MCMC algorithms used to solve these problems. Index Terms Bayesian inference, data augmentation, high-dimensional problems, Markov chain Monte Carlo, variable splitting.