This paper presents the parameter identification in a Bayesian setting for the elastoplastic problem, mathematically speaking the variational inequality of a second kind. The inverse problem is formulated in a probabilistic manner in which unknown quantities are embedded in a form of the probability distributions reflecting their uncertainty. With the help of the stochastic functional analysis the update procedure is introduced as a direct, purely algebraic way of computing the posterior, which is comparatively inexpensive to evaluate. Such formulation involves the process of solving the convex minimisation problem in a stochastic setting for which the extension of classical optimization algorithm in predictor-corrector form as the solution procedure is proposed. A validation study of identification procedure is done through a series of virtual experiments taking into account the influence of the measurement error and the order of approximation on the posterior estimate.