Very recently, a new age and state-dependent degradation process, named the Transformed Wiener (TW) process, has been proposed to describe degradation phenomena when the degradation growth of the units under study is not necessarily monotonically increasing and depends stochastically on the current degradation level. This paper suggests a Bayesian estimation approach for such a process, based on informative priors of its parameters, which allows one to incorporate into the estimation procedure the prior information on meaningful physical characteristics of the observed degradation process that is generally available to the analyst. Several different prior distributions are proposed, reflecting different degrees of knowledge on the observed phenomenon. A Monte Carlo Markov Chain technique is adopted for estimating the TW process parameters and some functions thereof. Finally, in order to show the feasibility of the proposed Bayesian estimation procedure and the flexibility of the TW process an example of application is developed.