Hybrid MSV-MGARCH models, in particular the MSF-SBEKK specification, proved useful in multivariate modelling of returns on financial and commodity markets. The initial MSF-MGARCH structure, called LN-MSF-MGARCH here, is obtained by multiplying the MGARCH conditional covariance matrix H_t by a scalar random variable g_t such that{ln g_t, t ∈ Z} is a Gaussian AR(1) latent process with auto-regression parameter φ. Here we alsoconsider an IG-MSF-MGARCH specification, which is a hybrid generalisation of conditionally Student t MGARCH models, since the latent process {g_t} is no longer marginally log-normal (LN), but for φ = 0 it leads to an inverted gamma (IG) distribution for g_t and to the t-MGARCH case. If φ =/ 0, the latent variables g_t are dependent, so (in comparison to the t-MGARCH specification) we get an additional source of dependence and one more parameter. Due to the existence of latent processes, the Bayesian approach, equipped with MCMC simulation techniques, is a natural and feasible statistical tool to deal with MSF-MGARCH models. In this paper we show how the distributional assumptions for the latent process together with the specification of the prior density for its parameters affect posterior results, in particular the ones related to adequacy of thet-MGARCH model. Our empirical findings demonstrate sensitivity of inference on the latent process and its parameters, but, fortunately, neither on volatility of the returns nor on their conditional correlation. The new IG-MSF-MGARCH specification is based on a more volatile latent process than the older LN-MSF-MGARCH structure, so the new one may lead to lower values of φ – even so low that they can justify the popular t-MGARCH model.