A general Bayesian framework for best-estimate plus uncertainty predictions of multidimensional continuous observables is presented. Parameterizing uncertainties in terms of multivariate normal distribution models, this Multivariate normal Bayesian model (MNBM) framework allows one to include both measured data and linear constraints in a mathematically consistent way. The resulting updating formulas are generalizations of the updating formulas of the Generalized Linear Least Squares (GLLS) framework, which is widely used for the generation of adjusted nuclear data libraries. While the GLLS methodology is restricted to first-order perturbation theory, there is no such restriction for the considered MNBM framework. This makes it possible to use Monte Carlo uncertainty propagation and to apply the updating formulas directly to the observables of interest without having to first update the input parameter distributions. After a general presentation of the MNBM framework and a brief discussion of its possible applications, the generation of bounding burnup-dependent axial burnup profiles of light water reactor fuel assemblies for the purpose of criticality safety analysis is discussed as an example application.