Predictive statistical mechanics is a form of inference from available data, without additional assumptions, for predicting reproducible phenomena. By applying predictive statistical mechanics to systems with Hamiltonian dynamics, a problem of predicting the macroscopic time evolution of the system in the case of incomplete information about the microscopic dynamics was considered. The goal of the research is to deepen the understanding, through the analysis of the fundamental principles of predictive statistical mechanics, of its general applicability in relation to other theories that seek to explain the phenomenon of irreversibility. Basic hypothesis is that this can be achieved with the detailed consideration of the role that information about the system has in the clarification of the problem of irreversibility. The hypothesis was tested through the analysis and further generalization of the results of the basic theoretical model. In a model of a closed Hamiltonian system that with the Liouville equation uses the concepts of information theory, analysis was conducted of the loss of correlation between the initial phase space paths and final microstates, and of the related loss of information about the state of the system. Applying the principle of maximum information entropy by maximizing the conditional information entropy, subject to the constraint given by the Liouville equation averaged over the phase space, allowed a definition of the rate of change of entropy without additional assumptions. Initial model was generalized, and with the introduction of the additional constraints which are equivalent to the hydrodynamic continuity equations, brought into direct connection with the known results from the nonequilibrium statistical mechanics and thermodynamics of irreversible processes. The results obtained in this work suggest the general applicability of the principles of predictive statistical mechanics and their importance for the theory of irreversibility.