Understanding and improving the predictive skill of imperfect models for complex systems in their response to external forcing is a crucial issue in diverse applications such as for example climate change science. Equilibrium statistical fidelity of the imperfect model on suitable coarse-grained variables is a necessary but not sufficient condition for this predictive skill, and elementary examples are given here demonstrating this. Here, with equilibrium statistical fidelity of the imperfect model, a direct link is developed between the predictive fidelity of specific test problems in the training phase where the perfect natural system is observed and the predictive skill for the forced response of the imperfect model by combining appropriate concepts from information theory with other concepts based on the fluctuation dissipation theorem. Here a suite of mathematically tractable models with nontrivial eddy diffusivity, variance, and intermittent non-Gaussian statistics mimicking crucial features of atmospheric tracers together with stochastically forced standard eddy diffusivity approximation with model error are utilized to illustrate this link.P redicting the long-range behavior of complex systems in nature in diverse disciplines ranging from climate change science (1, 2) to materials (3) and neuroscience (4) is an issue of central importance in contemporary engineering and science. Accurate predictions are hampered by the fact that the true dynamics of the system in nature are actually unknown due to inadequate scientific understanding or inadequate spatiotemporal resolution in the imperfect computer models used for these predictions; in other words, there are significant model errors compared to the true signal from nature. Recently, information theory has been utilized in different ways to systematically improve model fidelity and sensitivity (5, 6), to quantify the role of coarse-grained initial states in long-range forecasting (7,8), and to make an empirical link between model fidelity and forecasting skill (9, 10). Imperfect models for complex systems are constrained by their capability to reproduce certain statistics in a training phase where the natural system has been observed; for example, this training phase in climate science is roughly the 60-y dataset of extensive observations of the Earth's climate system. For long-range forecasting, it is natural to guarantee statistical equilibrium fidelity for an imperfect model, and a framework using information theory is a natural way to achieve this in an unbiased fashion (5-8, 10). First, equilibrium statistical fidelity for an imperfect model depends on the choice of coarse-grained variables utilized (5, 6); second, equilibrium model fidelity is a necessary but not sufficient condition to guarantee long-range forecasting skill (8). For example, ref. 11, Sect. 2.6, extensively discusses three very different strongly mixing chaotic dynamical models with 40 variables and with the same Gaussian equilibrium measure, the TBH, K-Z, and IL96 models, so that al...