!This talk will be on zoom only! We consider the problem of estimating expectations of quantities of interest of computationally expensive models. In this context, multifidelity Monte Carlo estimators, which rely on low-fidelity approximations of the high-fidelity model, allow to achieve variance reduction without increasing the computational cost. However, the performance of multi-fidelity estimators is strongly dependent on the correlation between the high-fidelity and low-fidelity models. In this talk, we propose a methodology to increase the correlation between the models and derive unbiased multifidelity estimators with smaller variance. In particular, we build a reduced shared parameterization based on transformations between probability distributions and low-dimensional manifolds that capture most of the variation of the models. To discover the low-dimensional manifolds, we employ both linear and nonlinear dimensionality reduction techniques, focusing on active subspaces and neural active manifolds, respectively. We present the limitations of the linear approach, and we show that they can be overcome in the nonlinear setting. Moreover, in the nonlinear framework, under idealized conditions, we provide a theoretical result that guarantees that the new correlation obtained through the reduced shared manifold is never smaller than the original correlation between the models. We highlight the advantages of the method through several numerical examples, and we show that it outperforms standard multifidelity estimators in terms of variance.