We present a novel geodynamic approach that can potentially tighten existing constraints on mantle rheology. This new approach, which we call probabilistic geodynamic modeling, is applied here to the rheology of the upper mantle. We combine the numerical modeling of plate-driven corner flow and the seismic observation of radial anisotropy, aiming to reduce substantial uncertainties associated with experimentally derived flow laws, but our results also highlight the complex competition among different deformation mechanisms under mantle conditions. Despite the remaining rheological uncertainty, our study suggests that significant background shear flow is required near the lithosphere-asthenosphere boundary to explain the strong radial anisotropy observed at 100-200 km depth underneath the Pacific plate, and the plausible nature of this background flow is characterized using our new probabilistic approach. Our analysis also provides a new insight into the asthenospheric water content and the grain size distribution in the upper mantle, but these results are also subject to nontrivial nonuniqueness. The merit of our probabilistic approach lies in its ability to assess the extent of such nonuniqueness, and we demonstrate this by quantifying the robustness of some of our results. Plain Language Summary A detailed understanding of how silicate minerals deform under various conditions is central to conducting realistic simulations of mantle dynamics, but the currently available experimental constraints on the flow laws of olivine, which is the dominant phase of the upper mantle, suffer from considerable uncertainties. We present a new kind of geodynamic modeling, which can reduce such uncertainties by assimilating geophysical observations. Unlike previous similar attempts, our approach properly handles parameter uncertainties such that the flow laws under consideration remain consistent with experimental deformation data. Most important, our new approach provides a unifying framework to combine experimental rock mechanics, observational geophysics, and theoretical geodynamics in a self-consistent manner.