Landscape evolution models that invert topography for rock uplift can improve our understanding of both tectonic and geomorphic processes when properly constrained with data. Here we present a flexible, data driven Bayesian approach to invert fluvial topography for tectonic and geomorphic model parameters and apply it to a case study, the uplifting footwall of the Corinth Rift, Greece. We invert transient river profiles and up flexed marine terraces to resolve seven unknown parameters in a regional to flexural uplift tectonic model and the stream power incision model. The best fit tectonic parameters are consistent with independent data and predict block uplift rates of ∼0.1 mm yr⁻¹ that changed to flexural uplift rates of ∼1.6 mm yr⁻¹ at ∼0.6 Ma, as the master normal fault initiated. Similarly, the best fit geomorphic parameters predict sediment flux consistent with the offshore record and erodibility consistent with previous studies. However, the drainage area exponent, m, of ∼2, and slope exponent, n, of ∼7, are unusually high, indicating a threshold channel steepness where fluvial topography is largely insensitive to rock uplift rate >0.05 mm yr⁻¹. Analysis indicates channels narrow to accommodate enhanced uplift rates, but channel narrowing only partially explains our results, suggesting that other processes not accounted for in the generic stream power model are also relevant to bedrock river incision in Corinth. Our results help clarify the tectonic and geomorphic evolution of the Corinth Rift, have important implications for studies that invert topography for rock uplift histories, and provide insight into potential limitations of some long term river incision models.