Recent innovation in trial design to improve study efficiency has led to the development of basket trials in which a single therapeutic treatment is tested on several patient populations, each of which forms a basket. In a common setting, patients across all baskets share a genetic marker and as such, an assumption can be made that all patients may have a homogeneous response to treatments. Bayesian information borrowing procedures utilize this assumption to draw on information regarding the response in one basket when estimating the response rate in others. This can improve power and precision of estimates particularly in the presence of small sample sizes, however, can come at a cost of biased estimates and an inflation of error rates, bringing into question validity of trial conclusions. We review and compare the performance of several Bayesian borrowing methods, namely: the Bayesian hierarchical model (BHM), calibrated Bayesian hierarchical model (CBHM), exchangeability-nonexchangeability (EXNEX) model and a Bayesian model averaging procedure. A generalization of the CBHM is made to account for unequal sample sizes across baskets. We also propose a modification of the EXNEX model that allows for better control of a type I error. The proposed method uses a data-driven approach to account for the homogeneity of the response data, measured through Hellinger distances. Through an extensive simulation study motivated by a real basket trial, for both equal and unequal sample sizes across baskets, we show that in the presence of a basket with a heterogeneous response, unlike the other methods discussed, this model can control type I error rates to a nominal level whilst yielding improved power.