This article compares the Bayesian standard of what constitutes convincing evidence to the frequentist reliance on the p-value to measure evidence. The author concludes that a much smaller p-value (0.005 or 0.001) is needed to be consistent with the Bayesian standard.
Valen E Johnson. Revised standards for statistical evidence. Proceedings of the National Academy of Sciences. 2013. doi:10.1073/pnas.1313476110.