Lower bounds on Bayes factors in favor of the null hypothesis in multinomial tests of point null hypotheses are developed. These are then applied to derive lower bounds on Bayes factors in both exact and asymptotic chi-squared testing situations. The general conclusion is that the lower bounds tend to be substantially larger than $P$-values, raising serious questions concerning the routine use of moderately small $P$-values (e.g., 0.05) to represent significant evidence against the null hypothesis.
"Lower Bounds on Bayes Factors for Multinomial Distributions, with Application to Chi-Squared Tests of Fit." Ann. Statist. 18 (3) 1295 - 1316, September, 1990. https://doi.org/10.1214/aos/1176347750