The article explores ways to test if the average values in different groups of inverse Gaussian populations are the same. Traditional methods assume the groups have the same variability, but the new approach doesn't make this assumption. By using Bayesian model selection techniques, the researchers found that the Bayes factor can be defined with a constant multiplier. They introduced objective Bayesian methods based on the fractional Bayes factor and the intrinsic Bayes factor. These methods were tested using simulations and real data, showing promising results for comparing means in inverse Gaussian populations with heterogeneity.