How to measure statistical evidence and its strength: Bayes factors or relative belief ratios?

Both the Bayes factor and the relative belief ratio satisfy the principle of evidence and are therefore valid measures of statistical evidence. Which of these measures of evidence is more appropriate? We argue here that there are questions concerning the validity of a commonly used definition of the Bayes factor based on a mixture prior, and when all is considered, the relative belief ratio has better properties as a measure of evidence. We further show that, when a natural restriction on the mixture prior is imposed, the Bayes factor equals the relative belief ratio obtained without using the mixture prior. Even with this restriction, this still leaves open the question of how the strength of evidence is to be measured. We argue here that the current practice of using the size of the Bayes factor to measure strength is not correct and present a solution. We also discuss and address several general criticisms of these measures of evidence.