Conditional on the two specific models being tested ($\mathcal{H}_0$ and $\mathcal{H}_1$), below is a breakdown of how the prior probabilities, the Bayes factor, and the posterior probabilities relate to each other:
Posterior distribution of $\delta$ under $\mathcal{H}_1$
Bayes factor
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Interpretation 1
Bayes factor
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Interpretation 2
Below is a list of important things to keep in mind when interpreting the Bayes factor.