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I have two normal distributions representing the posterior distribution of two alternate physical models.

In this physical problem I would like to know which model is more likely to produce a value greater than $n$. I can calculate the probability of a value greater than $n$ for both distributions using the normal cumulative distribution function.

Assuming both models had the same prior probability, can I calculate the evidence for each model using the Bayes factor calculated with the probabilities from the CDF?

It is a messy problem because I am trying to compare two models generated using different sets of observations (so I know the likelihood relative to that data), but what I would like to know the likelihood of each model producing values greater than $n$ which is a threshold value.

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  • $\begingroup$ Yes so I would like to calculate the Bayes factor to get an idea of the support or evidence for each of the two models. $\endgroup$ – Robin Jun 22 '20 at 6:37
  • $\begingroup$ But what is the connection with the posterior probability to exceed $n$? $\endgroup$ – Xi'an Jun 22 '20 at 9:33
  • $\begingroup$ The idea is that I have two distributions which I have found a value of $\mu$ and $\sigma$ and I want to find which model is more likely to produce a value greater than $n$. Ideally I would like to use a Bayes Factor to compare the two models because it is easy to discuss. $\endgroup$ – Robin Jun 22 '20 at 22:46
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    $\begingroup$ treat the observation as a Bernoulli $\mathbb P_\theta(X>n)$. $\endgroup$ – Xi'an Jun 23 '20 at 5:57

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