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Let's say I estimate two models, $M_{0}$ and $M_{1}$. The posterior odds ratio for for model $M_{0}$ against $M_{1}$ given the data, $y$, is,

$\frac{Pr\left(M_{0}\mid y\right)}{Pr\left(M_{1}\mid y\right)}=\frac{Pr\left(M_{0}\right)}{Pr\left(M_{1}\right)}\cdot\frac{f\left(y\mid M_{0}\right)}{f\left(y\mid M_{1}\right)}$

The first ratio in the expression above is the prior odds ratio while the second ratio is the Bayes factor.

I want to estimate two models and calculate the marginal likelihoods using the method described in Chib (1995), “Marginal Likelihood from the Gibbs Output”. Further, the two models are estimated using different prior information. I know that if I use same priors for both models the posterior odds ratio collapses to the Bayes factor,

$\frac{Pr\left(M_{0}\mid y\right)}{Pr\left(M_{1}\mid y\right)}=\frac{f\left(y\mid M_{0}\right)}{f\left(y\mid M_{1}\right)}$

My question is: If I don't have any information on which prior is more likely, i.e. neutral prior information, even though the priors are different can I set $\frac{Pr\left(M_{0}\right)}{Pr\left(M_{1}\right)}=1$ and calculate the posterior odds ratio by simply calculating the Bayes factor? Is this a valid approach?

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Yes, you can set $\frac{Pr(M_0)}{Pr(M_1)} = 1$ which results in $$\frac{Pr(M_0|y)}{Pr(M_1|y)} = \frac{f(y|M_0)}{f(y|M_1)}$$ and yes it is mathematically valid.

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  • $\begingroup$ Thanks for the answer. I might have been a bit too early asking the question as I read "Bayes Factors" by Kass and Raftery (1995) later that day which discuss the exact same thing. $\endgroup$ – Plissken Jul 12 '16 at 9:24
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Be careful with just arbitrarily setting the prior odds to 50/50, and be careful with using the Bayes factor as a decision rule for model comparison. If there is some knowledge that the prior odds are not 50/50, but it's uncertain, then the uncertainty is different than assuming exactly 50/50 prior odds. For a detailed example, see this blog post: http://doingbayesiandataanalysis.blogspot.com/2015/12/lessons-from-bayesian-disease-diagnosis_27.html

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