I tried to answer my own question Comparing two Bayesian models under disjoint prior supports using MCMC. Here is my intent. I am not confident in what I wrote so prefer to post it as a question : Is this a correct way to compute Bayes factor ?

I would like to compute the Bayes factor: $$ K = \frac{P(x|H_1)}{P(x|H_2)} =\frac{P(H_1|x)}{P(H_2|x)} \frac{P(H_2)}{P(H_1)} $$

I have a model consisting of three parameters $\theta_1$, $\theta_2$ and $\theta_3$. $H_1$ is $\{(\theta_1,\theta_2,\theta_3) \in [m,M]^3 \mbox{ such that } \theta_1<\theta_2<\theta_3\}$, $H_2$ is it complementary. Then suppose that I have a model to infer $P(\theta_1,\theta_2,\theta_3 | x)$ under the complete parameter space $H_1 \cup H_2 $ then $P(H_1|x)$ can simply be computed as: $$ P(H_1|x)=\int_m^M P(\theta_1,\theta_2,\theta_3|x) 1_{\theta_1>\theta_2>\theta_3}(\theta_1,\theta_2,\theta_3) d\theta_1 d\theta_2 d\theta_3 $$ by simply counting the posterior MCMC samples satisfying the condition and $P(H_2|x)=1-P(H_1|x)$.

So it remains to extract the associated $\frac{P(H_2)}{P(H_1)}$ by simply considering the subpart of $H_1$ over the overall parameter space: $$ P(H_1)=\int p(\theta_1,\theta_2,\theta_3) \cdot 1_{\theta_1>\theta_2>\theta_3}(\theta_1,\theta_2,\theta_3) d\theta_1 d\theta_2 d\theta_3 $$ which can be computed analytically for the prior $p(\theta_1,\theta_2,\theta_3)$ associated to my posterior and again $ P(H_2)=1-p(H_1)$.

Is this a correct way to compute Bayes factor ? If yes, is this method has a name?


1 Answer 1


This is a correct approach indeed, since you compare two subsets of the parameter space after selected a prior for the entire parameter set. This prior indeed defines single-handedly prior weights on both subsets $H_0$ and $H_1$, which can be computed by integration. (If the three parameters are exchangeable, the prior weight of $H_0$ is $1/3!$.)

  • $\begingroup$ Thanks again. Have you a reference in mind for this approach ? $\endgroup$
    – peuhp
    Feb 22, 2016 at 14:02
  • $\begingroup$ This is standard practice when considering null hypotheses that have a non zero weight under the original prior. There is no specific name or reference for this. $\endgroup$
    – Xi'an
    Feb 22, 2016 at 14:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.