Bayes Factor with data

Question

I have some normally distributed data x of length N. I want to test 2 hypotheses: that $\mu_1$=0 or that $\mu_2>0$. I can calculate the likelihood for all N data points for each hypothesis (for model 2 I use numerical integration to get it). Now I have N likelihoods under each model, how do I condense this to one Bayes Factor? I could avg it or count the number of times each model has a higher factor?

Answer 1

If you look at the definition of a Bayes factor (as, e.g., in my book), $\mathfrak{B}_{01}$ it is expressed as the ratio of two marginal likelihoods $$ m_0(x_1,\ldots,x_N)=\int_{\Theta_0} \ell_0(\theta_0|x_1,\ldots,x_N)\pi_0(\theta_0)\text{d}\theta_0 $$ and $$ m_1(x_1,\ldots,x_N)=\int_{\Theta_1} \ell_1(\theta_1|x_1,\ldots,x_N)\pi_1(\theta_1)\text{d}\theta_1 $$ [with generic notations]. Hence you have to integrate one likelihood per model for the entire dataset or sample. In each model you thus end up with a single numerical value.