I have a model which includes two exponential rate parameters. I would like to test whether a model with two individual rates describes some data better than a model for which both rates are the same. I think I could just estimate the model with two parameters and look at HDI around the difference. However, later I would like to compare more complex (an possibly not-nested) similar models. So I would like to use this as a toy example and calculate Bayes-Factors in JAGS.
Therefore, I would like to set up two models. One in which both rate parameters are drawn from the same normal distribution. And the other where they are drawn from independent normal distributions.
I'm unsure about the choice of priors, however. Do they need to be conjugate for the Bayes-Factors to be proper? (I suspect: Even if not in this toy example, when I use more complex models with different parameters they may have to be conjugate?).
How could I specify such conjugate priors (normal; unknown mean & unknown variance, I guess) for the rate parameters in JAGS?
Details and additional questions regarding the priors:
The model is as follows:
$y_i \sim B(N_i, p_i)$
$p_i $<- some_equation_with_two_exp_rate($v_1,v_2,i$)
Now I would like to specify different versions: One where both rates are equal ($v_1=v_2$) [and others, e.g., one that expresses $v_1>v_2$].
So first, for specifying the equality I was thinking of setting $v_1$ <- $x$ and $v2$ <- $x$ and $x$ ~ some_prior_dist. However, I think that wouldn't be correct, because the $v$'s represent two independent processes (with the same rate). So I tried adding hyperprior: $v_1$ ~ some_prior_dist($x,var$) and $v_2$ ~ some_prior_dist($x,var$) and $x$ ~ some_hyper_prior_dist($m,var2$).
For the distributions I mainly tried dnorm. Estimations seems strange: while I still get reasonable posteriors for the $v_1$ and $v_2$, m yields strange estimates (very broad distributions, as if not sufficiently constrained by the data). Furthermore, I'm unsure about which variances ($var,var2$ in the dummies above) to fix and which should get individual priors. Any hints?
Many thanks, Jan