In a Multi-level Bayesian Hierarchical Model, would higher level parameters be affected by how they are jointly modeled in lower levels?

Suppose we have a Multi-level Hierarchical Model where:

$$\begin{equation} Y_{0i} \sim Bin(\theta_{0i}, n_{0i}) \\ Y_{1i} \sim Bin(\theta_{1i}, n_{1i}) \\ \theta_{0i} \sim Unif(0,1) \\ log\left(\dfrac{\theta_{1i}}{\theta_{0i}}\right) \sim N\left(\mu, \sigma^2\right) \\ \mu \sim N(0, 0.1) \\ \sigma^2 \sim Gamma(1,1) \end{equation}$$

and another Hierarchical Model that is almost the same except for how the $$\theta$$'s are jointly modeled:

$$\begin{equation} Y_{0i} \sim Bin(\theta_{0i}, n_{0i}) \\ Y_{1i} \sim Bin(\theta_{1i}, n_{1i}) \\ \theta_{0i} \sim Unif(0,1) \\ \theta_{1i}-\theta_{0i}\sim N\left(\mu, \sigma^2\right) \\ \mu \sim N(0, 0.1) \\ \sigma^2 \sim Gamma(1,1) \end{equation}$$

I understand that inferences on the marginal posterior of $$\mu$$ and $$\sigma^2$$ will be different in both models. However, will the marginal posterior of $$\theta_{1i}$$ and $$\theta_{0i}$$ be different or will they be the same? It seems that since for example, $$P(\theta_{1i}|Y)$$ can be obtained by marginalizing out $$P(\theta_{1i}, \mu, \sigma^2|Y)$$, then it is the same. But is this the case?

• The second model is not compatible with the $\theta_{ij}$'s being probabilities and both models are incomplete because the joint on $(\theta_{0i},\theta_{1i})$ is not fully specified. – Xi'an Apr 4 at 8:12
• @Xi'an Sorry, I meant to model the control theta as a uniform, I've made the edit. When I use the second model, RJags appears to accept it. Is it perhaps simulating many times until a usable $\theta_{1i}$ (such that $\theta_{1i}-\theta_{0i}$) is retrieved? – user321627 Apr 4 at 8:23