# Joint or marginal?

My parameters of interest are say $\theta_{1:3}$ which I declare as multivariate normal as their prior distribution. I then get the posterior distributions $\theta_1$, $\theta_2$, $\theta_3$.

Can I consider $\theta_{1:3}$ as the joint distribution, or are these three marginal distributions? For example, can I validly compute $P(\theta_1 > 0\ \&\ \theta_2 < 0\ \&\ \theta_3 > 0.5)$ using the output from the posterior distributions?

• I think you should go over your terms a bit, $(\theta_1, \theta_2, \theta_3)$ is not a distribution, in this case it is a random variable, it has a distribution. When you say marginal, what does that mean, what are the other variables you are integrating out? (I'm pretty sure you are working with a Gibb's sampler and trying to interpret the output, but let's be clear about what's going on here). So let's define our terms and fill in some details then we can probably answer this. – Jonathan Lisic May 13 '15 at 23:53
• Sorry my language was loose. Theta[1:3] are random variables (in this case unknown parameters). I want P(theta[1:3] | Data). I use JAGS simulator to obtain this. JAGS gives back of course the simulated distributions of theta[1], theta[2] and theta[3]. What I'm not sure about is whether these are the 3 separate marginal distributions from the joint distribution of (theta[1:3]) or whether (theta[1], theta[2], theta[3]) are simulated observations from their joint distribution. – Mel May 14 '15 at 22:22