Summarizing samples of an interval I have a particular semiparametric model which I'm fitting via MCMC. One of the model parameters I have "semiparametric'ed" away (say $\alpha$) is known to lie between two other parameters, $\theta_1$ and $\theta_2$. Since I have a series of samples $(\theta_1^t, \theta_2^t)$ I also have a series of interval estimates for $\alpha$. What is a reasonable way to summarize these? I can think of doing something like putting down a grid and just recording for each gridpoint whether each sample covers it or not. But I worry about efficiency since I don't have a great sense of what a plausible range is, and I actually have hundreds to thousands of $\alpha$'s. 
(I'm also welcoming retags, since I can't seem to find any I like...)
 A: Let's see:
$$
P(\alpha) = \int P(\alpha, \theta_1, \theta_2) d\theta_1 d\theta_2
 = \int P(\alpha | \theta_1, \theta_2) P(\theta_1, \theta_2) d\theta_1 d\theta_2
$$
With the good help of Monte Carlo, we can approximate this as
$$
\frac{1}{n} \sum_t P(\alpha | \theta_1^t, \theta_2^t)
$$
With the grid trick, you are doing something like this, but then you are implicitly assuming that $P(\alpha | \theta_1, \theta_2)$ is uniform. Is that an OK assumption for you? If so, it's a good technique if you want to get a view on the distribution of $\alpha$. If you are only out for confidence intervals for $\alpha$, you can probably do better by ordering your intervalsbase on $\theta_1$, and then finding the last value that is only covered by the first $2.5\%$ of you intervals (if you're aiming for $95\%$ confidence), and then similar for the other side.
An OK range of values for your grid would be from the minimum $\theta_1$ up till the maximum $\theta_2$, no? If you mean by 'hundreds of thousands of $\alpha$s that you have that many parameters that you wish to get information about simultaneously, then yes, you are in trouble: you cannot abstract out those parameters, and then hope to automagically and easily get this kind of information back. Univariately though, you're safe.
