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Suppose I am a Bayesian working with multi-level data, $j$ and $t$.

I run a model using $t$ that calculates the posterior distribution of a parameter $\theta_j$ for each $j$, which I then use to calculate the posterior distribution a function $g(\theta_j)$ for each $j$. Thus, afterwards, if I ran $R$ draws in my MCMC, I end up with:

$\{g(\theta_j^r)\}_{r=1}^{R}$ for each $j$.

Now, I am interested in the variation in $E_{\theta_j}[g(\theta_j)]$ across $j$ units, which can be characterized by a distribution, which I will call $f(g)$.

Of course, my best guess of $f$ would be to plot the empirical distribution of the $E_{\theta_j}[g(\theta_j)]$'s. However, I want to characterize the uncertainty in $f$. How can I do this empirically given all my draws? A bootstrap?

Here is what I tried before I realized this was vastly over-estimating uncertainty:

Basically, I plotted the distribution of $g$'s at each value of $\theta$. Then took the credible intervals vertically. However, I think this over-estimates the uncertainty and actually gives an incorrect distribution. For instance, I know this because in my problem, the $E_{\theta_j}[g(\theta_j)]$ must be strictly positive, and when I plot their empirical distribution, they are. However, when I use the single-draw-at-a-time approach, we get a lot of negative mass with too much precision. Sure, maybe some of the draws might turn out negative, but not the expectation itself. Maybe a bootstrap would solve this? But I was hoping for a more bayesian-theoretically guided approach here.

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The problem with this question, upon further reflection, is that there is not a posterior distribution of $E_{\theta_j}[g(\theta_j))]$. If you are a true Bayesian, the expectation is just a number at this point. Therefore, it doesn't make sense fo try to characterize uncertainty for each expectation (and by extension, the distribution of the expectations).

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