# Priors on variable ordering and/or percentile ranking

Consider a set of variables $\mathbf{X}$ = $X_1 \ldots X_n$ where each variable is $\in [0,1]$.

I am modeling an inference problem on these variables. Among other things, I have the following prior knowledge about these variables: I know their relative ordering.

For example, for a set of 5 variables I may know that:

$X_5$ < $X_2$ < $X_1$ < $X_4$ < $X_3$

I would like to capture this with a multi-variate prior when I estimate my posterior on $\mathbf{X}$.

Are there any well-known parametric distributions, and hopefully one that is easy to sample from (e.g. for MCMC, or for ABC) that can help capture this relationship?

• You could multiply whatever prior you would use without the restriction (e.g., independent Betas or a single Dirichlet depending on the relationship between the $X_i$) by the indicator $\mathbf{1}(X_5 < \dots < X_3)$. You would just need to re-normalize the joint distribution and make sure your algorithm doesn't sample or calculate values outside of the support of this prior. – user44764 Mar 3 '16 at 0:04
• Thanks @Matt I agree and I thought already of doing that. I guess my thinking here was that parameterizing this knowledge using a known family may help with inference (see this other question). I am also interested in distributions that capture this relationship in a maximum-entropy sort of way (with minimal assumptions otherwise). – Amelio Vazquez-Reina Mar 3 '16 at 1:18
• Regarding your last comment, it seems to me that this question cannot be answered without exhibiting the likelihood. – peuhp Mar 3 '16 at 8:10
• You don't really give enough information. Many solutions are possible, such as providing a prior on $(X_5, X_2-X_5, X_1-X_2, X_4-X_1, X_3-X_4)$ in which all values must be positive and sum to a quantity less than $1$; or perhaps making the $X_i$ iid, drawing five values from their common distribution, and assigning $X_5$ to the the smallest component of $\mathbf{X}$, $X_2$ to the next smallest, etc. (This wouldn't even require computing a renormalizing constant.) But which of the many techniques would be appropriate depends on what else you know about $\mathbf{X}$. – whuber Mar 4 '16 at 18:29