Using Pairwise Differences Between Two Conditions as Data (Bayesian)?

Question

Suppose I have measurements for the expression-level of a "gene" from two groups of arbitrary (possibly different) sizes. Maybe one group is a control and the other treated.

$x$ = <4.5, 5.6, 4.3, 6> and $y$ = <1.2, 3.2, 1>

The data are log-transformed (say base 2) and I am interested in modeling the fold change. I do not care about the means of the two groups. I could take the 12 unique pairwise differences between members of the two groups

<4.5-1.2, 4.5-3.2, 4.5-1, 5.6-1.2, 5.6-3.2,5.6-1,4.3-1.2,4.3-3.2,4.3-1, 6-1.2, 6-3.2, 6-1>

and obtain

$d$ = <3.3, 1.2, 3.5, 4.4, 2.4, 4.6, 3.1, 1.1, 3.3, 4.8, 2.8, 5>

which are the observed logarithmic fold changes. I am wondering whether or not it is legitimate to treat these pairwise differences as the data and then do inference? I'm not sure if this is a reasonable question to ask, but I strongly suspect there may be problem. If this is wrong, what is the reason?

With a likelehood

$p(d_i| \mu, \tau) \sim N(\mu, \tau)$

and some priors

$\mu \sim N(0, 1)$

(precision) $\tau \sim Exp(0.5)$

I could estimate posterior distributions for the parameters $\mu$, $\tau$ or I might be interested in the posterior predictive distribution. Clearly a toy example, but in some contexts I might have better prior beliefs about the fold change than the mean abundances of the two conditions. For example, I might strongly believe that most genes are not expressed differently between the conditions, while having almost no prior information about the log-abundance itself. Or maybe, for the raw abundances, the variance or precision are tricky to model and depend strongly on the mean abundance. If I had many "genes" measured in parallel, I might use a hierarchical model.

Answer 1

This is an interesting idea. It seems related to Gini's mean difference, which is defined as the mean absolute difference over all possible pairs of observations. It's a measure of variability and its interpretation is intuitive: If you take two samples, one from each group, how different their measurements will be. It is just as meaningful to define for two samples.

However, with $D_i \sim N(\mu, \tau)$, you also propose (implicitly) that the pairwise differences are independent and this is an unreasonable assumption.