I have (x,y) data divided into five datasets: $D_1, D_2, \dots D_5$. The first dataset was generated randomly. Subsequent datasets were recursively generated by transforming the last dataset via a complicated transformation, so that $D_2$ was generated from $D_1$, $D_3$ from $D_2$, etc.
I want to evaluate the claim that these repeated transformations of the x/y pairs tends to structure originally unstructured data into linear data with a positive slope. What are possible ways of evaluating this claim?
Here is my tentative solution:
Perform bayesian linear regression on each dataset and obtain the prior likelihood that the (x,y) data are best described by a line with positive slope.
Plot these likelihoods against the dataset number (the subscripts) and determine whether there is a direct relationship via bayesian linear regression. A direct relationship would provide evidence that the transformations tend to structure unstructured data into linear data with a positive slope.
This made me pause because 1) I have never come across a bayesian analysis of bayesian analysis data and 2) it seems fairly complicated.