First of all, click on the canvas to represent a "draw". In fast mode you're looking at the exact posterior distribution implied by the draw with the prior that the distribution which generated the draw has a sparse (in the L0 sense) representation in the Haar basis. As it turns out, this problem is tractable (albeit with a 2^d factor in there). In the other mode, you're looking at the same model, but integrated over many affiner transforms with a sigmoid squash, weighted by their total evidence of course.

You could compute mutual information and detect non trivial relationships between pairs of variables in a large dataset

Sure go ahead. Your eyes might bleed if you look at the html/gui part of this, but the statistical code is decently clean.