To describe the log-likelihood computation in our model, let us consider a two scale pyramid for the moment. Given a (vectorized) j × j image I , denote by l = d(I) the coarsened image, and h = I − u(d(I)) to be the high pass. In this section, to simplify the computations, we use a slightly different u operator than the one used to generate the images displayed in Figure 3 of the paper. Namely, here we take d(I) to be the mean over each disjoint block of 2× 2 pixels, and take u to be the operator that removes the mean from each 2× 2 block. Since u has rank 3d/4, in this section, we write h in an orthonormal basis of the range of u, then the (linear) mapping from I to (l, h) is unitary. We now build a probability density p on Rd2 by p(I) = q0(l, h)q1(l) = q0(d(I), h(I))q1(d(I)); in a moment we will carefully define the functions qi. For now, suppose that qi ≥ 0, ∫ q1(l) dl = 1, and for each fixed l, ∫ q0(l, h) dh = 1. Then we can check that p has unit integral: ∫