Suppose have a probability distribution over space of $n$ binary variables. We can view logarithm of the density as a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$. Fourier expansion of the $f$ can be viewed as expansion of $f$ in terms of square free monomials.

Given a sample from this density, we'd like to find Fourier coefficients that approximate true density. For real-life densities, fit is usually done by adjusting a small number of low order coefficients to fit the sample and fixing the rest to 0. Why does it work better than adjusting high-order coefficients?