Suppose I have a distribution over independent sets on an $n\times n$ grid where the probability of independent set occupying nodes $(i_1,j_1),\ldots,(i_k,j_k)$ is proportional to $\lambda_{i_1,j_1}\cdot \lambda_{i_2,j_2} \cdots\lambda_{i_k,j_k}$. Let $p_v$ be the probability that vertex $v$ in this grid is occupied, how do I compute $p_v$ for every vertex $v$ in the grid?For instance, when $n=2$, the four probabilities are the ratios belowhttp://yaroslavvb.com/upload/independent-sets-all.pngMotivation: comparing accuracy of correlation decay based approximate counting with exact for some not-too-large graphsClarification 11/6: This is the same $p_v$ as in the first formula of page 3 of [Weitz, 2006]
Solving all-marginals problem for independent sets on grid
ds.algorithms;machine learning