## Archive for January, 2013

### bounding overlaps

A Venn diagram gives a schematic view of joint counts on a set of $$n$$ categories, e.g. $$c(S_1^n=s_1^n)$$ where $$s_i\in\{0,1\}$$. Each “patch” of the diagram corresponds to one of $$2^n$$ possible values of $$s_1^n$$.

If we have the total count $$C\triangleq \sum_{s_j: j\in \{1,…,n\}} c(S_1^n=s_1^n)$$, then we can take the counts as probabilities by normalizing with $$p(S_1^n=s_1^n)=c(S_1^n=s_1^n)/C$$.

Suppose we are given only singleton marginals $$p(S_i=1)\triangleq \sum_{s_j: j\in \{1,…,n\}\backslash i} p(S_1^n=s_1^n)$$, then we can bound the other probabilities by imposing universal constraints on probabilities to be between 0 and 1.
(Read the article)