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\\).<\/p>\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\\).<\/p>\n
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.
\n
\nTo get the bounds, we solve a linear programming problem. The standard form is \\(\\min_x f’x \\ \\ \\mathrm{s.t.}\\ Ax \\leq b\\). Here, \\(f\\) is a length-\\(2^n\\) vector that indicates the particular marginal probability we want to bound, by selecting among the \\(s_1^n\\)-sequences. \\(A\\) is \\([I_{2^n\\times 2^n}; B; \\mathbf{1}_{1\\times 2^n}]\\) where the columns correspond to all possible \\(s_1^n\\)-sequences, and where the rows of \\(B\\) are the \\(n\\) half-1 masks indicating the \\(n\\) singleton marginals. Finally, let \\(u=[\\mathbf{1}_{2^n\\times1}; p(S_1=s_1);…;p(S_n=s_n); 1]\\) and \\(l=[\\mathbf{0}_{2^n\\times1}; p(S_1=s_1);…;p(S_n=s_n); 1]\\).<\/p>\n
Hence to get the lower bound on the desired marginal, we solve \\(\\min_x f’ x \\ \\ \\mathrm{s.t.}\\ A x \\leq u\\ \\mathrm{and}\\ -A x \\leq -l\\). Likewise, for the upper bound, \\(-\\min_x -f’ x \\ \\ \\mathrm{s.t.}\\ A x \\leq u\\ \\mathrm{and}\\ -A x \\leq -l\\).<\/p>\n
For example, \\(n=2\\), \\(p(S_1=1)=0.6\\), \\(p(S_2=1)=0.5\\), then we can bound \\(p(S_1=1,S_2=1)\\) by \\(0.1\\le p(S_1=1,S_2=1)\\le 0.5\\).<\/p>\n
Of course, we can add additional constraints when we know them. In some cases with categories, we have that \\(p(S_1^n=0)=0\\), that is, everything must belong to at least one category. Then it would appear the doublet marginal \\(p(S_i=1,S_j=1)\\) satisfies
\n\\(\\max \\{0, p(S_i=1)+p(S_j=1)-1\\}\\le p(S_i=1,S_j=1)\\)
\n\\(\\le \\min \\{(\\sum_{i=1}^n p(S_i=1))-1, p(S_i=1), p(S_j=1)\\}\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
A Venn diagram gives a schematic view of joint counts on a set of categories, e.g. where . Each “patch” of the diagram corresponds to one of possible values of . If we have the total count , then we can take the counts as probabilities by normalizing with . Suppose we are given only […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[2943,2945,336,2942,2879,2921,2944,2946,2941,2521],"class_list":["post-983","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-marginal-probability","tag-masks","tag-math","tag-normalizing","tag-probabilities","tag-schematic-view","tag-sequences","tag-singleton","tag-universal-constraints","tag-vector"],"_links":{"self":[{"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=\/wp\/v2\/posts\/983","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=983"}],"version-history":[{"count":0,"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=\/wp\/v2\/posts\/983\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=983"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=983"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.yhuang.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}