On the Toric Algebra of Graphical Models Export

@techreport{citeulike:106398, abstract = {We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. This characterization generalizes the well-known Hammersley-Clifford Theorem. We show that for decomposable graphical models these conditions are equivalent to a set of statistical independence facts as in the Hammersley-Clifford Theorem but that for non-decomposable graphical models they are not. We also show that non-decomposable models can have non-rational maximum likelihood estimates. Finally, using these results, we provide a characterization of decomposable graphical models.}, author = {Geiger, Dan and Meek, Christopher and Sturmfels, Bernd}, citeulike-article-id = {106398}, citeulike-linkout-0 = {http://research.microsoft.com/research/pubs/view.aspx?tr_id=560}, comment = {* Hammersley-Clifford related independence equations to the form of positive distributions satisfying those equations (factored into clique potentials). * Distribution factors according to A if it's in the image of phi\_A * Factorization theorem -- P factors according to A iff the support of P is nice and all polynomials in an ideal basis of the toric ideal I\_A vanish at P * In addition to independence equations, "cross-product ratios" enter for non-decomposable models: * ie, simplest non-decomposable model (4 loop) has following condition (in addition to independence equations and other cpr constraints): p0100 p0111 p1001 p1010 = p0101 p0110 p1000 p1011}, journal = {Microsoft Research}, keywords = {algebra, exponential-families, geometry, graphical, loglinear}, month = {February}, posted-at = {2005-02-28 08:05:59}, priority = {0}, title = {On the Toric Algebra of Graphical Models}, url = {http://research.microsoft.com/research/pubs/view.aspx?tr_id=560}, volume = {MSR-TR-2002-47}, year = {2002} }

TY - RPRT ID - citeulike:106398 L3 - citeulike-article-id:106398 N2 - We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. This characterization generalizes the well-known Hammersley-Clifford Theorem. We show that for decomposable graphical models these conditions are equivalent to a set of statistical independence facts as in the Hammersley-Clifford Theorem but that for non-decomposable graphical models they are not. We also show that non-decomposable models can have non-rational maximum likelihood estimates. Finally, using these results, we provide a characterization of decomposable graphical models. JF - Microsoft Research TI - On the Toric Algebra of Graphical Models VL - MSR-TR-2002-47 KW - algebra KW - exponential-families KW - geometry KW - graphical KW - loglinear N1 - * Hammersley-Clifford related independence equations to the form of positive distributions satisfying those equations (factored into clique potentials). * Distribution factors according to A if it's in the image of phi_A * Factorization theorem -- P factors according to A iff the support of P is nice and all polynomials in an ideal basis of the toric ideal I_A vanish at P * In addition to independence equations, "cross-product ratios" enter for non-decomposable models: * ie, simplest non-decomposable model (4 loop) has following condition (in addition to independence equations and other cpr constraints): p0100 p0111 p1001 p1010 = p0101 p0110 p1000 p1011 AU - Geiger, Dan AU - Meek, Christopher AU - Sturmfels, Bernd PY - 2002/02// UR - http://research.microsoft.com/research/pubs/view.aspx?tr_id=560 ER -