Maximum Likelihood, permutohedra and Associativity Equations

We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge–Ampère domain and that the log-likelihood function generates its potential function at the identity. The tangent sheaf carries the structure of a pre-Lie algebra. We also show that the moduli space of diagonal matrices parameterizing the polyhedral spectrahedron satisfies the Associativity Equations, a notion central in mirror symmetry, and that its compactification is a toric variety associated to a permutohedron, reminiscent to Losev–Manin spaces. Finally we introduce Frobenius residuals: these are connected components of the compactified Frobenius manifold of diagonal matrices, generated by the Białynicki–Birula cells. We prove that the Maximum Likelihood degree is indexed by components lying on those Frobenius residuals.