Algebraic analysis for non-regular learning machines

Hierarchical learning machines are non-regular and non-identifiable statistical models, whose true parameter sets are analytic sets with singularities. Using algebraic analysis, we rigorously prove that the stochastic complexity of a non-identifiable learning machine is asymptotically equal to lambda (1) log n - (m(1) - 1) log log n + const., where n is the number of training samples. Moreover we show that the rational number XI and the integer mi can be algorithmically calculated using resolution of singularities in algebraic geometry. Also we obtain inequalities 0 < <lambda>(1) less than or equal to d/2 and 1 less than or equal to m(1) less than or equal to d, where d is the number of parameters.