Table 3 Four Types of Integrative Hypothesis Tests

Type-1 (model based IHT)

For Task A, each of two populations is modelled by a parametric model, with ε _A measured by the negative log-likelihood by Equation (32) or its extension to generalisation error. For Task B, a model based test is made to compare the difference between two parametric models, with ε _B by the corresponding p-value. For Task C, we get the classification by Equation (45), with ε _C by Equation (44) or the p-value by a BBT via a statistics obtained from Equation (10).

Type-2 (boundary based IHT)

A separating boundary is modelled by a hyperplane with its normal w, based on which Task D is handled by a boundary existence test by Equation (5) with ε _D measured by the corresponding p-value. For Task C we get the classification by Equation (46) with ε _C by Equation (44) or alternatively the corresponding p-value obtained by Equation (47), and for Task B we get the p-value by one of two BBT choices in Table 2.

Type-3 (mixing IHT)

Mix the above two types with two populations and their separating boundary all in parametric models. A basic one uses ε _A ,ε _B from Type-1 and ε _C ,ε _D from Type-2. The other uses ε _C ,ε _D from Type-2 while ε _A ,ε _B are modified by Equation (58).

Type-4 (Ying-Yang IHT)

Instead of mixing, the parametric models are jointly learned for two populations of samples and their separating boundary. One example is the BYY harmony learning based formulation to be introduced after Equation (60).