  Fuzzy logic is a powerful tool to model knowledge uncertainty, measurements
imprecision, and vagueness. However, there is another type of vagueness that
arises when data have multiple forms of expression that fuzzy logic does not
address quite well. This is the case for multiple instance learning problems
(MIL). In MIL, an object is represented by a collection of instances, called a
bag. A bag is labeled negative if all of its instances are negative, and
positive if at least one of its instances is positive. Positive bags encode
ambiguity since the instances themselves are not labeled. In this paper, we
introduce fuzzy inference systems and neural networks designed to handle bags
of instances as input and capable of learning from ambiguously labeled data.
First, we introduce the Multiple Instance Sugeno style fuzzy inference
(MI-Sugeno) that extends the standard Sugeno style inference to handle
reasoning with multiple instances. Second, we use MI-Sugeno to define and
develop Multiple Instance Adaptive Neuro Fuzzy Inference System (MI-ANFIS). We
expand the architecture of the standard ANFIS to allow reasoning with bags and
derive a learning algorithm using backpropagation to identify the premise and
consequent parameters of the network. The proposed inference system is tested
and validated using synthetic and benchmark datasets suitable for MIL problems.
We also apply the proposed MI-ANFIS to fuse the output of multiple
discrimination algorithms for the purpose of landmine detection using Ground
Penetrating Radar.
