Fuzzy Inference

A rule bkb_k is a product (AND) of several Membership Functions mfjmf_j that can be active or not via the learnable parameter pkjp_{kj}.

bk(x)=∏j=1M(mfj(x))pkj=exp⁡(∑j=1Mpkjln⁡(mfj(x)))b_k(x)=\prod_{j=1}^M\left(m f_j(x)\right)^{p_{k j}}=\exp \left(\sum_{j=1}^M p_{k j} \ln \left(m f_j(x)\right)\right)

The rewritten logarithmic form can be expressed as a Neural Network.