Two-layer neural networks
Let us look at the formal definition of a two-layer neural network. We follow the notations and description used by David MacKay (reference 1, 2, and 3 in the References section of this chapter). The input to the NN is given by 
. The input values are first multiplied by a set of weights to produce a weighted linear combination and then transformed using a nonlinear function to produce values of the state of neurons in the hidden layer:
A similar operation is done at the second layer to produce final output values 
:
The function 
is usually taken as either a
sigmoid function 
or 
. Another common function used for multiclass classification is softmax defined as follows:
This is a normalized exponential function.
All these are highly nonlinear functions exhibiting the property that the output value has a sharp increase as a function of the input. This nonlinear property gives neural networks more computational flexibility than standard linear or generalized linear models...
