Predicting output with a neural network

By combining layers of neurons together we create a stacked function that has non-linear transformations and trainable weights so it can learn to recognize complex relationships. To visualize this, let's transform the neural network from previous sections into a mathematical formula. First, let's take a look at the formula for a single layer:

The X variable is a vector that represents the input for the layer in the neural network. The w parameter represents a vector of weights for each of the elements in the input vector, X. In many neural network implementations, an additional term, b, is added, this is called the bias and basically increases or decreases the overall level of input required to activate the neuron. Finally, there's a function, f, which is the activation function for the layer.

Now that you've seen the formula for a single layer, let's put together additional layers to create the formula for the neural network:

Notice how the formula has changed. We now have the formula for the first layer wrapped in another layer function. This wrapping or stacking of functions continues when we add more layers to the neural network. Each layer introduces more parameters that need to be optimized to train the neural network. It also allows the neural network to learn more complex relationships from the data we feed into it. 

To make a prediction with a neural network, we need to fill all of the parameters in the neural network. Let's assume we know those because we trained it before. What's left is the input value for the neural network. 

The input is a vector of floating-point numbers that is a representation of the input of our neural network. The output is a vector that forms a representation of the predicted output of the neural network.