When predicting a poly-neuron output in neural nets, say, predicting multiple handwritten digits and giving an output neuron vector (0.1,...,0.9,0.1,...), many use sth like softmax (or sth like the energy dependent probability exponential formula in statistical mechanics) to normalize the output vector such that all the components of the output vector sum up to 1, and that the normalized output vector becomes a probability vector. I doubt the necessity of this normalization, for without which I can equally well predict as per the biggest vector component. Is there anything I overlooked?
You are correct, but this is not called normalization. You can simply use the highest probability output for category. This is what softmax does for you. For example 2 output neurons can have 0.1 for dog and 0.9 for cat as the loss. Softmax will it just convert it to [0,1] meaning no dog but a cat is on the image.
Yes, you have overlooked something. Let me explain this with an example:
When training a neural network, you use the loss function as a signal to backpropagation. Now let us say that you have a network which outputs three categories: Cat, Dog, and Toad. Let's say the prediction of the network in an iteration is [0.7, 0.6, 0.3], although the data is, in fact, an image of a dog, meaning the truth is: [0 1 0].
Without a softmax layer, you cannot really tell much the network got the prediction wrong. In this example you might think the difference is 0.7 - 0.6 = 0.1, however after running a softmax, you realize that it is in fact 0.03, since the network was very strong in differentiating between the Toad category vs. Dog and Cat, so the loss is not as big as it seems.
Now as an experiment, run a neural network without the softmax layer and see for yourself, how severe it can affect the training. Not only that, but normalization of the input batches also makes a huge difference in training a network.