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For example, you can reorder a softmax + nl (negative likelihood) to log_softmax + nll (negative log-likelihood)

Essentially changing log(softmax(x)) to softmax(log(x))

However, what are the rules to reordering logging of things?

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In general, you cannot reorder like that. The example you give is a very special case that only works because softmax is based on the exponential function, which is the inverse function of the natural logarithm.

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