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We often say that minimizing the (negative) cross-entropy error is the same as maximizing the likelihood. So can we say that NN are just an alternative way of performing Maximum Likelihood Estimation? if not, why?

In abstract terms, neural networks are models, or if you prefer, functions with unknown parameters, where we try to learn the parameter by minimizing loss function (not just cross entropy, there are many other possibilities). In general, minimizing loss is in most cases equivalent to maximizing some likelihood function, but as discussed in this thread, it's not that simple.

You cannot say that they are equivalent, because minimizing loss, or maximizing likelihood is a method of finding the parameters, while neural network is the function defined in terms of those parameters.

These are fairly orthogonal topics.

Neural networks are a type of model which has a very large number of parameters. Maximum Likelihood Estimation is a very common method for estimating parameters from a given model and data. Typically, a model will allow you to compute a likelihood function from a model, data and parameter values. Since we don't know what the actual parameter values are, one way of estimating them is to use the value that maximizes the given likelihood. Neural networks are our model, maximum likelihood estimation is one method for estimating the parameters of our model.

One slightly technical note is that often, Maximum Likelihood Estimation is not exactly used in Neural Networks. That is, there are a lot of regularization methods used that imply we're not actually maximizing a likelihood function. These include:

(1) Penalized maximum likelihood. This one is a bit of a cop-out, as it doesn't actually take too much effort to think of Penalized likelihoods as actually just a different likelihood (i.e., one with priors) that one is maximizing.

(2) Random drop out. In especially a lot of the newer architectures, parameter values will randomly be set to 0 during training. This procedure is more definitely outside the realm of maximum likelihood estimation.

(3) Early stopping. It's not the most popular method at all, but one way to prevent overfitting is just to stop the optimization algorithm before it converges. Again, this is technically not maximum likelihood estimation, it's really just an ad-hoc solution to overfitting.

(4) Bayesian methods, probably the most common alternative to Maximum Likelihood Estimation in the statistics world, are also used for estimating the parameter values of a neural network. However, this is often too computationally intensive for large networks.