I'm reading [this page](https://rdipietro.github.io/friendly-intro-to-cross-entropy-loss/) about cross-entropy loss and why it works as a maximum likelihood estimator.

He says:

> 
> Because we usually assume that our samples are independent and
>  identically distributed, the likelihood over all of our examples
>  decomposes into a product over the likelihoods of individual examples:
> 
> 
> 

Then he gives an example: if our NN predicts (0.4, 0.1, 0.5) as the probabilities of the three classes, and the correct value is (1.0, 0.0, 0.0), then he says that the "likelihood" of that single example is just 0.4.

As I understand it, L(x, theta) = f(x, theta) for a single observation. How are we supposed to measure f(x, theta) when its value is itself a distribution? That is, what does it mean to have an outcome of (0.4, 0.1, 0.5) given a distribution of (1.0, 0.0, 0.0)?

Edit: if the question were "what is the probability of outcome 1 given distribution (0.4, 0.1, 0.5)", then there's no confusion. I've been interpreting it as "what is the probability of (0.4, 0.1, 0.5) given (1.0, 0.0, 0.0)" which seems awkward.