Is negative log likelihood calculated in log space or exponential space? I have a question about calculating negative log likelihood in a machine learning model over a dataset which seems simple but I cannot find a solid answer/explanation online.
Is the NLL calculated as an average in log space?
$$
NLL = -\frac{1}{N} \sum_{i=0}^N \log p(y_i | x_i)
$$
or should it be done in exponential space?
$$
NLL = -\log \Big(\frac{1}{N} \sum_{i=0}^N p(y_i | x_i) \Big)
$$

Let's start from the beginning. The likelihood is defined as the joint probability of observing the data; I guess for your task, the probability of one observation has the symbol $p(y|x)$:
$$
L = \prod_{i=1}^N p(y_i | x_i)
$$
whence the negative log-likelihood is
$$
-\log(L) = -\sum_{i=1}^N \log\left( p(y_i | x_i)\right)
$$
and the choice to work with
$$\text{NLL}=-\frac{\log(L)}{N} = -\frac{1}{N}\sum_{i=1}^N \log\left( p(y_i | x_i)\right)$$
is an estimate of the cross-entropy of the model probability and the empirical probability in the data, which is the expected negative log probability according to the model averaged across the data.
Re-scaling by $\frac{1}{N}$ does not change the result of the optimization procedure, since multiplication by any positive scalar only changes the value of $L$ but not the location of the optima.
