# Do we use maximum likelihood or cross entropy Loss for training skip-gram model?

In the skip gram model, maximising the likelihood of the context words given the middle word is equivalent to minimising the objective function $$J(\theta)$$, where $$J(\theta) = -\frac{1}{T}\sum_{t=1}^T\sum_{-m \leq j \leq m}\log p(w_{t+j}|w_t)$$ and $$p(w_{t+j}|w_t) = \frac{exp(v_t\cdot u_{t+j}^T)}{\sum_{v \in V}exp(v_tu_v^T)}$$ where $$v_t$$ represents the word embedding for middle word $$w_t$$ and $$u_{t+j}$$ represents word embedding for context word $$w_{t+j}$$.

Suppose we use stochastic gradient descent, then at every word $$t$$, we can find gradients with $$\frac{\partial{J}}{\partial{v_t}}$$, $$\frac{\partial{J}}{\partial{u_1}}$$, ... , $$\frac{\partial{J}}{\partial{u_v}}$$.

I see cross entropy being used as the objective function instead of the above $$J(\theta)$$. Why is it used and how does it come into play ? For example this stack exchange post talks about cross entropy loss for training word2vec.

The equation you've shown at the top, though, is slightly wrong—it's missing a $$\log$$. It should be a double summation of log-probabilities.