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Why is the cross entropy formula different from the binary cross entropy?

There's no essential difference, information theoretically speaking any loss consisting of a negative log-likelihood (NLL) is a cross entropy between the empirical distribution defined by the training ...
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Why is the cross entropy formula different from the binary cross entropy?

The cross entropy loss with 2 classes is equivalent to the binary cross entropy loss: $$ \begin{equation} H_p(q) = - \sum_{i=1}^2 [q_i \log p_i] = q_1 \log p_1 + q_2 \log p_2 = q_1 \log p_1 + (1 - q_1)...
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