I am trying to write out the following log-likelihood: $$\mathcal L(\vec{x}, \vec{y}) = \sum_{i} \left[ k_{i}^{out} (\boldsymbol{A}^*) \ln x_i + k_i^{in} (\boldsymbol{A}^*) \ln y_i\right] - \sum_{i \neq j} \ln(1 + x_i y_i). \tag{1}$$
I read it as, in the first term, the sum over all $i$, which is logical. But then, in the second sum, it says, a sum over all $i$ but not $j$. Which in principle is fine, but how does one determine $j$ in that instance? It's also not that the second sum, should be included in the first sum, because both sums have a subscript $i$ Is this a mistake in the publication, or is this still solvable?
In case the second sum should be a double sum, is the equation below, also a double sum? (this is from the exact same paper, but I don't think a double sum would be logical here)
\begin{align} & \sum_{j \neq i} \dfrac{x_i^* y_j^*}{1 + x_I^* y_j^*} = k_i^{out}(\boldsymbol{A}^*); \qquad \forall i. \tag{2} \\ & \sum_{j \neq i} \dfrac{x_j^* y_i^*}{1 + x_j^* y_i^*} = k_i^{in}(\boldsymbol{A}^*); \qquad \forall i. \tag{3} \end{align}
