Values for $*$ symbol in linear algebra equation

Question

In the following equation shown below, I am wondering as to how to interpret the $*$ symbol. What values of $j$ in $p_{i*}$ should be used, and what values of $i$ should be used in $p_{*j}$ $$\begin{align} p_{ij}&= \frac{f_{ij}}{\sum_{i=1}^{n_r}\sum_{j=1}^{n_c}f_{ij}} \\[1ex] p_{i*}&= \frac{\sum_{j=1}^{n_c}f_{ij}}{\sum_{i=1}^{n_r}\sum_{j=1}^{n_c}f_{ij}} \\[1ex] p_{*j}&= \frac{\sum_{i=1}^{n_r}f_{ij}}{\sum_{i=1}^{n_r}\sum_{j=1}^{n_c}f_{ij}}\\[1ex] \operatorname{pmi}_{ij}&= \log\left(\frac{p_{ij}}{p_{i*}p_{*j}}\right) \\[1ex] x_{ij}&= \begin{cases} \operatorname{pmi}_{ij} & \text{if } \operatorname{pmi}_{ij} > 0 \\ 0 &\text{otherwise} \end{cases} \end{align}$$

Answer 1

Out of a list of frequencies ($f_{11}=24$ occurences for $i=1$ and $j=1$, $f_{21}=37$ occurences for $i=2$ and $j=1$...), one gets probabilities with normalization by the sum of frequencies. Forget the second and third equations.

Notation $p_{i*}=\sum_{j=1}^n p_{ij}$ denotes the marginal distribution associated with the joint probability distribution $p_{ij}$ for a fixed $i$. The same for notation $p_{*j}$. If, for all $i,j$ we have $p_{i*}\times p_{*j}=p_{ij}$ it means that variables $i$ and $j$ are independant. In this case $log\dfrac{p_{ij}}{p_{i*}\times p_{*j}}=log(1)=0$.