Why would split observed $x$ into two unobserved r.v $z_1,z_2$ consider a way to augmenting data in EM algorithm?

Question

I am reading the materials on the EM algorithm, and I am a bit confused about the example provided on the material I am currently reading. The example is considered a classical missing data problem solving by EM algorithm. Here is the content of the material I am reading :

197 animals are distributed into four categories $$ \left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(125,18,20,34) $$ and modeled with the multinomial distribution $$ \text { multinomial }\left(n ; \frac{1}{2}+\frac{\theta}{4}, \frac{1}{4}(1-\theta), \frac{1}{4}(1-\theta), \frac{\theta}{4}\right) $$ Estimation is easier if the $x_{1}$ cell is split into two cells, so we create the augmented model $$ \left(z_{1}, z_{2}, x_{2}, x_{3}, x_{4}\right) \sim \text { multinomial }\left(n ; \frac{1}{2}, \frac{\theta}{4}, \frac{1}{4}(1-\theta), \frac{1}{4}(1-\theta), \frac{\theta}{4}\right) $$ with $x_{1}=z_{1}+z_{2}$

In the introduction section of the EM algorithm, the setting is as follow:

Consider a sample of $n$ items, where $n_{1}$ of the items are observed while $n_{2}=n-n_{1}$ items are not observable Let $Z_{j}$ 's and $X_{i}$ 's be mutually independent and $$ \begin{array}{ll} X_{1}, \ldots, X_{n_{1}} \sim g(x \mid \theta) & -\text { Observed Data } \\ Z_{1}, \ldots, Z_{n_{2}} \sim f(z \mid \theta) & -\text { Unobserved Data } \end{array} $$ $(\boldsymbol{X}, \boldsymbol{Z})$ is called the complete data, $\boldsymbol{Z}$ the augmented data, and $\boldsymbol{X}$ the incomplete data.

Here, there is a clear cut between augmented data and incomplete data. However, here I am rather confused: it seems $(x_1,x_2,x_3,x_4)$ will be the incomplete data while $(z_1,z_2,x_2,x_3,x_4)$ will be consider the complete data. I have no previous exposure to the concept of augmented data and I am very confused why simply introducing two other random variables would work.

Also, I know the pdf for multinomial distribution with four outcomes have expression $$ f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\frac{n !}{x_{1} ! x_{2} ! x_{3} ! x_{4} !} \pi_{1}^{x_{1}} \pi_{2}^{x_{2}} \pi_{3}^{x_{3}} \pi_{4}^{x_{4}} $$

if we substitute $x_1$ with $z_1+z_2$, we have $(\pi_1+c)^{z_1}(\pi_1+c)^{z_2}$ but not $\pi_1^{z_1}c^{z_2}$, also, the factorial part of $x_1!=(z_1+z_2)!\neq z_1!z_2!$ How the substitution actually work here? What makes this substitution valid?

Answer 1

Here $g(x_1,x_2,x_3,x_4)$ is the distribution associated with observed data, whereas $f(z_1,z_2,x_2,x_3,x_4)$ is the distribution associated with the complete data. So, we are not substituting the variables $z_1,z_2$ into $g$. Rather, these are two different distributions and there is no reason why $g(x_1,x_2,x_3,x_4) = f(z_1,z_2,x_2,x_3,x_4), \text{where,} \quad x_1 = z_1 + z_2$ should hold in general.

The only relation between $g$ and $f$ is that $g(x_1,x_2,x_3,x_4) = \sum_{z_1,z_2}f(z_1,z_2,x_2,x_3,x_4)$ where the summation is over all possible $z_1,z_2 \geq 0$ such that $x_1 = z_1 + z_2$.