# 2.3.5 - Fusing and Partitioning Cells

2.3.5 - Fusing and Partitioning Cells

We can collapse a multinomial vector by fusing cells (i.e. by adding some of the cell counts $$X_j$$ together). If

$$X=(X_1,\ldots,X_k)\sim Mult(n,\pi)$$

where $$\pi = \left(\pi_1, \dots , \pi_k\right)$$, then

$$X^\ast=(X_1+X_2,X_3,X_4,\ldots,X_k)$$

is also multinomial with the same index $$n$$ and modified parameter $$\pi* = \left(\pi_1 + \pi_2, \pi_3, \dots , \pi_k\right)$$. In the multinomial experiment, we are simply fusing the events $$E_1$$ and $$E_2$$ into the single event "$$E_1$$ or $$E_2$$". Because these events are mutually exclusive,

$$P(E_1\text{ or }E_2)=P(E_1)+P(E_2)=\pi_1+\pi_2$$

We can also partition the multinomial by conditioning on (treating as fixed) the totals of subsets of cells. For example, consider the conditional distribution of $$X$$ given that...

$$X_1+X_2=z$$

$$X_3+X_4+\cdots+X_k=n-z$$

The subvectors $$\left(X_1, X_2\right)$$ and $$\left(X_3, X_4, \dots, X_k \right)$$ are conditionally independent and multinomial,

$$(X_1,X_2)\sim Mult\left[z,\left(\dfrac{\pi_1}{\pi_1+\pi_2},\dfrac{\pi_2}{\pi_1+\pi_2}\right)\right]$$

$$(X_3,\ldots,X_k)\sim Mult\left[n-z,\left(\dfrac{\pi_3}{\pi_3+\cdots+\pi_k},\cdots,\dfrac{\pi_k}{\pi_3+\cdots+\pi_k}\right)\right]$$

The joint distribution of two or more independent multinomials is called the "product-multinomial." If we condition on the sums of non-overlapping groups of cells of a multinomial vector, its distribution splits into the product-multinomial. The parameter for each part of the product-multinomial is a portion of the original $$\pi$$ vector, normalized to sum to one.

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