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


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...



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.