Move Score and Move Information of the Multinomial Probability Mass Function

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, let $l\in\mathbb{N}$ with $l\ge2$, and let $\mathbb{R}^l$ be Euclidean space; points $k\in\mathbb{R}^l$ are written $k=(k_1,\dots,k_l)$ with subscript coordinates, and $\delta_\gamma$ denotes the $\gamma$th standard basis vector as in Probability Simplex.

Let $\bar S=(\bar S_1,\dots,\bar S_l)$ be a point of the probability simplex $\Delta^l$ with $\bar S_\gamma>0$ for every $\gamma$, and put $s_{\min}=\min_{\gamma\in\{1,\dots,l\}}\bar S_\gamma$. For $n\in\mathbb{N}$ let $p_n=p_{n,\bar S}:\mathbb{R}^l\to\mathbb{R}$ be the multinomial probability mass function with $n$ trials and cell probabilities $\bar S$, and let $\mathsf{K}_n=\{k\in\mathbb{N}_0^{\,l}:\sum_{\gamma=1}^{l}k_\gamma=n\}$ be the set on which it is given by the multinomial formula, as in that definition.

Fix an index $\sigma\in\{1,\dots,l\}$ and put $\Gamma=\{1,\dots,l\}\setminus\{\sigma\}$, enumerated in increasing order. Let the moves be the $l-1$ points $a_\gamma=\delta_\gamma-\delta_\sigma$ ($\gamma\in\Gamma$) and let the weights be real numbers $w_\gamma$ ($\gamma\in\Gamma$), both families enumerated in the order of $\Gamma$ and regarded as moves $a$ and weights $w$ in the sense of Move Score of a Discrete Probability Mass Function (the number of moves, written $n$ in that definition, is $l-1$ here; in this statement the letter $n$ denotes only the number of trials). Define the extended weight vector $\tilde w=(\tilde w_1,\dots,\tilde w_l)\in\mathbb{R}^l$ by $\tilde w_\gamma=w_\gamma$ for $\gamma\in\Gamma$ and $\tilde w_\sigma=-\sum_{\gamma\in\Gamma}w_\gamma$, so that $\sum_{\gamma=1}^{l}\tilde w_\gamma=0$, and put

$$Q(\tilde w)=\sum_{\gamma=1}^{l}\frac{\tilde w_\gamma^{2}}{\bar S_\gamma},\qquad L(k)=\sum_{\gamma=1}^{l}\frac{\tilde w_\gamma}{\bar S_\gamma}k_\gamma\quad(k\in\mathbb{R}^l),\qquad c=\frac{\tilde w_\sigma}{\bar S_\sigma}.$$

Let $N\in\mathbb{N}$. Then the following hold.

1. (Probability mass function) $p_N$ is a discrete probability mass function on $\mathbb{R}^l$ with $\{k\in\mathbb{R}^l:p_N(k)>0\}=\mathsf{K}_N$, a finite set.

2. (Move score) For every $k\in\mathsf{K}_N$, the move score satisfies

$$\rho_{p_N,a,w}(k)=-\frac{\bar S_\sigma}{k_\sigma+1}\bigl(L(k)+c\bigr).$$

3. (Exits) For every $\gamma\in\Gamma$, $\{k\in\mathsf{K}_N:\ k+a_\gamma\notin\mathsf{K}_N\}=\{k\in\mathsf{K}_N:\ k_\sigma=0\}$, and

$$\sum_{k\in\mathsf{K}_N,\ k_\sigma=0}p_N(k)=(1-\bar S_\sigma)^{N}.$$

4. (Move information) The move information satisfies

$$\mathsf{J}(p_N;a,w)\le\frac{Q(\tilde w)}{N+1}+\frac{201\,Q(\tilde w)}{s_{\min}^{5}\,(N+1)(N+2)}.$$