🦁

Multinomial distributions

2024/01/28に公開

The number of trials N, \mathbf{x} = (x_v)_{v=1}^V, x_v represents the number of v occuring, \phi_v represents the probability of v occuring.

\begin{align*} \mathrm{Mult}(x|N, \set{\phi_v}_{v=1}^V) = \frac{N!}{x_1! x_2! \dots x_V!} \prod_{v=1}^V \phi_v^{x_v} \end{align*}
\begin{align*} E[x_v] &= N \phi_v \\ V[x_v] &= N \phi_v (1 - \phi_v) \end{align*}

Categorical distributions are defined as follows:

\begin{align*} \mathrm{Cat}(x|\set{\phi_v}_{v=1}^V) = \phi_x ,\quad x \in \set{1, 2, \dots, V} \end{align*}

The relationships between multinomial distributions and categorical and binomial distributions are as follows:

\begin{align*} \mathrm{Mult}(x|1, \set{\phi_v}_{v=1}^V) &= \mathrm{Cat}(x|\set{\phi_v}_{v=1}^V) \\ \mathrm{Mult}(x|N, \set{\phi_v}_{v=1}^2) &= \mathrm{Bin}(x|N, \phi_1) \\ \end{align*}

Discussion