The discrete multimodal distribution k is defined when the distribution has several modes, which can be shown as distinct peaks in the probability density function. The multivariate probability density function of a multimodal distribution can be written as follows:

$p\left(k\right)=\frac{k!}{k_1!\cdots k_N!}{\displaystyle \underset{i=1}{\overset{N}{\prod }}{y}_i^{k_i}},{\displaystyle \underset{i=1}{\overset{N}{\sum }}{k}_i=N}$

where k ∈ W ^D× 1 is a random vector which consists of whole numbers ( k _i ∈ 0,1,2,…) and p _i is the probability of i’th element, hence 0 ≤ p _i ≤ 1 and

${\displaystyle \underset{i=1}{\overset{N}{\sum }}{p}_i=1}.$