Maximum likelihood estimation for solving polynomial distribution parameters

The author: Hefei University School of Management Chanyang email:1563178220@qq.com content may not have, welcome to exchange.

no reprint without my permission . cause

This evening, when the teacher was watching Lda's math gossip, ask me a question, as shown in the following picture:



The polynomial distribution of the parameters, the use of a great estimate is how to ask for it? At that time, I did not know, so I found the information on the Internet, study a bit, hereby recorded.

Formula derivation

In many cases, it is assumed that a variable x x has K-K states, where K>2 k>2, each state assumes a probability of p1,p2,⋯,pk p_{1},p_{2},\cdots, p_{k}, and ∑ki=1pi=1 \sum _{i=1}^{k}p_{i }=1, independent of N-n experiments, with N1,n2,⋯,nk n_{1},n_{2},\cdots, n_{k} indicates the number of occurrences of each state, the number of occurrences follows the polynomial distribution:
P (N1,N2,⋯,NK|P1,P2,⋯,PK) =n!∏ki=1ni!∏i=1kpnii p\left (N_{1},n_{2},\cdots, N_{k}|p_{1},p_{2},\cdots, p_{k} \right) =\ frac{n!} {\prod _{i=1}^{k}n_{i}!} \prod _{i=1}^{k}p_{i}^{n_{i}}

The following maximum likelihood solution is used:

L (P1,P2,⋯,PK