This article is a summary of the research on "ML estimation in the bivariate passion distribution in the presence of missing values via the em algorithm" by K. Adamids & S. Loukas (1994).
Some time ago, I studied the estimation of the parameters of the joint normal distribution with missing data. It should be said that the research on the continuity assumption is not complete. Recently, I started to study a discrete distribution assumption - the parameter estimation method of the joint Poisson distribution, although It is difficult to find cases that obey the joint Poisson distribution in real life, but research for theoretical completeness is still necessary.
The EM algorithm has been used in early research as an effective method for estimating parameters with latent variables, typical applications such as mixture Gaussian distribution models. However, the definitions of latent variables are different for different distributions. Taking this paper as an example, the joint Poisson distribution is defined as:
assuming that X', Y', U are independent Poisson random variables with parameters a, b, d, respectively, And it satisfies:
and
at step E:
Assuming that
meets
then
at step M, we can obtain
by solving the maximum likelihood estimate of the probability density function: d sets an initial value, and brings it into the equation of step E to get the expectations of s1, s2, and s3, and then brings it into step M to update the values of the three parameters. Iterate like this until convergence. Assuming a=b=2, d=1, through experiments, we can get e-5 as the convergence condition, it can be seen that the final estimated result a=2.053, b=041, c=0.965, which is still very close to the original value of.
