I. DefinitionMaximum Likelihood Estimation is a method for estimating model parameters based on samples. The idea is that for a known sample, assume that it is subject to a certain model and estimates the unknown parameters in the model, so that the model has the highest probability of such samples. In this way, the estimated value of the unknown parameter is obtained.
Ii. ProcessFor example, we need to calculate the national population's weight. First, we assume that the national population's weight follows a normal distribution, but the mean and variance are unknown. Since we do not have so much manpower and material resources to make statistics, we can use the maximum likelihood estimation method to evaluate the mean and variance of this normal distribution.
1. List likelihood FunctionsAssume that the samples are independent and the same distribution, and the probability density function of the normal distribution is represented by unknown parameters, then this model can be expressed:
The likelihood function is expressed:
2. logarithm of the likelihood FunctionIn actual calculation, take the logarithm of the above two sides for convenience:
It is called the log likelihood. What we call the maximum likelihood is actually the maximum logarithm likelihood:
3. DerivativeSo when can we get the maximum value? The simplest method is to find the derivative and solve the likelihood function equation:
Maximum Likelihood Estimation)
