Maximum likelihood estimation and Bayesian estimation are commonly used parameter estimation methods in statistics, but there are some differences between them.

Firstly, maximum likelihood estimation is a parameter estimation method based on sample data, and the value of parameters is estimated by maximum likelihood function. Bayesian estimator is a parameter estimation method based on prior probability and posterior probability, which estimates the value of parameters by calculating the maximum value of posterior probability density function.

Secondly, the maximum likelihood estimator only considers the influence of sample data on parameters, but ignores the prior information. Therefore, in practical application, the maximum likelihood estimator may be biased by the sample data. Bayesian estimator fully considers the influence of prior information and sample data on parameters, so it can estimate the values of parameters more accurately.

In addition, maximum likelihood estimation usually needs to assume that the form of distribution is known, and the value of parameters needs to be solved by optimization algorithm. Bayesian estimator can directly calculate the maximum value of posterior probability density function through Bayesian formula, thus obtaining the estimated value of parameters.

Finally, maximum likelihood estimation and Bayesian estimation have different application ranges in practical applications. Maximum likelihood estimation is suitable for simple linear model, multiple linear regression and other scenarios; Bayesian estimator is suitable for complex nonlinear models, time series analysis and other scenarios.