Recently, I have seen a lot of Bayesian estimation problems on paper. Summarize the related knowledge of Bayesian estimation on the Internet.
Bayesian formula:
Maximum likelihood estimation:
In fact, I asked for the part framed by the red line
Maximum posterior estimate:
In fact, I asked for the part framed by the red line. There is one more parameter probability than the maximum likelihood estimate, and the parameter is also considered to have probability. Maximum a posteriori estimation (MAP), the biggest difference between it and maximum likelihood estimation is that it considers the distribution of the parameters themselves, which is the prior distribution.
Bayesian estimate:
At this time, the value of the parameter is not directly estimated, but the parameter is allowed to follow a certain probability distribution. That is, p (x) is also required.
Bayesian School:
The newly observed sample information will correct people's previous perceptions of things. In other words, before the new sample information is obtained, people's cognition is a priori distribution. After the new sample information X is obtained, people's cognition is a posterior distribution.
Disagreement between frequency and Bayesian school:
The frequency school believes that the parameter exists objectively and will not change. Although it is unknown, it is a fixed value; the Bayesian school believes that the parameter is a random value, because it is not observed, then it is no different from a random number, so the parameter There can also be distribution.
The frequency school cares most about the likelihood function, while the Bayesian school cares most about the posterior distribution. We will find that the posterior distribution is actually the likelihood function multiplied by the prior distribution and normalized to integrate it to 1. Therefore, many methods of the two are the same.
