Map (maximum a posteriori)

In statistics, the maximum posterior probability (MAP) estimation can be used for point estimation of unknown parameters. It is similar to the Fisher method of the maximum likelihood estimation maximum likelihood (ML, however, the maximum posterior probability is closely related to the prior distribution. Therefore, map can be regarded as normalization of ML estimation.

Suppose we want to use the observed VariablesXEstimate the parameter θ, and assume that the sampling distribution of X is F, then the conditional probability of X based on θ isF(X| θ) then there is a likelihood function, Estimation

 

 

It is used as the maximum likelihood estimation of θ.

Now, assuming that the prior probability of θ is known, we can treat θ as a random variable in Bayesian statistics. The posterior probability can be expressed:

 

 

 

Where G's defined domain is cosine, which is a direct application of Bayesian theorem. The maximum posterior probability based on θ is considered as the posterior distribution of random variable X:

 

In the above formula, the denominator of the posterior probability has nothing to do with θ, so it does not play any role in the optimization process. We can find that the prior distribution of θ of map is even, which is consistent with that of MLE.

 

Map calculation method:

1. When the posterior probability can form a closed table, it can be solved through the derivation method;

2. optimization through numerical computation. Common methods include the bounded gradient method and the Newton method. However, these methods often involve complicated first-and second-order derivatives;

3. the derivative of the posterior probability does not need to be calculated through the improved maximum Expectation Algorithm.

 

Rating:In Bayesian methods, methods like map that use a prior probability to solve a posterior pattern are rare. Map is a point estimation, the Bayesian method extracts data features based on the distribution. In particular, posterior probability is not a simple form of analysis. At this time, it can be simulated by Monte Carlo, but it is very difficult to obtain its distribution type.