The Bounded distribution is a method that greatly simplifies Bayesian analysis. The function of this function is to make the unknown parameters of these distributions give physical meaning before the test when the Bayesian formula contains multiple probability distributions, and continue to the test for convenience of analysis.
1. Bayesian FormulaThe Bayesian formula is as follows: it indicates an unknown parameter in the model and a sample. There are three important concepts:
Prior Distribution,
Likelihood Function, And
Posterior distribution.
It is a prior distribution, indicating that a probability distribution is considered as fit by experience before the samples are observed. For example, before throwing a coin, we think that the probability of both sides is 1/2.
It is a likelihood function that indicates the similarity between sample data and the probability model given the model parameters.
Is a posterior distribution, indicating the probability distribution of model parameters after observing a series of sample data. That is, the prior distribution is corrected, which is closer to the actual situation. In addition, because it is a sample, it is a definite value.
2. Definition of the bounded DistributionIn Bayesian formula, if the Prior Distribution and likelihood Function
So that the posterior distribution is in the same form as the prior distribution, then the Prior Distribution and the likelihood function are called the same..
3. Examples
3.1 beta distribution is bounded with binary Distribution
Beta probability functions are as follows:
It is a constant coefficient. Apart from the constant coefficient, the beta function has the same form as the two-item distribution function, that is. If the beta distribution is treated as a prior distribution and the binary distribution function as a likelihood function, the posterior distribution calculated using the Bayesian formula is in the same form as the prior distribution. Therefore, the Beta distribution is the same as the binary distribution.
3.2 Dirichlet distribution is bounded with multiple Distributions
The beta distribution is expanded to the multi-dimensional Dirichlet distribution (Dirichlet distribution). The multi-dimensional distribution is the multi-dimensional distribution. The Dirichlet distribution function is as follows:
Multiple distribution functions are as follows:
K indicates k dimension. And are both constant coefficients. They do not look at the constant coefficients. The Dirichlet function has the same form as the multiple distribution functions. Therefore, the posterior distribution calculated is the same as the prior distribution. That is, the Dirichlet distribution and multi-item distribution are bounded.
Adjoint distribution of Bayesian Formulas
