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Bayesian linear regression : ウィキペディア英語版
Bayesian linear regression

In statistics, Bayesian linear regression is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. When the regression model has errors that have a normal distribution, and if a particular form of prior distribution is assumed, explicit results are available for the posterior probability distributions of the model's parameters.
==Model setup==

Consider a standard linear regression problem, in which for i=1,...,n we specify the conditional distribution of ''y_i'' given a ''k \times 1'' predictor vector ''\mathbf_i'':
:y_ = \mathbf_^ \boldsymbol\beta + \epsilon_,
where ''\boldsymbol\beta'' is a ''k \times 1'' vector, and the \epsilon_i are independent and identical normally distributed random variables:
:\epsilon_ \sim N(0, \sigma^2).
This corresponds to the following likelihood function:
:\rho(\mathbf|\mathbf,\boldsymbol\beta,\sigma^) \propto (\sigma^)^ \exp\left(-\frac}(\mathbf- \mathbf \boldsymbol\beta)^(\mathbf- \mathbf \boldsymbol\beta)\right).
The ordinary least squares solution is to estimate the coefficient vector using the Moore-Penrose pseudoinverse:
: \hat = (\mathbf^\mathbf)^\mathbf^\mathbf
where \mathbf is the ''n \times k'' design matrix, each row of which is a predictor vector \mathbf_^; and \mathbf is the column n-vector (\; \cdots \; y_n )^.
This is a frequentist approach, and it assumes that there are enough measurements to say something meaningful about \boldsymbol\beta. In the Bayesian approach, the data are supplemented with additional information in the form of a prior probability distribution. The prior belief about the parameters is combined with the data's likelihood function according to Bayes theorem to yield the posterior belief about the parameters \boldsymbol\beta and \sigma. The prior can take different functional forms depending on the domain and the information that is available a priori.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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