Model Setup
Consider a standard linear regression problem, in which for we specify the conditional distribution of given a predictor vector :
where is a vector, and the is independent and identical normally distributed random variables:
This corresponds to the following likelihood function:
The ordinary least squares solution is to estimate the coefficient vector using the Moore-Penrose pseudoinverse:
where is the design matrix, each row of which is a predictor vector ; and is the column -vector .
This is a frequentist approach, and it assumes that there are enough measurements to say something meaningful about . 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 and . The prior can take different functional forms depending on the domain and the information that is available a priori.
Read more about this topic: Bayesian Linear Regression
Famous quotes containing the word model:
“She represents the unavowed aspiration of the male human being, his potential infidelity—and infidelity of a very special kind, which would lead him to the opposite of his wife, to the “woman of wax” whom he could model at will, make and unmake in any way he wished, even unto death.”
—Marguerite Duras (b. 1914)