Bayesian multivariate linear regression

In statistics, Bayesian multivariate linear regression is a Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator.

Consider a regression problem where the dependent variable to be predicted is not a single real-valued scalar but an m-length vector of correlated real numbers. As in the standard regression setup, there are n observations, where each observation i consists of k-1 explanatory variables, grouped into a vector ${\displaystyle \mathbf {x} _{i}}$ of length k (where a dummy variable with a value of 1 has been added to allow for an intercept coefficient). This can be viewed as a set of m related regression problems for each observation i:

where the set of errors ${\displaystyle \{\epsilon _{i,1},\ldots ,\epsilon _{i,m}\}}$ are all correlated. Equivalently, it can be viewed as a single regression problem where the outcome is a row vector ${\displaystyle \mathbf {y} _{i}^{\rm {T}}}$ and the regression coefficient vectors are stacked next to each other, as follows:

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