Bayesian multivariate linear regression

In statistics, Bayesian multivariate linear regression is aBayesian 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. In statistics, Bayesian multivariate linear regression is aBayesian 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 bepredicted is not a single real-valued scalar but an m-length vectorof correlated real numbers. As in the standard regression setup, thereare n observations, where each observation i consists of k-1explanatory variables, grouped into a vector x i {displaystyle mathbf {x} _{i}} of length k (where a dummy variable with a value of 1 has beenadded to allow for an intercept coefficient). This can be viewed as aset of m related regression problems for each observation i: where the set of errors { ϵ i , 1 , … , ϵ i , m } {displaystyle {epsilon _{i,1},ldots ,epsilon _{i,m}}} are all correlated. Equivalently, it can be viewed as a single regressionproblem where the outcome is a row vector y i T {displaystyle mathbf {y} _{i}^{ m {T}}} and the regression coefficient vectors are stacked next to each other, as follows: The coefficient matrix B is a k × m {displaystyle k imes m} matrix where the coefficient vectors β 1 , … , β m {displaystyle {oldsymbol {eta }}_{1},ldots ,{oldsymbol {eta }}_{m}} for each regression problem are stacked horizontally: The noise vector ϵ i {displaystyle {oldsymbol {epsilon }}_{i}} for each observation iis jointly normal, so that the outcomes for a given observation arecorrelated: