source: tol/tol_pkg/BysVecLinReg/doc/BysVecLinReg.wiki @ 1ea1fbe

BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with arbitrary constraining inequations

Vectorial linear regression equations are [[BR]]

[[LatexEquation(Y=A \cdot X + E)]] [[BR]]

where [[BR]]

 * [[LatexEquation(Y\in\mathbb{R}^{d\times N} )]] is the multivariant known output matrix, where each row is a different output vector [[LatexEquation(y_{n}\in\mathbb{R}^{N} )]]  [[BR]]
 * [[LatexEquation(X\in\mathbb{R}^{m\times N} )]] is the known and full rank input matrix, where each row is a different input vector [[LatexEquation(x_{n}\in\mathbb{R}^{N} )]] [[BR]]
 * [[LatexEquation(A\in\mathbb{R}^{d\times m} )]] has the unknown regression coefficients that we want to estimate [[BR]]
 * [[LatexEquation(E\in\mathbb{R}^{d\times N} )]] is the multivariant residuals, where each row is the residuals vector  [[LatexEquation(e_{n}\in\mathbb{R}^{N} )]] corresponding to output [[LatexEquation(y_{n} )]] 
 
All residuals inside the same row are incorrelated normal, but resiudals in the same column [[LatexEquation(j)]] are [[BR]] 

[[LatexEquation(e_{.,j} \sim N\left(0,V\right) E\in\mathbb{R}^{d\times d} \forall j=1 \ldots d )]][[BR]]

where [[LatexEquation(V)]] is symmetric positive definite and unknown, but the same for each column[[BR]]

When there are some restriction over parameters inside [[LatexEquation(A)]] we must to add the inequations of feasible region [[BR]]

[[LatexEquation(\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\})]] [[BR]] 

being [[BR]]

[[LatexEquation( F\left(A\right):\mathbb{R}^{d\times m}\longrightarrow\mathbb{R}^{r} )]]  [[BR]]

the constraining function. [[BR]]

The method used in this package is based on [https://www.tol-project.org/export/HEAD/tolp/trunk/tol_pkg/BysVecLinReg/doc/bayes-linear-minka.pdf  Bayesian linear regression Thomas Minka (2001)]