= BysVecLinReg = 

BysVecLinReg is an open source [wiki:TolPackageRulesAndComments TOL Package] published as partt of [wiki:OfficialTolArchiveNetwork Official Tol Archive Network]

BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with 
arbitrary constraining inequations and lineal constraining equations.

The method used to solve it in this package is based on 
[https://www.tol-project.org/export/HEAD/tolp/OfficialTolArchiveNetwork/BysVecLinReg/doc/bayes-linear-minka.pdf Bayesian linear regression Thomas Minka (2001)] using invariant scale prior over 
[[LatexEquation(A)]] and inverse prior over [[LatexEquation(V)]]

== Vectorial linear regression ==

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]]

Minka defines also the known data pair [[LatexEquation(D = left(Y,Xright))]] 
that will be used just to get more compact conditioninig expressions.

== Arbitrary constraining inequations ==

We will extend the model scope with arbitrary non null meassured restrictions 
over parameters inside [[LatexEquation(A)]] by means of adding a set of 
[[LatexEquation(r)]] inequations defining a 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 arbitrary constraining function. [[BR]]

== Invariant-scale prior over coefficient matrix ==

Although Minka not explicitly stated in any place, under the invariant prior
follows that [[LatexEquation(X)]] must be full-rank [[LatexEquation(m <= N)]]
because [[LatexEquation(X W X ^ T)]] must be nonsingular with
[[LatexEquation(W = \alpha I_{m})]], where [[LatexEquation(\alpha)]] is the
scale-invariant parameter governing the prior and estimated more
forward to maximize the evidence of the data, which depends on the assumptions
the model.