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= Package BysPrior =

BysPriorInf stands for Bayesian Prior Information and allows to define prior 
information handlers to be used in estimation systems (max-likelihood and 
bayesian ones).

A prior is a distribution function over a subset of the total set of variables 
of a model that expresses the knowledge about the phenomena behind the model.

The effect of a prior is to add the logarithm of its likelihood to the 
logarithm of the likelihood of the global model. So it can be two or 
more priors over some variables. For example, in order to stablish a 
truncated normal we can define a uniform over the feasible region and 
an unconstrainined normal.

In order to be estimated with [wiki:OfficialTolArchiveNetworkNonLinGloOpt NonLinGloOpt] 
(max-likelihood) and [wiki:OfficialTolArchiveNetworkBysSampler  BysSampler ]
(Bayesian sampler), each prior must define methods to calculate the logarithm
of the likelihood (except an additive constant), its gradient and its hessian,
and an optional set of constraining inequations, in order to define the feasible 
region. Each inequation can be linear or not and the gradient must 
be also calculated. Note that this implies that priors should be continuous and 
two times differentiable and restrictions must be continuous and differerentiable,
but this an admisible restricion in almost all cases.

== Non informative priors ==

Let [[LatexEquation( \beta )]] a uniform random variable in a region 
[[LatexEquation(\Omega\in\mathbb{R}^{n} )]] which likelihood function is [[BR]]

[[LatexEquation(lk\left(\beta\right) \propto 1 )]]

Since the logarithm of the likelihood but a constant is zero, when 
log-likelihood is not defined for a prior, the default assumed will be the 
uniform distribution, also called non informative prior.

=== Domain prior ===
The easiest way, but one of the most important, to define a non informative 
prior, is to stablish a domain interval for one or more variables. 

In this cases, you mustn't to define the log-logarithm nor the constraining 
inequation functions, but simply it's needed to fix the lower and upper 
bounds:[[BR]][[BR]]

[[LatexEquation( \beta\in\Omega\Longleftrightarrow l_{k}\leq\beta_{i_k}\leq u_{k}\wedge-\infty\leq l_{k}<u_{k}\leq\infty\forall k=1\ldots r )]]

If both lower and upper bounds are non finite, then we call it the neutral 
prior, that is equivalent to don't define any prior.

=== Polytope prior ===
A polytope prior is defined by a system of compatible linear inequalities [[BR]]

[[LatexEquation( A\beta\leq a\wedge A\in\mathbb{R}^{r\times n}\wedge a\in\mathbb{R}^{r} )]]

An special and common case of polytope region is the defined by order relations like

[[LatexEquation( \beta_{i}}\leq\beta_{j}})]]

We can implement this type of prior by means of a set of [[LatexEquation( r )]] 
inequations but, since [wiki:OfficialTolArchiveNetworkNonLinGloOpt NonLinGloOpt] 
doesn't have any special behaviour for linear inequations, it could be an 
inefficient implementation. However we can define just one non linear inequation 
that is equivalent to the full set of linear inequations. If we define 

[[LatexEquation( d\left(\beta\right)=A\beta-a=\left(d_{k}\left(\beta\right)\right)_{k=1\ldots r} )]]

then

[[LatexEquation( D_{k}\left(\beta\right)=\begin{cases} 0 & \forall d_{k}\left(\beta\right)\leq0\\ d_{k}\left(\beta\right) & \forall d_{k}\left(\beta\right)>0\end{cases} )]]

is a continuous function in [[LatexEquation( \mathbb{R}^{n}  )]] and

[[LatexEquation( D_{k}^{3}\left(\beta\right)=\begin{cases} 0 & \forall d_{k}\left(\beta\right)\leq0\\ d_{k}^{3}\left(\beta\right) & \forall d_{k}\left(\beta\right)>0\end{cases}  )]]

is a continuous and differentiable in [[LatexEquation( \mathbb{R}^{n} )]]

[[LatexEquation( \frac{\partial D_{k}^{3}\left(\beta\right)}{\partial\beta_{i}}=\begin{cases} 0 & \forall d_{k}\left(\beta\right)\leq0\\ 3d_{k}^{2}\left(\beta\right)A_{ki} & \forall d_{k}\left(\beta\right)>0\end{cases}  )]]

The feasibility condition can then be defined as a single continuous nonlinear 
inequality and differentiable everywhere

[[LatexEquation( g\left(\beta\right)=\underset{k=1}{\overset{r}{\sum}}D_{k}^{3}\left(\beta\right)\leq0 )]]

The gradient of this function is

[[LatexEquation( \frac{\partial g\left(\beta\right)}{\partial\beta_{i}}=3\underset{k=1}{\overset{r}{\sum}}D_{k}^{2}\left(\beta\right)A_{ki} )]]

== Multinormal prior ==


== Inverse chi-square prior ==