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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 NonLinGloOpt (max-likelihood) and 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.

== Chained priors ==

A prior can depend on a set of parameters that can be defined as constant 
values or another subset of the model variables. For example we can define 
hierarquical structures among the variables using a latent variable that is
the average of a normal prior for a subset of variables. We also can consider
that the varianze of these normal prior is another variable and to define 
an inverse chi-square prior over this one.

== 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 non informative 
prior information 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\leq u_{k}\wedge-\infty\leq l_{k}<u_{k}\leq\infty )]]

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

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

We can define this type of prior bye means of a set of [[LatexEquation( r )]] 
inequations due NonLinGloOpt doesn't have any special behaviour for linear 
inequations, and 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 twice 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 ==

== Scalar bounded normal prior ==

== Inverse chi-square prior ==