Changes between Version 1 and Version 2 of OfficialTolArchiveNetworkBysPrior

Timestamp:

Dec 24, 2010, 3:23:01 PM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysPrior

@@ -20 +20 @@
 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 and hessian must 
+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 but this an admisible restricion in almost all cases.
+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 ==
@@ -46 +56 @@
 
 === Polytope prior ===
-A polytope is defined by a system of arbitrary linear inequalities
+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 ==
+
+