Changes between Version 9 and Version 10 of OfficialTolArchiveNetworkBysPrior

Timestamp:

Dec 26, 2010, 12:46:58 AM (15 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysPrior

@@ -78 +78 @@
 that is equivalent to the full set of linear inequations. 
 
-||If we define [[BR]][[BR]] [[LatexEquation( d\left(x\right)=Ax-a=\left(d_{k}\left(x\right)\right)_{k=1\ldots r} )]] then [[br]][[br]] [[LatexEquation( D_{k}\left(x\right)=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ d_{k}\left(x\right) & \forall d_{k}\left(x\right)>0\end{cases} )]] [[br]][[br]] is a continuous function in [[LatexEquation( \mathbb{R}^{n}  )]] and [[br]][[br]] [[LatexEquation( D_{k}^{3}\left(x\right)=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ d_{k}^{3}\left(x\right) & \forall d_{k}\left(x\right)>0\end{cases}  )]] [[br]][[br]] is continuous and differentiable in [[LatexEquation( \mathbb{R}^{n} )]] [[br]][[br]] [[LatexEquation( \frac{\partial D_{k}^{3}\left(x\right)}{\partialx_{i}}=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ 3d_{k}^{2}\left(x\right)A_{ki} & \forall d_{k}\left(x\right)>0\end{cases}  )]] || [[Image(source:/tolp/OfficialTolArchiveNetwork/BysPrior/doc/image004.png)]] ||
+||If we define [[BR]][[BR]] [[LatexEquation( d\left(x\right)=Ax-a=\left(d_{k}\left(x\right)\right)_{k=1\ldots r} )]] then [[br]][[br]] [[LatexEquation( D_{k}\left(x\right)=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ d_{k}\left(x\right) & \forall d_{k}\left(x\right)>0\end{cases} )]] [[br]][[br]] is a continuous function in [[LatexEquation( \mathbb{R}^{n}  )]] and [[br]][[br]] [[LatexEquation( D_{k}^{3}\left(x\right)=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ d_{k}^{3}\left(x\right) & \forall d_{k}\left(x\right)>0\end{cases}  )]] [[br]][[br]] is continuous and differentiable in [[LatexEquation( \mathbb{R}^{n} )]] [[br]][[br]] [[LatexEquation( \frac{\partial D_{k}^{3}\left(x\right)}{\partial x_{i}}=\begin{cases} 0 & \forall d_{k}\left(x\right)\leq0\\ 3d_{k}^{2}\left(x\right)A_{ki} & \forall d_{k}\left(x\right)>0\end{cases}  )]] || [[Image(source:/tolp/OfficialTolArchiveNetwork/BysPrior/doc/image004.png)]] ||
 
 The feasibility condition can be defined as a single nonlinear 
@@ -87 +87 @@
 The gradient of this function is
 
-[[LatexEquation( \frac{\partial g\left(x\right)}{\partialx_{i}}=3\underset{k=1}{\overset{r}{\sum}}D_{k}^{2}\left(x\right)A_{ki} )]]
+[[LatexEquation( \frac{\partial g\left(x\right)}{\partial x_{i}}=3\underset{k=1}{\overset{r}{\sum}}D_{k}^{2}\left(x\right)A_{ki} )]]
 
 == Random priors ==
@@ -113 +113 @@
 The gradient is
 
- [[LatexEquation( \left(\frac{\partial L\left(x\right)}{\partialx_{i}}\right)_{i=1\ldots n}=-\Sigma^{-1}\left(x-\mu\right) )]]
+ [[LatexEquation( \left(\frac{\partial L\left(x\right)}{\partial x_{i}}\right)_{i=1\ldots n}=-\Sigma^{-1}\left(x-\mu\right) )]]
 
 and the hessian 
 
- [[LatexEquation( \left(\frac{\partial^{2}L\left(x\right)}{\partialx_{i}\partialx_{j}}\right)_{i,j=1\ldots n}=-\Sigma^{-1} )]]
+ [[LatexEquation( \left(\frac{\partial^{2}L\left(x\right)}{\partial x_{i}\partial x_{j}}\right)_{i,j=1\ldots n}=-\Sigma^{-1} )]]
  
 
@@ -168 +168 @@
 If we know the first and second derivatives of the transformation
 
-[[LatexEquation( \frac{\partial\gamma_{k}}{\partialx_{i}}  )]]
+[[LatexEquation( \frac{\partial\gamma_{k}}{\partial x_{i}}  )]]
 
-[[LatexEquation( \frac{\partial^{2}\gamma_{k}}{\partialx_{i}\partialx_{j}}  )]]
+[[LatexEquation( \frac{\partial^{2}\gamma_{k}}{\partial x_{i}\partial x_{j}}  )]]
 
 the we can calculate the original gradient and the hessian after the gradient 
 and the hessian of the transformed prior as following
 
-[[LatexEquation( \frac{\partial L\left(x\right)}{\partialx_{i}}=\underset{k=1}{\overset{K}{\sum}}\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial\gamma_{k}}{\partialx_{i}}  )]]
+[[LatexEquation( \frac{\partial L\left(x\right)}{\partial x_{i}}=\underset{k=1}{\overset{K}{\sum}}\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial\gamma_{k}}{\partial x_{i}}  )]]
 
-[[LatexEquation( \frac{\partial L^{2}\left(x\right)}{\partialx_{i}\partialx_{j}}=\underset{k=1}{\overset{K}{\sum}}\left(\frac{\partial^{2}L^{*}\left(\gamma\right)}{\partial\gamma_{k}\partialx_{j}}\frac{\partial\gamma_{k}}{\partialx_{i}}+\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial^{2}\gamma_{k}}{\partialx_{i}\partialx_{j}}\right)=\underset{k=1}{\overset{K}{\sum}}\left(\frac{\partial^{2}L^{*}\left(\gamma\right)}{\partial\gamma_{k}\partial\gamma_{k}}\frac{\partial\gamma_{k}}{\partialx_{i}}\frac{\partial\gamma_{k}}{\partialx_{j}}+\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial^{2}\gamma_{k}}{\partialx_{i}\partialx_{j}}\right)  )]]
+[[LatexEquation( \frac{\partial L^{2}\left(x\right)}{\partial x_{i}\partial x_{j}}=\underset{k=1}{\overset{K}{\sum}}\left(\frac{\partial^{2}L^{*}\left(\gamma\right)}{\partial\gamma_{k}\partial x_{j}}\frac{\partial\gamma_{k}}{\partial x_{i}}+\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial^{2}\gamma_{k}}{\partial x_{i}\partial x_{j}}\right)=\underset{k=1}{\overset{K}{\sum}}\left(\frac{\partial^{2}L^{*}\left(\gamma\right)}{\partial\gamma_{k}\partial\gamma_{k}}\frac{\partial\gamma_{k}}{\partial x_{i}}\frac{\partial\gamma_{k}}{\partial x_{j}}+\frac{\partial L^{*}\left(\gamma\right)}{\partial\gamma_{k}}\frac{\partial^{2}\gamma_{k}}{\partial x_{i}\partial x_{j}}\right)  )]]
 
 Thus it is possible to define a variety of information a priori from a