Changes between Version 23 and Version 24 of OfficialTolArchiveNetworkBysPrior

Timestamp:

Jun 14, 2011, 3:50:52 PM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysPrior

@@ -136 +136 @@
  [[LatexEquation( \left(\frac{\partial^{2}L\left(x\right)}{\partial x_{i}\partial x_{j}}\right)_{i,j=1\ldots n}=-C\Sigma^{-1}C^{T} )]]
 
+==== Hierarquical relation of simple homogenity ====
+
+If we want to express that a certain group of variables [[LatexEquation( \left\{ \beta_{i}\right\} _{i=1\ldots k} )]] are independent and normally distributed with unknow average and fixed standard deviation the usual way is to define a new latent variable [[LatexEquation( \alpha )]] representing the average
+
+   [[LatexEquation( \beta_{i}\sim N\left(\alpha,\sigma^{2}I\right) )]]
+
+If there is not posible to use a hierarquical simulation engine we can rewrite these relations removing the latent variable and setting that each variable must be around the average of the rest of them with certain covariance matrix
+
+   [[LatexEquation( \beta_{i+1}-\frac{1}{k-1}\underset{j\neq i}{\sum}\beta_{j}\sim N\left(0,\Sigma\right) )]]
+
+Then it's posible to write this as an special case of multinormal prior over a linear combination of variables taking
+
+   [[LatexEquation( C=\frac{1}{n-1}\left(\begin{array}{ccccc}n-1 & -1 & \cdots & -1 & -1\\-1 & n-1 & \ddots & \vdots & \vdots\\\vdots & \ddots & \ddots & -1 & -1\\-1 & \cdots & -1 & n-1 & -1\end{array}\right)\in\mathbb{R}^{\left(k-1\right)\times k} )]]
+
+   [[LatexEquation( \mu=C\beta=\left(\begin{array}{c}0\\0\\\vdots\\0\end{array}\right)\in\mathbb{R}^{k} )]]
+
+   [[LatexEquation( \Sigma=\sigma^{2}CC^{T}=\frac{n\sigma^{2}}{\left(n-1\right)^{2}}\left(\begin{array}{cccc}n-1 & -1 & \cdots & -1\\-1 & n-1 & \ddots & \vdots\\\vdots & \ddots & \ddots & -1\\-1 & \cdots & -1 & n-1\end{array}\right)\in\mathbb{R}^{\left(k-1\right)\times\left(k-1\right)} )]]
+
 
 === Transformed prior ===