Changes between Version 21 and Version 22 of OfficialTolArchiveNetworkBysPrior

Timestamp:

Jun 14, 2011, 8:08:56 AM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysPrior

@@ -115 +115 @@
 we can express telling that it have a normal distribution with average in these value.
 This type of prior knowledge can be extended to higher dimensions by the multinormal
-distribution
+distribution over a linear combination of variables
 
- [[LatexEquation(  x\sim N\left(\mu,\Sigma\right) )]] 
+ [[LatexEquation(  C x\sim N\left(\mu,\Sigma\right) )]] 
+ [[LatexEquation(  C\in\mathbb{R}^{n\times k}\wedge\mu\in\mathbb{R}^{k}\wedge\Sigma\in\mathbb{R}^{k\times k}\wedge\mathrm{rank}\left(\Sigma\right)=k )]] 
 
 which likelihood function is
 
- [[LatexEquation(  lk\left(x\right)=\frac{1}{\left(2\pi\right)^{n}\left|\Sigma\right|^{\frac{1}{2}}}e^{^{-\frac{1}{2}\left(x-\mu\right)^{T}\Sigma^{-1}\left(x-\mu\right)}} )]]
+ [[LatexEquation(  lk\left(x\right)=\frac{1}{\left(2\pi\right)^{k}\left|\Sigma\right|^{\frac{1}{2}}}e^{^{-\frac{1}{2}\left(Cx-\mu\right)^{T}\Sigma^{-1}\left(Cx-\mu\right)}} )]]
 
 The log-likelihood is
  
- [[LatexEquation( L\left(x\right)=\ln\left(lk\left(x\right)\right)=-\frac{n}{2}\ln\left(2\pi\right)-\frac{1}{2}\ln\left(\left|\Sigma\right|\right)-\frac{1}{2}\left(x-\mu\right)^{T}\Sigma^{-1}\left(x-\mu\right) )]]
+ [[LatexEquation( L\left(x\right)=\ln\left(lk\left(x\right)\right)=-\frac{k}{2}\ln\left(2\pi\right)-\frac{1}{2}\ln\left(\left|\Sigma\right|\right)-\frac{1}{2}\left(Cx-\mu\right)^{T}\Sigma^{-1}\left(Cx-\mu\right) )]]
  
 The gradient is
 
- [[LatexEquation( \left(\frac{\partial L\left(x\right)}{\partial x_{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(Cx-\mu\right) )]]
 
 and the hessian