Graphical models for protein structure: Difference between revisions

Revision as of 08:56, 20 July 2010

Revision as of 05:55, 21 July 2010

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===Guassian graphical models of protein structures===

Gaussian Graphical Models are multivariate probability distributions encoding a network of dependencies among variables. Let <math>\Theta=[\theta_1, \theta_2, ..\dots, \theta_n]</math> be a set of <math>n</math> variables, such as <math>n</math> [[dihedral angles]], and let <math>f(\Theta=D)</math> be the value of the [[probability density function]] at a particular value ''D''. A multivariate Gaussian graphical model defines this probability as follows:

:<math>f(\Theta=D) = \frac{1}{Z} \exp\left\{-\frac{1}{2}(D-\mu)^T\Sigma^{-1}(D-\mu)\right\}</math>

Where <math>Z = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}</math> is the closed form for the [[partition function]]. The parameters of this distribution are <math>\mu</math> and <math>\Sigma</math>. <math>\mu</math> is the vector of [[mean values]] of each variable, and <math>\Sigma^{-1}</math>, the inverse of the [[covariance matrix]], also known as the [[precision matrix]]. Precision matrix contains the pairwise dependencies between the variables. A zero value in <math>\Sigma^{-1}</math> means that conditioned on the values of the other variables, the two corresponding variable are independent of each other.