# Gaussian Markov Random Field

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**See:** Markov Random Field, Multivariate Gaussian Distribution, Finite Random Vector.

## References

### 2009

- (Frick, 2009) ⇒ Hannah Frick. (2009). “Theory of Gaussian Markov Random Fields." Seminar Spatial Statistics, Winter 2009/2010

### 2005

- (Rue & Held, 2005) ⇒ Havard Rue, and Leonhard Held. (2005). “Gaussian Markov Random Fields: Theory and Applications." CRC Press. ISBN:1584884320
- This monograph considers
*Gaussian Markov random fields*(GMRFs) covering both theory and applications. A GMRF is really a simple construct: It is just a (finite-dimensional) random vector following a multivariate normal (or Gaussian) distribution. However, we will be concerned with more restrictive versions where the GMRF satisfies additional*conditional independence*assumption, hense the term*Markov*. Conditional independence is a powerful concept. Let [math]\displaystyle{ \mathbf{x} = (x_1,x_2,x_3)^T }[/math] be a random vector, then [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math] are conditionally independent given [math]\displaystyle{ x_3 }[/math] if, for known value of [math]\displaystyle{ x_3 }[/math], discover [math]\displaystyle{ x_2 }[/math] tells you nothing new about the distribution of [math]\displaystyle{ x_1 }[/math].

- This monograph considers