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Unformatted text preview: gaussian identities sam roweis (revised July 1999) 0.1 multidimensional gaussian a d-dimensional multidimensional gaussian (normal) density for x is: N ( μ , Σ ) = (2 π )- d/ 2 | Σ |- 1 / 2 exp- 1 2 ( x- μ ) T Σ- 1 ( x- μ ) (1) it has entropy: S = 1 2 log 2 h (2 πe ) d | Σ | i- const bits (2) where Σ is a symmetric postive semi-definite covariance matrix and the (unfortunate) constant is the log of the units in which x is measured over the “natural units” 0.2 linear functions of a normal vector no matter how x is distributed, E[ Ax + y ] = A (E[ x ]) + y (3a) Covar[ Ax + y ] = A (Covar[ x ]) A T (3b) in particular this means that for normal distributed quantities: x ∼ N ( μ , Σ ) ⇒ ( Ax + y ) ∼ N ( A μ + y , AΣA T ) (4a) x ∼ N ( μ , Σ ) ⇒ Σ- 1 / 2 ( x- μ ) ∼ N ( , I ) (4b) x ∼ N ( μ , Σ ) ⇒ ( x- μ ) T Σ- 1 ( x- μ ) ∼ χ 2 n (4c) 1 0.3 marginal and conditional distributions let the vector z = [ x T y T ] T be normally distributed according to:...
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This note was uploaded on 09/07/2011 for the course COGS 118A taught by Professor Staff during the Fall '10 term at UCSD.
- Fall '10