Fisher matrix for multivariate normal distributionby hdb Tags: distribution, fisher, matrix, multivariate, normal 

#1
Mar1711, 01:47 PM

P: 3

The fisher information matrix for multivariate normal distribution is said at many places to be simplified as:
[tex]\mathcal{I}_{m,n} = \frac{\partial \mu^\mathrm{T}}{\partial \theta_m} \Sigma^{1} \frac{\partial \mu}{\partial \theta_n}.\ [/tex] even on http://en.wikipedia.org/wiki/Fisher_...l_distribution I am trying to come up with the derivation, but no luck so far. Does anyone have any ideas / hints / references, how to do this? Thank you 



#2
May1811, 10:47 PM

P: 5

Using matrix derivatives one has [tex] D_x(x^T A x) = x^T(A+A^T) [/tex] from which it follows that [tex] D_{\theta} \log p(z ; \mu(\theta) , \Sigma) = (z\mu(\theta))^T \Sigma^{1} D_{\theta} \mu(\theta) [/tex] For simplicity let's write [tex] D_{\theta} \mu(\theta) = H [/tex] The FIM is then found as [tex] J = E[ ( D_{\theta} \log p(z ; \mu(\theta) , \Sigma))^T D_{\theta} \log p(z ; \mu(\theta) , \Sigma)] = E[ H^T R^{1} (z  \mu(\theta))^T (z  \mu(\theta)) R^{1} H] = H^T R^{1} R R^{1} H = H^T R^{1} H [\tex] which is equivalent to the given formula. Notice that this formula only is valid as long as [tex] \Sigma [\tex] does not depend on [tex] \theta [\tex]. I'm still struggling to find a derivation of the more general case where also [tex] \Sigma [\tex] depends on [tex] \theta [\tex].
For some reason my tex code is not correctly parsed. I cannot understand why. 



#3
May1811, 11:00 PM

P: 5

Actually the general proof can apparently be found in Porat & Friedlander: Computation of the Exact Information Matrix of Gaussian Time Series with Stationary Random Components, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol ASSP34, No. 1, Feb. 1986.




#4
May1911, 08:16 PM

P: 2,490

Fisher matrix for multivariate normal distribution 



#5
May2011, 03:56 AM

P: 3

Klein, A., and H. Neudecker. “A direct derivation of the exact Fisher information matrix of Gaussian vector state space models.” Linear Algebra and its Applications 321, no. 13 


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