How is MAP Computed for a Multivariate Gaussian?

[mort.motes]

The standard multivariate gaussian is given by:

http://upload.wikimedia.org/math/1/c/d/1cd250fc27ef7b7a9da469416333d07f.png

taken from:

http://en.wikipedia.org/wiki/Multivariate_normal_distribution

The parameters can be estimated using:

http://en.wikipedia.org/wiki/Estima...tion_for_the_multivariate_normal_distribution

and:

http://en.wikipedia.org/wiki/Normal_distribution#Estimation_of_parameters

But given those parameters how are MAP (maximum posterior) or the "mode" computed? Again from wiki: "The parameter μ is at the same time the mean, the median and the mode of the normal distribution"

In a bayes context MAP can somtimes be estimated as:

MAP = ML*Prior

 

[Valkarie]

where ML is the maximum likelihood estimate and Prior is the prior estimate.However, for a multivariate gaussian, it is not as straightforward as for a single variable gaussian. For a multivariate gaussian, MAP can be found by using an iterative optimization algorithm such as gradient descent. This is because the mode of a multivariate gaussian is the point at which the gradient of the log-likelihood of the model is zero.