@TECHREPORT{Murphy98fittinga, author = {Kevin P. Murphy}, title = {Fitting a Conditional Gaussian Distribution}, institution = {}, year = {1998} }

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Abstract

Introduction We consider the problem of nding the Maximum Likelihood (ML) estimates of the parameters of a conditional Gaussian node Y with continuous parent X and discrete parent Q, i.e., p(yjx; Q = i) = cj i j 1 2 exp 1 2 (y B i x) 0 1 i (y B i x) where c = (2) d=2 is a constant and jyj = d. The j'th row of B i is the regression vector for the j component of y given that Q = i. To allo

... covariance matrix has d(d+1) 2 parameters, we are often interested in placing restrictions on the form of this matrix (an alternative is to use a Dirichlet-Normal-Wishart prior and do MAP estimation [DeG70]). We consider the following kinds of restrictions. i is tied across all states, i.e., i = for all i. i is diagonal. i is spherical (isotropic), i.e., i = 2 i I . 2 Eqn Cov. B= Ti...

...version of this is the Kalmanslter.) Mixture of Gaussians. X does not exist, Q is hidden, and Y is observed. (The temporal version of this is an HMM with MOG outputs.) Mixture of factor analyzers [G=-=H96-=-]. i is diagonal, Q and X are hidden, Y is observed. (The temporal version of this is a switching Kalmanslter.) We need to allow the possibility of all nodes being hidden (including Y ) to handle e.g...

...stimation of the transition parameters in a switching Kalman filter model (where Y corresponds to the current hidden state, X to the previous hidden state, and Q to the current hidden switch variable =-=[Mur98b]-=-). In addition, since we are interested in estimating time-invariant dynamic models (such as Kalman filters whose parameters are tied across all time slices), we express all the estimates in terms of ...

... estimation of the transition parameters in a switching Kalmanslter model (where Y corresponds to the current hidden state, X to the previous hidden state, and Q to the current hidden switch variable =-=[Mur98b]-=-). In addition, since we are interested in estimating time-invariant dynamic models (such as Kalmanslters whose parameters are tied across all time slices), we express all the estimates in terms of ex...