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where ``\boldsymbol{Z}^{(t)}`` is the completed response matrix from conditional mean and ``\boldsymbol{M}_i^{*(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot (\boldsymbol{\Omega}^{-(t)} \boldsymbol{P}^T \boldsymbol{C}^{(t)}\boldsymbol{P}\boldsymbol{\Omega}^{-(t)})] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)``, while ``\boldsymbol{R}^{*(t)}`` is the ``n \times d`` matrix such that ``\text{vec}\ \boldsymbol{R}^{*(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Z}^{(t)} - \boldsymbol{X} \boldsymbol{B}^{(t)})``. Additionally, ``\boldsymbol{P}`` is the ``nd \times nd`` permutation matrix such that ``\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}``, where ``\boldsymbol{y}_{\text{obs}}`` and ``\boldsymbol{y}_{\text{mis}}`` are vectors of observed and missing response values, respectively, in column-major order, and the block matrix ``\boldsymbol{C}^{(t)}`` is ``\boldsymbol{0}`` except for a lower-right block consisting of conditional variance. As seen, the two MM updates are of similar form.
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where ``\boldsymbol{Z}^{(t)}`` is the completed response matrix from conditional mean and ``\boldsymbol{M}_i^{*(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot (\boldsymbol{\Omega}^{-(t)} \boldsymbol{P}^T \boldsymbol{C}^{(t)}\boldsymbol{P}\boldsymbol{\Omega}^{-(t)})] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)``, while ``\boldsymbol{R}^{*(t)}`` is the ``n \times d`` matrix such that ``\text{vec}\ \boldsymbol{R}^{*(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Z}^{(t)} - \boldsymbol{X} \boldsymbol{B}^{(t)})``. Additionally, ``\boldsymbol{P}`` is the ``nd \times nd`` permutation matrix such that ``\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}``, where ``\boldsymbol{y}_{\text{obs}}`` and ``\boldsymbol{y}_{\text{mis}}`` are vectors of observed and missing response values, respectively, in column-major order, and the block matrix ``\boldsymbol{C}^{(t)}`` is ``\boldsymbol{0}`` except for a lower-right block consisting of conditional variance. As seen, the MM updates are of similar form to the non-missing response case.
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# Special case: ``m = 2``
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When there are ``m = 2`` variance components such that ``\boldsymbol{\Omega} = \boldsymbol{\Gamma}_1 \otimes \boldsymbol{V}_1 + \boldsymbol{\Gamma}_2 \otimes \boldsymbol{V}_2``, repeated inversion of the ``nd \times nd`` matrix ``\boldsymbol{\Omega}`` per iteration can be avoided and reduced to one ``d \times d`` generalized eigen-decomposition per iteration. Without loss of generality, if we assume ``\boldsymbol{V}_2`` to be positive definite, the generalized eigen-decomposition of the matrix pair ``(\boldsymbol{V}_1, \boldsymbol{V}_2)`` yields generalized eigenvalues ``\boldsymbol{d} = (d_1, \dots, d_n)^T`` and generalized eigenvectors ``\boldsymbol{U}`` such that ``\boldsymbol{U}^T \boldsymbol{V}_1 \boldsymbol{U} = \boldsymbol{D} = \text{diag}(\boldsymbol{d})`` and ``\boldsymbol{U}^T \boldsymbol{V}_2 \boldsymbol{U} = \boldsymbol{I}_n``. Similarly, if we let the generalized eigen-decomposition of ``(\boldsymbol{\Gamma}_1^{(t)}, \boldsymbol{\Gamma}_2^{(t)})`` be ``(\boldsymbol{\Lambda}^{(t)}, \boldsymbol{\Phi}^{(t)})`` such that ``\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_1^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{\Lambda}^{(t)} = \text{diag}(\boldsymbol{\lambda^{(t)}})`` and ``\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_2^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{I}_d``, then the MM updates in each iteration become
@@ -57,7 +57,12 @@ When there are ``m = 2`` variance components such that ``\boldsymbol{\Omega} = \
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where ``\tilde{\boldsymbol{X}} = \boldsymbol{U}^T \boldsymbol{X}``, ``\tilde{\boldsymbol{Y}} = \boldsymbol{U}^T \boldsymbol{Y}``, ``\boldsymbol{L}_1^{(t)}`` is the Cholesky factor of ``\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}(\boldsymbol{D}(\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}``, ``\boldsymbol{L}_2^{(t)}`` is the Cholesky factor of ``\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}((\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}``, ``\boldsymbol{N}_1^{(t)} = \boldsymbol{D}^{1/2}\{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Lambda}^{(t)}\boldsymbol{\Phi}^{-(t)}``, and ``\boldsymbol{N}_2^{(t)} = \{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Phi}^{-(t)}``. ``\oslash`` denotes the Hadamard quotient.
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In this setting, the Fisher information matrix is equivalent to
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For the sake of completeness, we note that the EM updates become
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