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The second variational energy concerns the population precision �� and is given by equation(32) I2��=lnpk�̦˦�q�̦� equation(33) =m2ln��?��2��j=1m�̦�j?�̦�2+�Ǧ�j?1+�Ǧ�?1+a0?1ln��?��b0+cwhere c represents a term that is constant with respect to ��. The above expression already has the form of a log-Gamma distribution with parameters equation(34) a��=a0+12mand equation(35) b��=1b0+12��j=1m�̦�j?�̦�2+�Ǧ�j?1+�Ǧ�?1?1. From this we can see that the shape parameter a ?�� is a weighted sum of prior shape a ?0 and data m ?. When viewing the second parameter as a ��rate�� coefficient b ?��??1 (as opposed to a shape coefficient b ?��), it becomes clear that the posterior rate really is a weighted sum of: the prior rate (b ?0??1); the dispersion of subject-specific means; their variances ( �Ǧ�j?1); and our uncertainty about the population mean (�Ǧ�??1). The S1PR1 variational energy of the third partition concerns the model parameters representing subject-specific latent accuracies. This energy is given by equation(36) I3��=lnpk�̦˦�q�̦� equation(37) =��j=1mkjln�Ҧ�j+nj?kjln1?�Ҧ�j?12a��b�˦�j?�̦�2+c. Rigosertib Since an analytical expression for the maximum of this energy does not exist, we resort to an iterative Newton�CRaphson scheme based on a quadratic Taylor-series approximation to the variational energy I3(��). For this, we begin by considering the Jacobian equation(38) dI3��d��j=?I3��?��j=kj?nj�Ҧ�j+a��b�˦̦�?��and the Hessian equation(39) d2I3��d��2jk=?2I3��?��j?��k=?��jknj�Ҧ�j1?�Ҧ�j+a��b��where the Kronecker delta operator ��jk is 1 if j?=?k and 0 otherwise. As noted before, the absence of off-diagonal elements in the Hessian is not based on an assumption of conditional independence selleckchem of subject-specific posteriors; it is a consequence of the mean-field separation in Eq. (11). Each GN iteration performs the update equation(40) ��*����*?d2I3��d��2��=��*?1��dI3��d�Ѧ�=��*until the vector ��* converges, i.e., ����?current???��?previous��2?