Bayesian Restricted Maximum Likelihood Sample Clauses

Bayesian Restricted Maximum Likelihood. ‌ Compared with ▇▇▇▇▇▇▇▇’▇ method, an implementation of restricted maximum like- lihood (ReML) (▇▇▇▇▇▇▇▇, 1977), a modified ikelihood-based estimation method by applying the maximum-likelihood principles to the residuals (see more details in Section 3.3.1), is embedded in the Statistical Parametric Mapping (SPM) software 1 in MATLAB. The SPM software package, including a suite of MATLAB functions and subroutines (with some externally compiled C routines), is designed for the brain imaging data analysis, and is freely available to the neuroimaging community. Previous studies showed that this Bayesian ReML approach in SPM is more accur- ate than ▇▇▇▇▇▇▇▇’▇ method in heritability estimation, producing the estimates with lower bias and smaller variance (▇▇▇▇▇▇▇ et al., 2009). SPM uses a non-standard ReML implementation, a Bayesian version of ReML, for the estimation of variance components (hyperparameters in the hierarchical model) with a Gaussian prior, where the log transformation can be further employed to en- force the non-negative constraints on variance components 2. In Bayesian inference, the parameters are treated as random variables with a Gaussian distribution, and a linear hierarchical model for timeseries Y from all subjects can be constructed as Y = X(1)θ(1) + s(1), θ(1) = X(2)θ(2) + s(2), where X(i) is the specified design matrix at level i, θ(i) are parameters at level i, and the errors s(i) at level i are distributed as N(0, C(i)). The covariance matrix C(i) can be written as C(i) = Σ j ρj (i) Q(i), where the hyperparameters ρj (i) are variance 1▇▇▇▇://▇▇▇.▇▇▇.▇▇▇.▇▇▇.▇▇.▇▇/spm/ 2For reference, the SPM user-recommended configuration for the Gaussian prior distribution of log-hyperparameters is with hyperprior expectation hE = log(Var(Y)) 1 and hyperprior covariance hC = exp(8), both of which were shown to be preferable to other settings in simulations in terms of estimation accuracy and model selection. components at level i and Q(i) is the corresponding basis set. This two-level model is identical to a non-hierarchical model: Y = X(1)X(2)θ(2) + X(1)s(2) + s(1), and can be re-written as Y = Xθ + s(1), X = [X(1), X(1)X(2)], θ = [s(2), θ(2)]T.