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The BIC is an approximation to the Bayesian marginal likelihood which integrates out over uncertainty in parameter coefficient values.
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In this study, o is the inferential focus and w is a nuisance parameter which can be integrated out as follows: where where I is the indicator function.
Model parameters are estimated by maximizing the marginal likelihood which is obtained by integrating out the unobserved random effects b i from the likelihood function [ 16].
The probability density function (pdf) given the π0 ij 's and π1 ij 's of the observations of the i th study is given by: (10) Inference on the parameters is obtained through the standard likelihood method based on the marginal density for the data, which is calculated by integrating out the random effects B = (α, ξ1, ⋯, ξ J -1)'.
They showed that under certain assumptions, the missing data mechanism can be ignored and inference can be based solely on the likelihood function of the observed data, which is calculated by integrating out missing values in the likelihood function obtained had data been complete.
From (11), it is not difficult to see that ({E{tau (U,V |V}}), in which the random variable V is integrated out, has the same mean with ({tau (U,V }), yet a reduced variance.
They are obtained via the projection technique [9,10] upon integrating out such basis vectors which correspond to high energy excitations, or rather which have very large diagonal matrix elements in a suitable basis.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com