Bayesian Information Quadratic Inference Function Sample Clauses

Bayesian Information Quadratic Inference Function. (BIQIF) Wang and Qu(2009) [81] proposed the Bayesian Information Quadratic Inference Function (BIQIF). The BIQIF approach modifies the Bayesian information criterion (BIC) by replacing the negative of two times the log-likelihood with the quadratic inference function (QIF) [69]. This substitution is motivated by the observation that the role of QIF in the semi-parametric setting is similar to the role of the negative of two times the log-likelihood in the parametric setting. Consider selecting an appropriate marginal regression model from among a class of candidate models that correspond to select different subsets of covariates. Let M be the class of candidate models being considered. Each member of M can be identified with a unique set m, where m is a subset of {1, . . . , q} and contains the indices of the covariates that are included in the candidate model. Then, let β (m) denote the (q + 1) × 1 vector which sets the corresponding components of β to zero if they are not selected by this model. Further, let B (m) be the corresponding parameter space. Let QN (β) be the QIF as defined in Qu [69]. The QIF-based BIC then selects the model in M which minimizes BIQIF (m) = QN {βˆ (m)} + |β (m) |log(N ), { } where QN βˆ (m) = infβ∈B(m) {QN (β)} and |β (m) | denotes the number of non- zero elements in β (m). As in BIC, BIQIF contains a term to penalize the lack of fit of the model and a term to penalize the complexity of the model. Although BIQIF approach to model selection has several appealing properties, the quadratic inference function is not on the same scale as the negative of two times the log-likelihood. More specifically, the asymptotic distribution of QN {βˆ} is χ2 with r − q degrees of freedom, where r is the dimension of the extended score equation, gN , and q is the dimension of β, where q < r. Thus, while the quadratic inference function may be useful for a likelihood-ratio-type test, it is not generally applicable as a measure of model fit for information criteria.
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