20 Pages

E. Implementing The Bayesian Approach

The starting point for the estimation is the marginalized posterior probability distribution

g(€, rlU) ex L(UI€, r)g(€I1J)· (E.2)


The first, second, and cross derivatives of the logarithm of this posterior density are needed by the Newton-Raphson procedure. The first derivative of the marginalized likelihood component of Equation E.2 with respect to ai is obtained by following Equations 7.10 and 7.11 with Ci = o. The derivative is

8 n JaOti {log[L(UI~, r)]} = e" tr [UirPi (OJ )](Orbi)[P(Oj IUj,~, r)]dO.(E.3) The posterior term [P(XkIUj,~,r)] corresponds to nik (Bock & Aitkin, 1981; Harwell, Baker, & Zwart, 1988). Substituting nik and putting Equation E.3 in numerical quadrature form yields

8 qa. {log[L(UI~,r)]} = e" ~)fik - nikPi(Xk)](Xk - bi ). at k=l

The derivative of the prior term with respect to ai is appended to the derivative of the likelihood component, yielding the Bayes modal expression

aa . [g(~, rIU)] = L1 = e" i)fik -nikPi(Xk)](Xk-bi ) - (Oti -/,,) .(E.5) at k=l G'o

The expression for the first derivative of the likelihood component with respect to bi can be obtained by inspection of the three-parameter model result (Equation 7.17). The numerical quadrature form of this term is

8 8 q abo [g(~, rIU)] = L 2 = abo {log[L(UI~, r)]} = -e" Z=[fik-nikPi(Xk)].(E.6)

No prior distribution was imposed on the difficulty parameter; hence, there is no appending term, and Equation E.6 also is the Bayes modal estimation expression.