Theorem 1

Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f, the KL divergence of ${\Theta }_{\tilde{g}}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _{\tilde{g}}$$\end{document} w.r.t. ${\Theta }_f$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _{f}$$\end{document}, i.e.,

$\begin{array}{c}\Delta \left({\Theta }_f\text{;}{\Theta }_{\tilde{g}}\right)={\int }_{R}ln\frac{f\left(\theta \right)}{\tilde{g}\left(\theta \right)}f\left(\theta \right)\text{d}\theta ,\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\Theta _f{\text { ; }}\Theta _{\tilde{g}}) = \int _{\mathbb {R}} \ln \frac{f(\theta )}{\tilde{g}(\theta )}f(\theta ){\text {d}}\theta , \end{aligned}$$\end{document}

is always smaller than or equal to EKL divergence. That is,

$\begin{array}{c}\Delta \left({\Theta }_f\text{;}{\Theta }_{\tilde{g}}\right)\le E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\Theta _f{\text { ; }}\Theta _{\tilde{g}}) \le \mathbb {E}(\Delta (\Theta _f{\text { ; }}\Theta _{g} \mid \mathbf {X}_f)). \end{aligned}$$\end{document}

Proof

We start with rewriting the logarithm of the ratio of $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} over f

$\begin{array}{cc}ln\frac{\tilde{g}\left(\theta \right)}{f\left(\theta \right)}& =ln\left(\frac{{\sum }_{x}g(\theta \mid X=x)P(X_f=x)}{{\sum }_{x}f(\theta \mid X=x)P(X_f=x)}\right)\\ & =ln\left(\sum _x,\frac{g(\theta \mid X=x)P(X_f=x)}{f(\theta \mid X=x)P(X_f=x)},\frac{f(\theta \mid X=x)P(X_f=x)}{{\sum }_{x}f(\theta \mid X=x)P(X_f=x)}\right)\\ & =ln\left(\sum _x,\frac{g(\theta \mid X=x)}{f(\theta \mid X=x)},P,(X=x\mid \theta )\right)\\ & \ge \sum _{x}ln\frac{g(\theta \mid X=x)}{f(\theta \mid X=x)}P(X=x\mid \theta ),\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ln \frac{\tilde{g}(\theta )}{f(\theta )}&= \ln \left\{ \frac{\sum _{\mathbf {x}}g(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})}{\sum _{\mathbf {x}}f(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})}\right\} \\&=\ln \left\{ \sum _{\mathbf {x}}\frac{g(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})}{f(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})} \frac{f(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})}{\sum _{\mathbf {x}}f(\theta \mid \mathbf {X}=\mathbf {x})P(\mathbf {X}_f=\mathbf {x})}\right\} \\&=\ln \left\{ \sum _{\mathbf {x}}\frac{g(\theta \mid \mathbf {X}=\mathbf {x})}{f(\theta \mid \mathbf {X}=\mathbf {x})} P(\mathbf {X}=\mathbf {x} \mid \theta )\right\} \\&\ge \sum _{\mathbf {x}} \ln \frac{g(\theta \mid \mathbf {X}=\mathbf {x})}{f(\theta \mid \mathbf {X}=\mathbf {x})} P(\mathbf {X}=\mathbf {x} \mid \theta ), \end{aligned}$$\end{document}

using Jensen’s inequality. Thus, we obtain

$\begin{array}{cc}ln\frac{f\left(\theta \right)}{\tilde{g}\left(\theta \right)}& \le \sum _{x}ln\frac{f(\theta \mid X=x)}{g(\theta \mid X=x)}P(X=x\mid \theta ).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ln \frac{f(\theta )}{\tilde{g}(\theta )}&\le \sum _{\mathbf {x}} \ln \frac{f(\theta \mid \mathbf {X}=\mathbf {x})}{g(\theta \mid \mathbf {X}=\mathbf {x})} P(\mathbf {X}=\mathbf {x} \mid \theta ). \end{aligned}$$\end{document}

Integrating both sides of this expression w.r.t. f gives the desired result:

$\begin{array}{cc}{\int }_{R}ln\frac{f\left(\theta \right)}{\tilde{g}\left(\theta \right)}f\left(\theta \right)\text{d}\theta & \le {\int }_R\sum _{x}ln\frac{f(\theta \mid X=x)}{g(\theta \mid X=x)}P(X=x\mid \theta )f\left(\theta \right)\text{d}\theta \\ & =\sum _x{\int }_{R}ln\frac{f(\theta \mid X=x)}{g(\theta \mid X=x)}f(\theta \mid X=x)\text{d}\theta P(X_f=x).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\mathbb {R}}\ln \frac{f(\theta )}{\tilde{g}(\theta )} f(\theta ) \text {d} \theta&\le \int _{\mathbb {R}}\sum _{\mathbf {x}} \ln \frac{f(\theta \mid \mathbf {X}=\mathbf {x})}{g(\theta \mid \mathbf {X}=\mathbf {x})} P(\mathbf {X}=\mathbf {x} \mid \theta ) f(\theta ) \text {d}\theta \\&= \sum _{\mathbf {x}}\int _{\mathbb {R}} \ln \frac{f(\theta \mid \mathbf {X}=\mathbf {x})}{g(\theta \mid \mathbf {X}=\mathbf {x})} f(\theta \mid \mathbf {X}=\mathbf {x}) \text {d}\theta \text { }P(\mathbf {X}_f=\mathbf {x}). \end{aligned}$$\end{document}

It follows that $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} converges in law to f if $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} converges to f in EKL. Proving convergence in EKL will be the burden of the ensuing sections.$\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 1

Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f, the EKL divergence of ${\Theta }_f\mid X$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _f \mid \mathbf {X}$$\end{document} and ${\Theta }_g\mid X$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _g \mid \mathbf {X}$$\end{document}, w.r.t. $f(\Theta \mid X)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\Theta \mid \mathbf {X})$$\end{document} and $P\left(X_f\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X}_f)$$\end{document}, equals prior divergence minus marginal divergence, that is,

$\begin{array}{c}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right)=\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)-\Delta \left(X_f\text{;}{X}_g\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f{\text { ; }}\Theta _g\mid \mathbf {X}_f))=\Delta (\Theta _f{\text { ; }}\Theta _g) - \Delta (\mathbf {X}_f{\text { ; }}\mathbf {X}_g). \end{aligned}$$\end{document}

Proof

Using the definition of the posterior, and given the IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X}\mid \theta )$$\end{document}, we rewrite the EKL divergence as follows:

$\begin{array}{cc}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right)=& \sum _x{\int }_{R}ln\left(\frac{\frac{P(X=x\mid \theta )f\left(\theta \right)}{P(X_f=x)}}{\frac{P(X=x\mid \theta )g\left(\theta \right)}{P(X_g=x)}}\right)f(\theta \mid X=x)\text{d}\theta P(X_f=x)\\ =& \sum _x{\int }_{R}ln\left(\frac{f\left(\theta \right)}{g\left(\theta \right)},\frac{P(X_g=x)}{P(X_f=x)}\right)f(\theta \mid X=x)\text{d}\theta P(X_f=x),\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_f)) =&\sum _{\mathbf {x}} \int _{\mathbb {R}} \ln \left( \frac{\frac{P(\mathbf {X}=\mathbf {x} \mid \theta )f(\theta )}{P(\mathbf {X}_f=\mathbf {x})}}{\frac{P(\mathbf {X}=\mathbf {x} \mid \theta )g(\theta )}{P(\mathbf {X}_g=\mathbf {x})}}\right) f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \text { }P(\mathbf {X}_f=\mathbf {x})\\ =&\sum _{\mathbf {x}} \int _{\mathbb {R}} \ln \left( \frac{f(\theta )}{g(\theta )} \frac{P(\mathbf {X}_g=\mathbf {x})}{P(\mathbf {X}_f=\mathbf {x})}\right) f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \text { }P(\mathbf {X}_f=\mathbf {x}), \end{aligned}$$\end{document}

where $P\left(X_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X}_g)$$\end{document} is the distribution of the data under the prior g. Using properties of the logarithm, we obtain

$\begin{array}{cc}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right)=& \sum _x{\int }_{R}ln\left(\frac{f\left(\theta \right)}{g\left(\theta \right)}\right)f(\theta \mid X=x)\text{d}\theta P(X_f=x)\\ & +\sum _x{\int }_{R}ln\left(\frac{P(X_g=x)}{P(X_f=x)}\right)f(\theta \mid X=x)\text{d}\theta P(X_f=x).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_f)) =&\sum _{\mathbf {x}} \int _{\mathbb {R}} \ln \left( \frac{f(\theta )}{g(\theta )}\right) f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \text { }P(\mathbf {X}_f=\mathbf {x})\\&+ \sum _{\mathbf {x}} \int _{\mathbb {R}} \ln \left( \frac{P(\mathbf {X}_g=\mathbf {x})}{P(\mathbf {X}_f=\mathbf {x})}\right) f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \text { }P(\mathbf {X}_f=\mathbf {x}). \end{aligned}$$\end{document}

If we sum over the possible values of $X$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}$$\end{document} in the first term and integrate over $\Theta$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} in the second term, respectively, we obtain

$\begin{array}{cc}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right)& ={\int }_{R}ln\left(\frac{f\left(\theta \right)}{g\left(\theta \right)}\right)f\left(\theta \right)\text{d}\theta +\sum _{x}ln\left(\frac{P(X_g=x)}{P(X_f=x)}\right)P(X_f=x)\\ & ={\int }_{R}ln\left(\frac{f\left(\theta \right)}{g\left(\theta \right)}\right)f\left(\theta \right)\text{d}\theta -\sum _{x}ln\left(\frac{P(X_f=x)}{P(X_g=x)}\right)P(X_f=x)\\ & =\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)-\Delta \left(X_f\text{;}{X}_g\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_f))&= \int _{\mathbb {R}} \ln \left( \frac{f(\theta )}{g(\theta )}\right) f(\theta )\text {d}\theta + \sum _{\mathbf {x}}\ln \left( \frac{P(\mathbf {X}_g=\mathbf {x})}{P(\mathbf {X}_f=\mathbf {x})}\right) P(\mathbf {X}_f=\mathbf {x})\\&= \int _{\mathbb {R}} \ln \left( \frac{f(\theta )}{g(\theta )}\right) f(\theta )\text {d}\theta - \sum _{\mathbf {x}}\ln \left( \frac{P(\mathbf {X}_f=\mathbf {x})}{P(\mathbf {X}_g=\mathbf {x})}\right) P(\mathbf {X}_f=\mathbf {x})\\&=\Delta (\Theta _f\text { ; }\Theta _g) - \Delta (\mathbf {X}_f\text { ; }\mathbf {X}_g). \end{aligned}$$\end{document}

It follows that EKL divergence of the posterior distribution is equal to the difference between prior divergence $\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (\Theta _f\text { ; }\Theta _g)$$\end{document} and marginal divergence $\Delta \left(X_f\text{;}{X}_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (\mathbf {X}_f\text { ; }\mathbf {X}_g)$$\end{document} (i.e., divergence of $P\left(X_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X}_g)$$\end{document} w.r.t. $P\left(X_f\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X}_f)$$\end{document}).$\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 1 implies that $E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_f)\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g\mid \mathbf {X}_f))$$\end{document} equals zero if and only if prior divergence is equal to marginal divergence. Since the divergences are finite and non-negative, we find that

$\begin{array}{c}\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)\ge \Delta \left(X_f\text{;}{X}_g\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\Theta _f\text { ; }\Theta _g) \ge \Delta (\mathbf {X}_f\text { ; }\mathbf {X}_g). \end{aligned}$$\end{document}

We will now prove that $\Delta \left(X_f\text{;}{X}_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (\mathbf {X}_f\text { ; }\mathbf {X}_g)$$\end{document} is a monotone non-decreasing sequence in the number of items n with $\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (\Theta _f\text { ; }\Theta _g)$$\end{document} as an upper bound. To this aim, we consider what happens to marginal divergence when an item is added ( i.e., n is increased to $n+1$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}). To fix the notation, let $X_1\text{,}{X}_2\text{,}...$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1\text {, } X_2\text {, } ...$$\end{document} denote an infinite sequence of item responses, with $X_n$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_n$$\end{document} the n-th element and $X_n$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}_n$$\end{document} a vector consisting of the first n elements of this sequence.

Lemma 2

Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f, the marginal divergence for $n+1$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} observations is larger than or equal to marginal divergence for n observations:

$\begin{array}{c}\Delta (X_{f,n+1};X_{g,n+1})\ge \Delta (X_{f,n};X_{g,n}).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\mathbf {X}_{f,n+1}; \mathbf {X}_{g,n+1}) \ge \Delta (\mathbf {X}_{f,n}; \mathbf {X}_{g,n}). \end{aligned}$$\end{document}

Proof

The marginal divergence for $n+1$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} items is

$\begin{array}{c}\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right)=\sum _{x_{n+1}}ln\left(\frac{P(X_{f,n+1}=x_{n+1})}{P(X_{g,n+1}=x_{n+1})}\right)P(X_{f,n+1}=x_{n+1}).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\mathbf {X}_{f,n+1}\text { ; }\mathbf {X}_{g,n+1}) = \sum _{\mathbf {x}_{n+1} }\ln \left( \frac{P(\mathbf {X}_{f,n+1} =\mathbf {x}_{n+1})}{P(\mathbf {X}_{g,n+1} =\mathbf {x}_{n+1})}\right) P(\mathbf {X}_{f,n+1} =\mathbf {x}_{n+1}). \end{aligned}$$\end{document}

Conditioning on the first n observations and factoring the distribution, we obtain

$\begin{array}{cc}\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right)=& \sum _{x_n}\sum _{x_{n+1}}ln\left(\frac{P(X_{f,n+1}=x_{n+1}\mid X_n=x_n)}{P(X_{g,n+1}=x_{n+1}\mid X_n=x_n)},\frac{P(X_{f,n}=x_n)}{P(X_{g,n}=x_n)}\right)\\ & \times P(X_{f,n+1}=x_{n+1}\mid X_n=x_n)P(X_{f,n}=x_n).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\mathbf {X}_{f,n+1}\text { ; }\mathbf {X}_{g,n+1}) =&\sum _{\mathbf {x}_{n}}\sum _{x_{n+1}} \ln \left( \frac{P(X_{f,n+1}=x_{n+1} \mid \mathbf {X}_n=\mathbf {x}_n)}{P(X_{g,n+1}=x_{n+1}\mid \mathbf {X}_n=\mathbf {x}_n)} \frac{P(\mathbf {X}_{f,n}= \mathbf {x}_n)}{P(\mathbf {X}_{g,n}=\mathbf {x}_n)}\right) \\&\times P(X_{f,n+1}=x_{n+1}\mid \mathbf {X}_n=\mathbf {x}_n)P(\mathbf {X}_{f,n}=\mathbf {x}_n). \end{aligned}$$\end{document}

This is equal to

$\begin{array}{cc}\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right)& =\sum _{x_n}\sum _{x_{n+1}}ln\left(\frac{P(X_{f,n+1}=x_{n+1}\mid X_n=x_n)}{P(X_{g,n+1}=x_{n+1}\mid X_n=x_n)}\right)\\ & \times P(X_{f,n+1}=x_{n+1}\mid X_n=x_n)P(X_{f,n}=x_n)\\ & +\sum _{x_n}ln\left(\frac{P(X_{f,n}=x_n)}{P(X_{g,n}=x_n)}\right)P(X_{f,n}=x_n)\\ & =E\left(\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right|X_{f,n})\right)+\Delta \left(X_{f,n}\text{;}{X}_{g,n}\right),\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\mathbf {X}_{f,n+1}\text { ; }\mathbf {X}_{g,n+1})&= \sum _{\mathbf {x}_{n}} \sum _{x_{n+1}} \ln \left( \frac{P(X_{f,n+1}=x_{n+1}\mid \mathbf {X}_n=\mathbf {x}_n)}{P(X_{g,n+1}=x_{n+1}\mid \mathbf {X}_n=\mathbf {x}_n)}\right) \\&\quad \times P(X_{f,n+1}=x_{n+1}\mid \mathbf {X}_n=\mathbf {x}_n)P(\mathbf {X}_{f,n}=\mathbf {x}_n)\\&\qquad + \sum _{\mathbf {x}_{n}} \ln \left( \frac{P(\mathbf {X}_{f,n}=\mathbf {x}_n)}{P(\mathbf {X}_{g,n}=\mathbf {x}_n)}\right) P(\mathbf {X}_{f,n}=\mathbf {x}_n)\\&= \mathbb {E}(\Delta (X_{f,n+1}\text { ; }X_{g,n+1}|\mathbf {X}_{f,n})) + \Delta (\mathbf {X}_{f,n}\text { ; }\mathbf {X}_{g,n}), \end{aligned}$$\end{document}

a result closely related to the chain rule of KL divergence (Cover & Thomas, Reference Cover and Thomas1991, p. 23). Since $E\left(\Delta (X_{f,n+1}\text{;}{X}_{g,n+1}\mid X_{f,n})\right)\ge 0$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}(\Delta (X_{f,n+1}\text { ; }X_{g,n+1}\mid \mathbf {X}_{f,n})) \ge 0$$\end{document}, we see that

$\begin{array}{c}\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right)\ge \Delta \left(X_{f,n}\text{;}{X}_{g,n}\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\mathbf {X}_{f,n+1}\text { ; }\mathbf {X}_{g,n+1}) \ge \Delta (\mathbf {X}_{f,n}\text { ; }\mathbf {X}_{g,n}). \end{aligned}$$\end{document}

$\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Using Lemmas 1 and 2, we can now state Theorem 2.

Theorem 2

(Monotonicity Theorem) Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f, $E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n})\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}(\Delta (\Theta _f{\text { ; }}\Theta _g\mid \mathbf {X}_{f,n}))$$\end{document} is monotone non-increasing in the number of items n.

Proof

From Lemmas 1 and 2, we obtain

$\begin{array}{cc}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n+1})\right)=\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)& -\Delta \left(X_{f,n+1}\text{;}{X}_{g,n+1}\right)\\ =\Delta \left({\Theta }_f\text{;}{\Theta }_g\right)& -E\left(\Delta (X_{f,n+1}\text{;}{X}_{g,n+1}\mid X_{f,n})\right)\\ & -\Delta \left(X_{f,n}\text{;}{X}_{g,n}\right),\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g\mid \mathbf {X}_{f,n+1})) = \Delta (\Theta _f\text { ; }\Theta _g)&- \Delta (\mathbf {X}_{f,n+1}\text { ; }\mathbf {X}_{g,n+1})\\ = \Delta (\Theta _f\text { ; }\Theta _g)&- \mathbb {E}(\Delta (X_{f,n+1}\text { ; }X_{g,n+1} \mid \mathbf {X}_{f,n}))\\&- \Delta (\mathbf {X}_{f,n}\text { ; }\mathbf {X}_{g,n}), \end{aligned}$$\end{document}

and Lemma 1 shows that the difference of the first and the last terms is equal to the EKL divergence for n items. Thus, we have

$\begin{array}{c}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n+1})\right)=E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n})\right)-E\left(\Delta (X_{f,n+1}\text{;}{X}_{g,n+1}\mid X_{f,n})\right).\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_{f,n+1})) = \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_{f,n})) - \mathbb {E}(\Delta (X_{f,n+1}\text { ; }X_{g,n+1} \mid \mathbf {X}_{f,n})). \end{aligned}$$\end{document}

This implies a sequence of EKL divergences which adheres to the (in-)equality:

$\begin{array}{c}0\le E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n+1})\right)\le E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n})\right)\le \Delta \left({\Theta }_f\text{;}{\Theta }_g\right),\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0 \le \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_{f,n+1})) \le \mathbb {E}(\Delta (\Theta _f\text { ; }\Theta _g \mid \mathbf {X}_{f,n})) \le \Delta (\Theta _f\text { ; }\Theta _g), \end{aligned}$$\end{document}

i.e., a monotone non-increasing sequence in n with lower bound 0. Since prior divergence is finite by assumption, it is an upper bound for this sequence. $\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Theorem 3

(Convergence Theorem) Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f,

$\begin{array}{c}\underset{n\to \infty }{lim}E\left(\Delta ({\Theta }_f\text{;}{\Theta }_g\mid X_{f,n})\right)=0\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {E}(\Delta (\Theta _f{\text { ; }}\Theta _g\mid \mathbf {X}_{f,n})) = 0 \end{aligned}$$\end{document}

if the sequence of posteriors converges to a degenerate distribution.

Proof

We start with a direct proof of (2) (suppressing the dependence on n). Note first that,

(3)$\begin{array}{cc}\forall x:ln\frac{P(X_f=x)}{P(X_g=x)}& =-ln\frac{P(X_g=x)}{P(X_f=x)}\\ & =-ln\frac{{\int }_{R}P(X=x\mid \theta )g\left(\theta \right)\text{d}\theta }{{\int }_{R}P(X=x\mid \theta )f\left(\theta \right)\text{d}\theta }\\ & =-ln{\int }_R\frac{P(X=x\mid \theta )g\left(\theta \right)}{P(X=x\mid \theta )f\left(\theta \right)}\frac{P(X=x\mid \theta )f\left(\theta \right)}{{\int }_{R}P(X=x\mid \theta )f\left(\theta \right)\text{d}\theta }\text{d}\theta \\ & =-ln{\int }_R\frac{g\left(\theta \right)}{f\left(\theta \right)}f(\theta \mid X=x)\text{d}\theta \\ & \le -{\int }_{R}ln\frac{g\left(\theta \right)}{f\left(\theta \right)}f(\theta \mid X=x)\text{d}\theta ={\int }_{R}ln\frac{f\left(\theta \right)}{g\left(\theta \right)}f(\theta \mid X=x)\text{d}\theta \end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \forall \mathbf {x}: \ln \frac{P(\mathbf {X}_f=\mathbf {x})}{P(\mathbf {X}_g=\mathbf {x})}&= -\ln \frac{P(\mathbf {X}_g=\mathbf {x})}{P(\mathbf {X}_f=\mathbf {x})}\nonumber \\&= -\ln \frac{\int _\mathbb {R} P(\mathbf {X}=\mathbf {x} \mid \theta )g(\theta )\text {d}\theta }{\int _\mathbb {R} P(\mathbf {X}=\mathbf {x}\mid \theta )f(\theta )\text {d}\theta }\nonumber \\&= -\ln \int _\mathbb {R}\frac{P(\mathbf {X}=\mathbf {x}\mid \theta )g(\theta ) }{P(\mathbf {X}=\mathbf {x}\mid \theta )f(\theta )} \frac{P(\mathbf {X}=\mathbf {x}\mid \theta )f(\theta )}{\int _\mathbb {R} P(\mathbf {X}=\mathbf {x}\mid \theta )f(\theta )\text {d}\theta } \text {d}\theta \nonumber \\&= -\ln \int _\mathbb {R}\frac{g(\theta ) }{f(\theta )} f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \nonumber \\&\le -\int _\mathbb {R}\ln \frac{g(\theta ) }{f(\theta )} f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta = \int _\mathbb {R}\ln \frac{f(\theta ) }{g(\theta )} f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \end{aligned}$$\end{document}

using Jensen’s inequality in the last line. Taking expectations w.r.t. $P_f\left(X\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_f(\mathbf {X})$$\end{document} gives the inequality in (2). Similarly, we obtain

$\begin{array}{c}\forall x:ln\frac{P(X_f=x)}{P(X_g=x)}=ln\frac{{\int }_{R}P(X=x\mid \theta )f\left(\theta \right)\text{d}\theta }{{\int }_{R}P(X=x\mid \theta )g\left(\theta \right)\text{d}\theta }=ln{\int }_R\frac{f\left(\theta \right)}{g\left(\theta \right)}g(\theta \mid X=x)\text{d}\theta ,\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \forall \mathbf {x}: \ln \frac{P(\mathbf {X}_f=\mathbf {x})}{P(\mathbf {X}_g=\mathbf {x})} = \ln \frac{\int _\mathbb {R} P(\mathbf {X}=\mathbf {x} \mid \theta )f(\theta )\text {d}\theta }{\int _\mathbb {R} P(\mathbf {X}=\mathbf {x}\mid \theta )g(\theta )\text {d}\theta } = \ln \int _\mathbb {R}\frac{f(\theta ) }{g(\theta )} g(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta , \end{aligned}$$\end{document}

such that

(4)$\begin{array}{cc}-ln{\int }_R\frac{g\left(\theta \right)}{f\left(\theta \right)}f(\theta \mid X=x)\text{d}\theta & =ln{\int }_R\frac{f\left(\theta \right)}{g\left(\theta \right)}g(\theta \mid X=x)\text{d}\theta \\ & \le {\int }_{R}ln\frac{f\left(\theta \right)}{g\left(\theta \right)}f(\theta \mid X=x)\text{d}\theta .\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\ln \int _\mathbb {R}\frac{g(\theta ) }{f(\theta )} f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta&= \ln \int _\mathbb {R}\frac{f(\theta ) }{g(\theta )} g(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta \nonumber \\&\le \int _\mathbb {R}\ln \frac{f(\theta ) }{g(\theta )} f(\theta \mid \mathbf {X}=\mathbf {x})\text {d}\theta . \end{aligned}$$\end{document}

Since f is absolutely continuous w.r.t. g, we obtain that both $\frac{f\left(\theta \right)}{g\left(\theta \right)}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{f(\theta ) }{g(\theta )}$$\end{document} and $ln\frac{f\left(\theta \right)}{g\left(\theta \right)}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ln \frac{f(\theta ) }{g(\theta )}$$\end{document} are uniformly integrable. Convergence in probability of both posteriors (w.r.t. f and g as prior) is then sufficient to guarantee the equality in (3) (e.g., Venkatesh, Reference Venkatesh2013, pp. 480–481), since under these conditions we may change the order of limits and integration.$\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The Convergence Theorem relies on posterior consistency. The regularity conditions that imply posterior consistency can be found in many places. For unidimensional monotone IRT models, the regularity conditions for strong consistency (i.e., almost sure convergence) can be found in Chang and Stout (Reference Chang and Stout1993, pp. 42–43). As a courtesy to the reader, we list their conditions in Appendix 1. Chang and Stout (Reference Chang and Stout1993, pp. 43–45) argued that in practice these conditions are “very general and appropriate hypotheses” (p. 51). Similar conditions can be found in Chang (Reference Chang1996) for polytomous IRT models.

Combining Theorem 1, the Monotonicity Theorem, and the Convergence Theorem, we arrive at our final result.

Theorem 4

(Monotone Convergence Theorem) Given an IRT model $P(X\mid \theta )$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {X} \mid \theta )$$\end{document} and assuming that the support of g contains the support of f and the sequence of posteriors converges to a degenerate distribution, then $\Delta \left({\Theta }_f\text{;}{\Theta }_{\tilde{g}}\right)\to 0$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (\Theta _f{\text { ; }}\Theta _{\tilde{g}}) \rightarrow 0$$\end{document}, monotonically, and furthermore, ${\Theta }_{\tilde{g}}\stackrel{L}{⟶}{\Theta }_f$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _{\tilde{g}} \overset{\mathcal {L}}{\longrightarrow }\Theta _f$$\end{document}.

Proof

Under the stated assumptions, the Convergence Theorem implies that the EKL divergence converges to zero as n tends to infinity. Convergence is monotone by Theorem 2. From Theorem 1, we consequently obtain

$\begin{array}{c}\Delta \left({\Theta }_f\text{;}{\Theta }_{\tilde{g}}\right)\to 0.\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta (\Theta _f\text { ; }\Theta _{\tilde{g}}) \rightarrow 0 . \end{aligned}$$\end{document}

Since convergence in KL divergence implies convergence in law (DasGupta, Reference DasGupta2008, p. 21), we have

$\begin{array}{c}{\Theta }_{\tilde{g}}\stackrel{L}{⟶}{\Theta }_f.\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta _{\tilde{g}} \overset{\mathcal {L}}{\longrightarrow }\Theta _f. \end{aligned}$$\end{document}

$\square$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

In summary, the Monotone Convergence Theorem states that (under mild regularity conditions) the marginal distribution of PVs $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is a consistent estimator of the true ability distribution f.

5.1. What can we learn from Plausible Values?

What can we learn about the “correct” population model $f\left(\theta \right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta )$$\end{document} when we are using PVs from the “wrong” posterior $g(\theta \mid X=x)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\theta \mid \mathbf {X}=\mathbf {x})$$\end{document}? A common misconception is that the marginal distribution of PVs equals the population model (i.e., $\tilde{g}=g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g} = g$$\end{document}) and nothing can be learned from PVs over that which is already known from the population model (prior distribution) (e.g., Kreiner & Christensen, Reference Kreiner and Christensen2014). This is true, if and only if, the population model is the true ability distribution (i.e., $g=f$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g = f$$\end{document}). This is not likely and in practice we expect to see that $\tilde{g}\ne g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g} \ne g$$\end{document}.

Example 2

(PISA) To illustrate that the PV distribution may diverge from the prior in applications, we analyze data from the 2006 PISA cycle. More specifically, we used the $n=26$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 26$$\end{document} items intended to assess reading ability in booklet 6 made by $N=$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N =$$\end{document} 1738 Canadian students (see Appendix 2 for details of this analysis). A single PV was generated for each student using the One Parameter Logistic Model (OPLM; Verhelst & Glas, Reference Verhelst, Glas, Fischer and Molenaar1995) as IRT model, and a standard normal distribution as prior. The ecdf of N draws from the prior distribution g (solid gray line) and the ecdf of the generated PVs using $n=26$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=26$$\end{document} items are shown in Fig. 2 (solid black line). The marginal distribution of the PVs is clearly different from the specified prior distribution.

Figure 2. Ecdf of PVs ($\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document}) and N draws from a standard normal prior distribution (i.e., $g\left(\theta \right)=\varphi \left(\theta \right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\theta )=\phi (\theta )$$\end{document}) in the PISA example.

If the population model is misspecified (i.e., $g\ne f$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \ne f$$\end{document}), we can still learn a lot from looking at the PV distribution. The PV distribution provides a consistent estimate of the true ability distribution, which is at least as plausible as the population model which figures as a prior. Specifically, it follows from the Monotonicity Theorem that, if $g\ne f$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \ne f$$\end{document}, and hence $\tilde{g}\ne g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g} \ne g$$\end{document}, the marginal distribution of PVs $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is closer to f than g is; as we saw in Example 1. Moreover, we can use PVs to evaluate the fit of the population model by testing the hypothesis $H_0:\tilde{g}=g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0 : \tilde{g} = g$$\end{document} against $H_1:\tilde{g}\ne g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1 : \tilde{g} \ne g$$\end{document}. If $H_0$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} is rejected, there is no reason to be interested in g: $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is our best guess of what the true distribution of ability would look like.

Example 3

(PISA continued) We use the PISA example to illustrate that we can test the hypothesis $H_0:\tilde{g}=g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0: \tilde{g} = g$$\end{document} against $H_1:\tilde{g}\ne g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1: \tilde{g} \ne g$$\end{document} using real data with a relatively small number of observations, and that the power of this test is increasing with n. To this aim, we randomly assigned each student two items out of the 26 items that were available. Figure 2 shows the ecdf of the PVs using $n=2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 2$$\end{document} items (dashed line). It is clear that even with two items, the marginal distribution of PVs differs from the specified prior distribution and $H_0$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} does not hold (this test is performed in the next example, see Table 1). Figure 2 also shows that the PV distributions diverge from the prior distribution as n increases, thereby increasing the probability to reject $H_0$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} if it is wrong.

Table 1. Average values of KS test statistic using PISA data to compare $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} with the prior distributions used to generate $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document}.

Values over 0.046 are significant at an $\alpha$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }$$\end{document} level of 0.05.

5.2. Choose a flexible population model

The population model is formally a prior and, under the conditions of the Monotone Convergence Theorem, becomes irrelevant as the number of items becomes large. Essentially, this is an instance of the common finding that the data overrule the prior when the number of observations increases. In practice, however, there is a natural limit to the number of items that can be administered which raises the question how we can favor convergence without increasing the number of items.

The answer comes from Lemma 1 which suggests that convergence of the PV distribution to the true distribution of ability is faster if prior divergence is reduced. Thus, for a given n, we would like prior divergence to be as small as possible (i.e., we would like g to resemble f). When little or nothing is known about f, we may achieve this using a flexible prior; that is, one that easily adapts to different shapes. Otherwise, we may look at the PV distribution found in previous editions of the study to improve the prior. Convergence is also improved if we adopt an empirical Bayesian approach and estimate the parameters of the prior so that it adapts itself to the data as much as possible (see, for instance; White, Reference White1982). Using, for instance, a normal prior in Example 1 would help to discover the bimodality of the true ability distribution with less items.

Example 4

(PISA Continued) We use the previously established OPLM model with three prior distributions ordered in terms of flexibility:

1. 1. A standard normal distribution $N\left(0\text{,}1\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}(0\text {, }1)$$\end{document}.

2. 2. A normal distribution $N\left(\mu \text{,}{\sigma }^2\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}(\mu \text {, }\sigma ^2)$$\end{document} with mean $\mu$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and variance ${\sigma }^2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}.

3. 3. A PCA regression prior $N\left(\hat{\Lambda }\beta \text{,}{\sigma }^2\right)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}(\widehat{\varvec{\Lambda }}\varvec{\beta }\text {, }\sigma ^2)$$\end{document}, where $\hat{\Lambda }$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\varvec{\Lambda }}$$\end{document} constitutes the principal component scores estimated on student covariates assessed in the PISA student questionnaire. We use the first 50 principal components explaining roughly $60\%$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$60~\%$$\end{document} of the variance in the student questionnaire.

The parameters of the prior distribution are estimated using the Gibbs sampler (Geman & Geman, Reference Geman and Geman1984) with non-informative hyper-priors (Gelman, Carlin, Stern, & Rubin, Reference Gelman, Carlin, Stern and Rubin2004).

For each prior distribution, we test the hypothesis $H_0:\tilde{g}=g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0 : \tilde{g} = g$$\end{document} against $H_1:\tilde{g}\ne g$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1 : \tilde{g} \ne g$$\end{document} using the two-sample Kolmogorov-Smirnov (KS) test. For the second and third prior, we ran an additional 1000 iterations of the Gibbs sampler. In each iteration, we generated one PV for each person, generate a sample of size N from the prior, and compute the KS test statistic. Thus, we obtained 1000 replications for the test statistic, which were then averaged. The results are shown in Table 1 and confirm that prior divergence decreases as more flexible prior distributions are used.

Our main concern is whether or not the PV distribution converges to the true ability distribution. Since we do not know the true ability distribution, we compare our results with the best guess that we have, i.e., the distribution of PVs obtained by using $n=26$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 26$$\end{document} items and the PCA regression prior. We repeated the procedure to obtain Table 1, but instead of comparing the generated PVs with draws from the prior, we compared the generated PVs with the PVs generated using $n=26$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 26$$\end{document} items and the PCA regression prior. The results in Table 2 show that the PV distributions converge to a single (true) distribution as n increases and/or the prior becomes more flexible.

Table 2. Average values of KS test statistic using PISA data to compare $\tilde{g}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} using different prior distributions with the best guess.

It is important to note that there is a limit to the amount of parameters that we can estimate, and thus the amount of flexibility that we can achieve in practice. This can be seen in Example 4. For $n=2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2$$\end{document}, Table 2 seems to suggest that the normal prior works better than the more flexible PCA regression prior. This counter intuitive result only holds for $n=2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2$$\end{document} and is due to the poor estimation of hyper-parameters that results when both N and n are small. The normal prior has just two parameters, $\mu$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and ${\sigma }^2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}, whereas the PCA regression prior has 52 parameters, $\beta =\{{\beta }_0\text{,}{\beta }_1\text{,}...\text{,}{\beta }_{50}\}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\beta } = \{\beta _0\text {, }\beta _1\text {, }...\text {, }\beta _{50}\}$$\end{document} and ${\sigma }^2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}. Since the standard errors accumulate for the generated PV distributions, we expect to observe larger variations in the generated PV distributions using the PCA regression prior. These larger variations are reflected in the value of the KS test statistic.

5.3. What if we miss a covariate?

A remarkable feature of Example 1 is that the PV distribution reveals the difference between boys and girls even though sex was not included as a covariate in the population model. This is consistent with our results. Given the conditions of the Monotone Convergence Theorem, the distribution of plausible values

$\begin{array}{c}\tilde{g}(\theta \mid z_1\text{,}{z}_2)=\sum _{x}g(\theta \mid x\text{,}{z}_2)P_f(x\mid z_1\text{,}{z}_2)\end{array}$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{g}(\theta \mid z_1\text {, } z_2)=\sum _{\mathbf {x}} g(\theta \mid \mathbf {x}\text {, }z_2)P_f(\mathbf {x}\mid z_1\text {, }z_2) \end{aligned}$$\end{document}

is a consistent estimator of the population distribution $f(\theta \mid z_1\text{,}{z}_2)$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta \mid z_1\text {, }z_2)$$\end{document}, for sets of covariates $z_1$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_1$$\end{document} and $z_2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_2$$\end{document}; even when $z_2$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_2$$\end{document} is the empty set (i.e., if we miss all covariates). It also means that a secondary analyst who happens to observe the student’s sex will, when n is sufficiently large, recover the difference between boy and girls even if the PVs have been generated with a population model that contains no covariates at all.

Figure 3. Plausible value distributions of boys and girls with and without gender as a covariate in the PISA example.