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Gibbs Samplers for Logistic Item Response Models via the Pólya–Gamma Distribution: A Computationally Efficient Data-Augmentation Strategy

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Abstract

Fully Bayesian estimation of item response theory models with logistic link functions suffers from low computational efficiency due to posterior density functions that do not have known forms. To improve algorithmic computational efficiency, this paper proposes a Bayesian estimation method by adopting a new data-augmentation strategy in uni- and multidimensional IRT models. The strategy is based on the Pólya–Gamma family of distributions which provides a closed-form posterior distribution for logistic-based models. In this paper, an overview of Pólya–Gamma distributions is described within a logistic regression framework. In addition, we provide details about deriving conditional distributions of IRT, incorporating Pólya–Gamma distributions into the conditional distributions for Bayesian samplers’ construction, and random drawing from the samplers such that a faster convergence can be achieved. Simulation studies and applications to real datasets were conducted to demonstrate the efficiency and utility of the proposed method.

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Author information

Authors and Affiliations

  1. 309DE LB Gorgas Library, University Libraries, The University of Alabama, Tuscaloosa, USA

    Zhehan Jiang

  2. The University of Kansas, Lawrence, USA

    Jonathan Templin

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  1. Zhehan Jiang
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  2. Jonathan Templin
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Correspondence to Zhehan Jiang.

Appendix I

Appendix I

Proof

The conditional posterior of \({\varvec{\beta }}\) conditioning on a set of Pólya–Gamma random variables \({{\varvec{w}}}=\left( {w_1 ,w_2 ,\ldots ,w_N } \right) \) is:

$$\begin{aligned} p({\varvec{\beta }} |{{\varvec{w}}}, {{\varvec{y}}})\propto p\left( {\varvec{\beta }} \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-{{\varvec{X}}}{\varvec{\beta }}} \right) ^{T}{\varvec{\varOmega }} \left( {{{\varvec{z}}}-{{\varvec{X}}}{\varvec{\beta }}} \right) } \right\} , \end{aligned}$$

where \({{\varvec{z}}}=\left( {\frac{k_1 }{w_1 },\frac{k_2 }{w_2 },\ldots ,\frac{k_N }{w_N }} \right) \), \({\varvec{\varOmega }} =diag\left( {{w}_1 ,w_2 ,\ldots ,w_N } \right) \), and the prior for \({\varvec{\beta }}\) is \(p\left( {\varvec{\beta }} \right) \) and \({{\varvec{X}}}\) is the predictor matrix consisted of known scalars. The \({{\varvec{y}}}\) is reparametrized to k by calculating \(y-\frac{n}{2}\), where n =1 in a binomial likelihood. Let \({\varvec{\beta }} =\theta _e \), \({{\varvec{X}}}\)= (\({{\varvec{a}}}\), \({{\varvec{b}}}\)), and \(p\left( {\theta _e } \right) \) be the prior for \(\theta _e \). Given \({{\varvec{a}}}\), \({{\varvec{b}}}\), and \({\varvec{\varOmega }}_{{\varvec{e}}}\) are constant, the conditional posterior distribution for \(\theta _e \) can be obtained as:

$$\begin{aligned}&p\left( {\theta _e {|}{{\varvec{a}}},{{\varvec{b}}},{{\varvec{w}}}, {{\varvec{y}}}_e } \right) \propto p\left( {\theta _e } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-{{\varvec{a}}}\left( {\theta _e -{{\varvec{b}}}} \right) } \right) ^{T}{\varvec{\varOmega }}_{{\varvec{e}}} \left( {{{\varvec{z}}}-{{\varvec{a}}}\left( {\theta _e -{{\varvec{b}}}} \right) } \right) } \right\} ,\\&\quad \propto p\left( {\theta _e } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-{{\varvec{a}}}\theta _e +{{{\varvec{a}}}}'{{{\varvec{b}}}}} \right) ^{T}} {\varvec{\varOmega }}_{{\varvec{e}}}\left( {{{\varvec{z}}}-{{\varvec{a}}}\theta _e +{{{\varvec{a}}}}'{{\varvec{b}}}} \right) \right\} ,\\&\quad \propto p\left( {\theta _e } \right) \exp \left\{ {-\frac{1}{2}\left( {\left( {{{\varvec{z}}}+{{{\varvec{a}}}}'{{\varvec{b}}}} \right) }-{{\varvec{a}}}\theta _e \right) ^{T}{\varvec{\varOmega }} _{\mathrm{e}} \left( {\left( {{{\varvec{z}}}+{{{\varvec{a}}}}'{{\varvec{b}}}} \right) } -{{\varvec{a}}}\theta _e\right) } \right\} . \end{aligned}$$

\(\square \)

Let \({{\varvec{z}}}_{{\varvec{e}}} ={{\varvec{z}}}+{{{\varvec{a}}}}'{{\varvec{b}}}\), and therefore, \({{\varvec{z}}}_e =(\frac{a_1 b_1 w_{1e} +k_{1e} }{w_{1e} }\), ..., \(\frac{a_I b_I w_{Ie} +k_{Ie} }{w_{Ie} })\); the formula above can be simplified as \(p\left( {\theta _e {|}{{\varvec{a}}},{{\varvec{b}}},{{\varvec{w}}}, {{\varvec{y}}}_e } \right) \propto p\left( {\theta _e } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}_{{\varvec{e}}} -{{\varvec{a}}}\theta _e } \right) ^{T}{\varvec{\varOmega }} _{\mathrm{e}} \left( {{{\varvec{z}}}_{{\varvec{e}}} -{{\varvec{a}}}\theta _e } \right) } \right\} \) with both \({\varvec{\varOmega }} _{\mathrm{e}} =diag\left( {w_{1e} ,\ldots ,w_{Ie} } \right) \) and \({{\varvec{k}}}_e ={{\varvec{y}}}_e -\frac{1}{2}\). This finding is given in Eq. 7.

Let \({\varvec{\beta }} =b_i \), \({{\varvec{X}}}\)= (\({\varvec{\theta }},a_i )\), and \(p\left( {b_i } \right) \) be the prior for \(b_i \). Given \({\varvec{\theta }}\), \(a_i\), and \({\varvec{\varOmega }}_{{\varvec{b}}} \) are constant, the conditional posterior distribution for \(b_i \) can be obtained as:

$$\begin{aligned}&f\left( {b_i {|}{\varvec{\theta }}, a_i ,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-a_i \left( {{\varvec{\theta }} -\mathbf{1}b_i } \right) } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{b}}} \left( {{{\varvec{z}}}-a_i \left( {{\varvec{\theta }} -\mathbf{1}b_i } \right) } \right) } \right\} ,\\&\quad \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-a_i {\varvec{\theta }} +\mathbf{1}a_i b_i } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{b}}} \left( {{{\varvec{z}}}-a_i {\varvec{\theta }} +\mathbf{1}a_i b_i } \right) } \right\} ,\\&\quad \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {\left( {{{\varvec{z}}}-a_i {\varvec{\theta }}} \right) +\mathbf{1}a_i b_i } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{b}}} \left( {\left( {{{\varvec{z}}}-a_i {\varvec{\theta }}} \right) +\mathbf{1}a_i b_i } \right) } \right\} . \end{aligned}$$

Let \({{\varvec{z}}}_{{\varvec{b}}} ={{\varvec{z}}}-a_i {\varvec{\theta }}\), and therefore, \({{\varvec{z}}}_b =(\frac{k_{i1} -a_i \theta _1 w_{i1} }{w_{i1} }\), ..., \(\frac{k_{iN} -a_i \theta _{N} w_{iN} }{w_{iN} })\); the formula above can be simplified as \(f\left( {b_i {|}{\varvec{\theta }} ,a_i,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}_b +\mathbf{1}a_i b_i } \right) ^{T}{\varvec{\varOmega }} _{ab} \left( {{{\varvec{z}}}_b +\mathbf{1}a_i b_i } \right) } \right\} \) with both \({\varvec{\varOmega }} _{\mathrm{b}} =diag\left( {w_{i1} ,\ldots ,w_{iN} } \right) \) and \({{\varvec{k}}}_i ={{\varvec{y}}}_{{\varvec{i}}} -\frac{1}{2}\). This finding is given in Eq. 8.

Let \({\varvec{\beta }}=a_i \), \({{\varvec{X}}} = ({\varvec{\theta }},b_i )\), and \(p\left( {a_i } \right) \) be the prior for \(a_i \). Given \({{\varvec{\theta }}} \), \(b_i\), and \({\varvec{\varOmega }}_{{\varvec{a}}}\) are constant, the conditional posterior distribution for \(a_i \) can be obtained as:

$$\begin{aligned}&f\left( {a_i {|}{\varvec{\theta }},b_i ,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \propto p\left( {a_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-a_i \left( {{\varvec{\theta }} -\mathbf{1}b_i } \right) } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{a}}} \left( {{{\varvec{z}}}-a_i \left( {{\varvec{\theta }} -\mathbf{1}b_i } \right) } \right) } \right\} ,\\&\quad \propto p\left( {a_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-\left( {{\varvec{\theta }} -\mathbf{1}b_i } \right) a_i } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{a}}} \left( {{{\varvec{z}}}-\left( {{\varvec{\theta }}-\mathbf{1}b_i } \right) a_i } \right) } \right\} . \end{aligned}$$

As specified previously, \({{\varvec{z}}}=\left( {\frac{k_1 }{w_1 },\frac{k_2 }{w_2 },\ldots ,\frac{k_N }{w_N }} \right) \), \({{\varvec{k}}}_i ={{\varvec{y}}}_{{\varvec{i}}} -\frac{1}{2}\), and \({\varvec{\varOmega }}_{{\varvec{a}}} \) is identical to \({\varvec{\varOmega }} _{{\varvec{b}}} \). This finding is given in Eq. 9.

In the multidimensional 2-PL model, \({\varvec{\Theta }}\) is the N x K matrix of latent traits where d is the dth column of the \({\varvec{\Theta }}\) matrix. Therefore, \({\varvec{\Theta }}_{{\varvec{d}}} \) is essentially a vector where \({\varvec{\Theta }}_{-{{\varvec{d}}}} \) is the \({\varvec{\Theta }}\) matrix excluding the \({\varvec{\Theta }}_{{\varvec{d}}} \). For each examine, \({\varvec{\theta }}_{{\varvec{e}}} =\left( {\theta _{\mathrm {e}1} ,\theta _{\mathrm {e}2} ,\ldots \theta _{ek} } \right) \) is the K-dimensional column vector of latent traits for an examinee e. Let \({\varvec{\beta }} ={\varvec{\theta }} _{{\varvec{e}}} \), \({{\varvec{X}}}\) = (\({{\varvec{a}}},{{\varvec{b}}})\), and \(p\left( {{\varvec{\theta }} _{{\varvec{e}}} } \right) \) be the prior for \({\varvec{\theta }}_{{\varvec{e}}} \). Given \({{\varvec{a}}}\), \({{\varvec{b}}}\), and \({\varvec{\varOmega }} _{{\varvec{e}}} \) are constant, the conditional posterior distribution for \({\varvec{\theta }}_{{\varvec{e}}} \) can be obtained as:

$$\begin{aligned}&f\left( {{\varvec{\theta }} _e {|}{{\varvec{a,b,y}}}_e ,{{\varvec{w}}}} \right) \propto p\left( {{\varvec{\theta }} _{{\varvec{e}}} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} -{{\varvec{b}}}} \right) } \right) ^{T}{\varvec{\Omega }}_{\mathrm{e}} \left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} -{{\varvec{b}}}} \right) } \right) } \right\} ,\\&\quad \propto p\left( {{\varvec{\theta }} _{{\varvec{e}}} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}+{{\varvec{b}}}-{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} } \right) ^{T}{\varvec{\Omega }}_{\mathrm{e}} \left( {{{\varvec{z}}}+{{\varvec{b}}}-{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}}} \right) } \right\} ,\\&\quad \propto p\left( {{\varvec{\theta }} _{{\varvec{e}}} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}+{{\varvec{b}}}-{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} } \right) ^{T}{\varvec{\Omega }}_{\mathrm{e}} \left( {{{\varvec{z}}}+{{\varvec{b}}}-{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} } \right) } \right\} . \end{aligned}$$

Let \({{\varvec{z}}}_{{\varvec{e}}} ={{\varvec{z}}}+{{\varvec{b}}}\), and therefore, \({{\varvec{z}}}_e =(\frac{b_1 w_{1e} +k_{1e} }{w_{1e} }\), ..., \(\frac{b_I w_{Ie} +k_{Ie} }{w_{Ie} })\); the formula above can be simplified as \(f\left( {{\varvec{\theta }} _e {|}{\varvec{\Sigma }},{{\varvec{a,b,y}}}_e } \right) \propto p\left( {{\varvec{\theta }} _{{\varvec{e}}} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}_{{\varvec{e}}} -{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} } \right) ^{T}{\varvec{\varOmega }} _{\mathrm{e}} \left( {{{\varvec{z}}}_{{\varvec{e}}} -{{\varvec{a}}}^{T}{\varvec{\theta }} _{{\varvec{e}}} } \right) } \right\} \) with both \({\varvec{\varOmega }} _{\mathrm{e}} =diag\left( {w_{1e} ,\ldots ,w_{Ie} } \right) \) and \({{\varvec{k}}}_e ={{\varvec{y}}}_e -\frac{1}{2}\). This finding is given in line 1 of Eq. 12.

Let \({\varvec{\beta }} =b_i \), \({{\varvec{X}}}\) = (\({\varvec{\Theta }},{{\varvec{a}}}_{{\varvec{i}}} )\), and \(p\left( {b_i } \right) \) be the prior for \(b_i \). Given \({\varvec{\Theta }}\), \({{\varvec{a}}}_{{\varvec{i}}}\), and \({\varvec{\varOmega }}_{{\varvec{b}}}\) are constant, the conditional posterior distribution for \(b_i \) in the multidimensional 2-PL model can be obtained as:

$$\begin{aligned}&f\left( {b_i {|}{{\varvec{a}}}_{{\varvec{i}}} ,{\varvec{\Theta }},{{\varvec{y}}}_i ,{{\varvec{w}}}} \right) \propto p\left( {b_{{\varvec{i}}} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}_{{\varvec{i}}}^T {\varvec{\Theta }}-\mathbf{1}b_{{\varvec{i}}} } \right) } \right) ^{T}{\varvec{\Omega }}_{\mathrm{b}} \left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}_{{\varvec{i}}}^T {\varvec{\Theta }}-\mathbf{1}b_{{\varvec{i}}} } \right) } \right) } \right\} ,\\&\quad \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-{{\varvec{a}}}^{T}{\varvec{\Theta }}+\mathbf{1}b_{{\varvec{i}}} } \right) ^{T}{\varvec{\Omega }}_{\mathrm{b}} \left( {{{\varvec{z}}}-{{\varvec{a}}}^{T}{\varvec{\Theta }}+\mathbf{1}b_{{\varvec{i}}} } \right) } \right\} , \end{aligned}$$

Let \({{\varvec{z}}}_{{\varvec{b}}} ={{\varvec{z}}}-{{\varvec{a}}}_{{\varvec{i}}}^T {\varvec{\Theta }}\), and therefore, \({{\varvec{z}}}_b =(\frac{k_{i1} -a_{i1} \theta _{11} w_{i1} -a_{i2} \theta _{12} w_{i1} -\ldots -a_{iK} \theta _{1K} w_{i1} }{w_{i1} }\), ..., \(\frac{k_{iN} -a_{i1} \theta _{N1} w_{iN} -a_{i2} \theta _{N3} w_{iN} -\ldots -a_{iK} \theta _{NK} w_{iN} }{w_{iN} })\); the formula above can be simplified as \(f\left( {b_i {|}{\varvec{\theta }},a_i ,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \propto p\left( {b_i } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}_b +\mathbf{1}b_i } \right) ^{T}{\varvec{\varOmega }}_b \left( {{{\varvec{z}}}_b +\mathbf{1}b_i } \right) } \right\} \) with both \({\varvec{\varOmega }} _{\mathrm{b}} =diag\left( {w_{i1} ,\ldots ,w_{iN} } \right) \) and \({{\varvec{k}}}_i ={{\varvec{y}}}_{{\varvec{i}}} -\frac{1}{2}\). This finding is given in line 2 of Eq. 12.

Let \({\varvec{\beta }} =a_{id} \), \({{\varvec{X}}}\) = (\({\varvec{\Theta }},{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) } ,b_i )\), and \(p\left( {a_{id} } \right) \) be the prior for \(a_{id} \). Given \({\varvec{\Theta }}\), \({{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) } \), \(b_i \), and \({\varvec{\varOmega }}_{{\varvec{a}}} \) are constant, the conditional posterior distribution for \(a_{id} \) can be obtained as:

$$\begin{aligned}&f\left( {a_{id} {|}{\varvec{\Theta }},{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) } ,b_i ,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \\&\quad \propto p\left( {a_{id} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) }^T {\varvec{\Theta }}_{-d} +a_{id} {\varvec{\Theta }}_d -\mathbf{1}b_i } \right) } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{a}}} \left( {{{\varvec{z}}}-\left( {{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) }^T {\varvec{\Theta }}_{-d} +a_{id} {\varvec{\Theta }}_d -\mathbf{1}b_i } \right) } \right) } \right\} ,\\&\quad \propto p\left( {a_i } \right) \exp \left\{ {-\frac{1}{2}\left( {\left( {{{\varvec{z}}}-{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) }^T {\varvec{\Theta }}_{-d} +\mathbf{1}b_i } \right) -a_{id} {\varvec{\Theta }}_d } \right) ^{T}{\varvec{\varOmega }} _{{\varvec{a}}} \left( {\left( {{{\varvec{z}}}-{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) }^T {\varvec{\Theta }}_{-d} +\mathbf{1}b_i } \right) -a_{id} {\varvec{\Theta }}_d } \right) } \right\} . \end{aligned}$$

Let \({{\varvec{z}}}_{ad} ={{\varvec{z}}}-{{\varvec{a}}}_{{{\varvec{i}}}\left( {-d} \right) }^T {\varvec{\Theta }}_{-d} +\mathbf{1}b_i \) and (1) if d = 1, \({{\varvec{z}}}_b =(\frac{k_{i1} -a_{i2} \theta _{12} w_{i1} -\ldots -a_{iK} \theta _{1K} w_{i1} +b_i w_{i1} }{w_{i1} }\), ..., \(\frac{k_{iN} -a_{i2} \theta _{N2} w_{iN} -\ldots -a_{iK} \theta _{NK} w_{i1} +b_i w_{iN} }{w_{iN} })\); (2) if d = K, \({{\varvec{z}}}_b =(\frac{k_{i1} -a_{i1} \theta _{11} w_{i1} -\ldots -a_{i\left( {K-1} \right) } \theta _{1\left( {K-1} \right) } w_{i1} +b_i w_{i1} }{w_{i1} }\), ..., \(\frac{k_{iN} -a_{i1} \theta _{N1} w_{iN} -\ldots -a_{i\left( {K-1} \right) } \theta _{N\left( {K-1} \right) } w_{iN} +b_i w_{iN} }{w_{iN} })\); and (3) if \(d \in \) (1, K), \({{\varvec{z}}}_b =(\frac{k_{i1} -a_{i1} \theta _{11} w_{i1} -\ldots -a_{i\left( {d-1} \right) } \theta _{1\left( {d-1} \right) } w_{i1} -a_{i\left( {d+1} \right) } \theta _{1\left( {d+1} \right) } w_{i1} -\ldots -a_{iK} \theta _{1K} w_{i1} +b_i w_{i1} }{w_{i1} }\), ..., \(\frac{k_{iN} -a_{i1} \theta _{N1} w_{iN} -\ldots -a_{i\left( {d-1} \right) } \theta _{N\left( {d-1} \right) } w_{iN} -a_{i\left( {d+1} \right) } \theta _{N\left( {d+1} \right) } w_{iN} -\ldots -a_{iK} \theta _{NK} w_{i1} +b_i w_{iN} }{w_{iN} })\). Therefore, the posterior above can be simplified as \(f\left( {a_{id} {|}{\varvec{\Theta }},{{\varvec{a}}}_{{{\varvec{i}}}\left( {-{{\varvec{d}}}} \right) } ,b_i ,{{\varvec{y}}}_{{\varvec{i}}} ,{{\varvec{w}}}} \right) \propto p\left( {a_{id} } \right) \exp \left\{ {-\frac{1}{2}\left( {{{\varvec{z}}}_{ad} -a_{id} {\varvec{\Theta }}_d } \right) ^{T} {\varvec{\varOmega }} _{{\varvec{a}}} \left( {{{\varvec{z}}}_{ad} -a_{id} {\varvec{\Theta }}_d } \right) } \right\} \) with both \({\varvec{\varOmega }} _{{\varvec{a}}} =diag\left( {w_{i1} ,\ldots ,w_{iN} } \right) \) and \({{\varvec{k}}}_i ={{\varvec{y}}}_{{\varvec{i}}} -\frac{1}{2}\). This finding is given in line 3 of Eq. 12.

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Jiang, Z., Templin, J. Gibbs Samplers for Logistic Item Response Models via the Pólya–Gamma Distribution: A Computationally Efficient Data-Augmentation Strategy. Psychometrika 84, 358–374 (2019). https://doi.org/10.1007/s11336-018-9641-x

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