Bayesian pooling for analyzing categorical data from small areas
Section 4. Conclusion

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In this study, we construct hierarchical parametric Bayesian pooling models and their nonparametric versions using the Ferguson (1973) Dirichlet process prior to pool the data. The pooling methodologies developed here are useful for analyzing survey data. We used the grid method to draw the parameters with nonstandard posterior densities and support that lies in a finite interval. However, we used the Metropolis-Hastings algorithm to draw the parameters with support in an infinite interval.

The Dirichlet process is assumed for the parameter of interest ${\pi }_i,$ for $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I,$ in our models. We apply the slice sampling algorithm for the specification of Dirichlet process prior, which is an extension of the widely used stick-breaking prior proposed by Ishwaran and James (2001). Five parametric models are modeled in a finite-dimensional parameter space, and three nonparametric versions have an infinite-dimensional parameter space. The eight hierarchical Bayesian models are also distinguished according to the type of effects in the model parameters. For the basic model (2.1), we can construct more effective and efficient models that allow for a borrowing effect from neighboring areas in small-area estimations. However, exchangeable priors in a hierarchical Bayesian model may cause an overshrinkage problem. To compensate for this problem, the effect of a parameter is divided into two elements, as shown in the basic model (2.2), called the hierarchical Bayesian global-local pooling model. The model allows for grouping of similar experiments (area) and the borrowing of information in each area.

To compare the eight models using real data, we use the BMD data provided by the NHANES III. BMD is statistically correlated with the probability of fractures, which are an important public health problem, especially in elderly women. Therefore, BMD is an important indicator in diagnoses of osteoporosis, where patients might benefit from early management to improve their bone strength. For each sample, we assign an indicator based on three categories (normal, osteopenia, osteoporosis) before analyzing the data. The resulting hierarchical models with a pooling prior for BMD data outperformed the other models. To compare the models’ performances, we calculated the DIC and the LCPO. Here, we found the best performance in the global $–$ local pooling model. Although the nonparametric versions of the models have an infinite-dimensional parameter space, they showed similar values for the two comparison measures to those of the parametric pooling model with a finite-dimensional parameter space. Therefore, we should be careful in interpreting the results.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2016R1D1A1B03932261). This research was also supported by a grant from the Simons Foundation (#353953, Balgobin Nandram).

Appendix

A. Computations for parameteric models

Let $n\mathrm{<mo>=</mo>}{\left(n_1\mathrm{,}\dots ,n_I\right)}^{\prime }$ be the response matrix and $\pi \mathrm{<mo>=</mo>}{\left({\pi }_1\mathrm{,}\dots ,{\pi }_I\right)}^{\prime }$ be the proportion parameter matrix for (2.1). In an adaptive pooling model, let ${\Omega }_i\mathrm{<mo>=</mo>}\left({\pi }_i\mathrm{,}\mu \mathrm{,}\tau \right),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I.$ Here, no pooling and complete pooling are special cases of adaptive pooling, with parameters $\mu$ and $\tau .$ The full conditional posterior density of the parameters for the given data is obtained in the usual way by combining the likelihood and the priors, as follows:

$$\begin{array}{ll}\pi \left({\Omega }_1\mathrm{,}\dots ,{\Omega }_I|n\right)\hfill & \propto \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^{I}f\left(n|{\pi }_i\right)\pi \left({\pi }_i\right)}\right\}\pi \left(\mu \mathrm{,}\tau \right)\hfill \\ \hfill & \propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left\{\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{n_{ik}+{\mu }_k\tau -1}}\right\}\frac{\left(K-1\right)!}{{\left(\tau +1\right)}^2}\mathrm{,}}\hfill \end{array}$$

where $D\left(\mu \tau \right)\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\tau \right)/\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\tau }\right)}.$

To run a Gibbs sampler, we draw values as follows:

$$\pi \left(\mu |n\mathrm{,}\pi \mathrm{,}\tau \right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left\{\frac{\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\tau }\right)}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\tau \right)}}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}\right\}\mathrm{.}}(\text{A}.1)$$

Let ${\mu }_{(k)}\mathrm{,}k\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,K$ denote the vector of parameters other than the $k^{\text{th}}$ component ${\mu }_k.$ Then, we obtain the conditional posterior density of ${\mu }_k,$ given ${\mu }_{(k)},$ in each stage. Here, we need to estimate the $K-1$ components of parameter $\mu$ sequentially. Then, we can calculate the $K^{\text{th}}$ component value of $\mu$ using ${\mu }_K\mathrm{<mo>=</mo>}1-{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\mu }_k}.$ Using the conditional posterior density (3.1), we can draw ${\mu }_k\mathrm{,}k\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,K-1$ using the grid method, with support $1-{\displaystyle {\sum }_{k^{\prime }\mathrm{<mo>=</mo>\; 1,}{k}^{\prime }\ne k}^{K-1}{\mu }_{k^{\prime }}}.$

$$\pi \left(\tau |n\mathrm{,}\pi \mathrm{,}\mu \right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left\{\frac{\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\tau }\right)}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\tau \right)}}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}\right\}\frac{1}{{\left(\tau +1\right)}^2}\mathrm{.}}$$

We can use the grid method for $\tau$ in this case as well. Because the grid method can be used for closed support, we transform $\tau$ to $\rho \mathrm{<mo>=</mo>}1/\left(1+\tau \right)\mathrm{,}0\mathrm{<mo><</mo>}\rho \mathrm{<mo><</mo>}1.$ The absolute Jacobian is $1/{\rho }^2.$ Then, the conditional posterior density of $\rho$ can be expressed as follows:

$$\pi \left(\rho |n\mathrm{,}\pi \mathrm{,}\mu \right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left\{\frac{\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\frac{1-\rho }{\rho }}\right)}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\frac{1-\rho }{\rho }\right)}}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\frac{1-\rho }{\rho }-1}}\right\}.}$$

The full conditional posterior density in the case of restricted pooling is obtained from the likelihood and the priors, constructed from $\pi \left(\mu \mathrm{,}\tau \right),$ and from an additional prior for $\varphi \mathrm{,}i\mathrm{<mo>=</mo>}1,\dots ,I\text{:}$

$$\pi \left({\Omega }_1\mathrm{,}\dots ,{\Omega }_I|n\right)\mathrm{<mo>=</mo>}\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^{I}f\left(n|{\pi }_i\right)\pi \left({\pi }_i\right)}\right\}\pi \left(\mu \mathrm{,}\tau \right)\pi \left(\varphi \right)\mathrm{,}$$

where ${\Omega }_i\mathrm{<mo>=</mo>}\left({\pi }_i\mathrm{,}\mu \mathrm{,}\tau \mathrm{,}\varphi \right),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I.$

For the Gibbs sampler, we consider the latent variables $z_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I$ from a Bernoulli distribution with parameter $\varphi .$ Here, the joint posterior density is given as follows:

$$\begin{array}{ll}\pi \left({\Omega }_1\mathrm{,}\dots ,{\Omega }_I|n\right)\hfill & \propto \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^{I}f\left(n|{\pi }_i\right)\pi \left({\pi }_i|z_i\right)\pi \left(z_i\right)}\right\}\pi \left(\mu \mathrm{,}\tau \right)\pi \left(\varphi \right)\hfill \\ \hfill & \propto \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I{\left[\varphi \frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau +n_{ik}-1}}\right]}^{z_i}{\left[\left(1-\varphi \right){\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{n_{ik}}}\right]}^{1-z_i}}\right\}\hfill \\ \hfill & \times \frac{\left(K-1\right)!}{{\left(\tau +1\right)}^2}{I}_{\left(\varphi \in \left(\frac{1}{2}\mathrm{,}1\right)\right)}.\hfill \end{array}$$

Then, our Gibbs sampler is described as follows:

$$\pi \left(\varphi |n\mathrm{,}\mu \mathrm{,}\tau \mathrm{,}z\right)\propto {\varphi }^{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^I{z}_i}}{\left(1-\varphi \right)}^{I-{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^I{z}_i}}\mathrm{,}\frac{1}{2}\le \varphi \le 1.$$

In other words, given $\mu \mathrm{,}\tau \mathrm{,}z,$ and the data, $\varphi$ follows a truncated beta distribution with parameters ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^I{z}_i}$ and $I-{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^I{z}_i},$ and the lower bound of $\varphi$ is $1/2.$

$$p_i\mathrm{<mo>=</mo>}\left(\varphi {\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}/D\left(\mu \tau \right)\right)/\left(\varphi {\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}/D\left(\mu \tau \right)+\left(1-\varphi \right)I_{\left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}\mathrm{<mo>=</mo>\; 1}}\right)}/K\right).$$

$$\pi \left({\pi }_i|n\mathrm{,}{z}_i\mathrm{,}\mu \mathrm{,}\tau \right)\propto {\left[{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau +n_{ik}-1}}\right]}^{z_i}{\left[{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{1+n_{ik}-1}}\right]}^{1-z_i}\mathrm{.}$$

In the case of $z_i\mathrm{<mo>=</mo>}1,$ we can generate the value of ${\pi }_i$ from a Dirichlet distribution with parameters $\mu$ and $\tau .$ In other cases, we can interpret that as the uncertainty of the modeling. That is, for ${\pi }_i$ given $z_1\mathrm{<mo>=</mo>}0,$ $n$ draws its value from a uniform Dirichlet distribution with a $K\times 1$ parameter vector, where each component has the value one.

$$\pi \left(\mu |n\mathrm{,}\pi \mathrm{,}\tau \right)\propto {\displaystyle \prod _{z_i\mathrm{<mo>=</mo>\; 1}}\left\{\frac{\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\tau }\right)}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\tau \right)}}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}\right\}}.$$

$$\pi \left(\tau |n\mathrm{,}\pi \mathrm{,}\mu \right)\propto {\displaystyle \prod _{z_i\mathrm{<mo>=</mo>\; 1}}\left\{\frac{\Gamma \left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\tau }\right)}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_k\tau \right)}}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}\right\}\frac{1}{{\left(\tau +1\right)}^2}\mathrm{.}}$$

Of course, the generating process for parameters $\mu$ and $\tau$ is similar to that of the parameter in adaptive pooling. In addition, the data used for $\mu$ and $\tau$ are $z_i\mathrm{<mo>=</mo>}\mathrm{1,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I.$

Otherwise, the parameter vector corresponding to area $i$ in global-local pooling consists of $\theta \mathrm{,}\eta \mathrm{,}z\mathrm{,}\varphi \mathrm{,}{\sigma }^2\mathrm{,}{\sigma }_1^2\mathrm{,}\dots ,$ and ${\sigma }_{K-1}^2.$ Then, the full conditional posterior density for the given data in the model is as follows:

$$\begin{array}{ll}\pi \left({\Omega }_1\mathrm{,}\dots ,{\Omega }_I|n\right)\hfill & \propto \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^{I}f\left(n_i|{\eta }_i\mathrm{,}\theta \right)}\right\}\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\pi \left({\eta }_i|z_i\mathrm{,}{\sigma }^2\mathrm{,}{\sigma }_1^2\mathrm{,}\dots ,{\sigma }_{K-1}^2\right)}\right\}\pi \left(\text{}{\sigma }^2\text{}\right)\hfill \\ \hfill & \times \pi \left({\sigma }_1^2\mathrm{,}\dots ,{\sigma }_{K-1}^2\right)\pi \left(\text{}\theta \text{}\right)\pi \left(z|\varphi \right)\pi \left(\text{}\varphi \text{}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\frac{n_{i\cdot }\mathrm{<mo>!</mo>}}{{\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K{n}_{ik}!}}\left\{{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\left(\frac{e^{\theta +{\eta }_{ik}}}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{il}}}}\right)}^{n_{ik}}}\right\}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{ik}}}}\right)}^{n_{iK}}}\right\}\hfill \\ \hfill & \times \left\{{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left[{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{\sqrt{2\pi {\sigma }^2}}\text{exp}\left(-\frac{1}{2{\sigma }^2}{\eta }_{ik}^2\right)}\right]}}^{z_i}{\left[{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{\sqrt{2\pi {\sigma }_k^2}}\text{exp}\left(-\frac{1}{2{\sigma }_k^2}{\eta }_{ik}^2\right)}\right]}^{1-z_i}\right\}\hfill \\ \hfill & \times \frac{1}{{\left(1+{\sigma }^2\right)}^2}\left\{{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{{\left(1+{\sigma }_k^2\right)}^2}}\right\}\frac{1}{\pi \left(1+{\theta }^2\right)}\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I{\varphi }^{z_i}{\left(1-\varphi \right)}^{1-z_i}}\right\}{I}_{\left(1/2\le \varphi \le 1\right)}\mathrm{,}\hfill \end{array}$$

where ${\Omega }_i\mathrm{<mo>=</mo>}\left(\theta \mathrm{,}{\eta }_i\mathrm{,}{z}_i\mathrm{,}\varphi \mathrm{,}{\sigma }^2\mathrm{,}{\sigma }_1^2\mathrm{,}\dots \mathrm{,}{\sigma }_{K-1}^2\right),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I,$ $\eta \mathrm{<mo>=</mo>}{\left({\eta }_1\mathrm{,}\dots ,{\eta }_I\right)}^{\prime }$ and $z\mathrm{<mo>=</mo>}{\left(z_1\mathrm{,}\dots ,z_I\right)}^{\prime }.$

Then, the Gibbs sampler is as follows:

$$\pi \left(\theta |\text{others}\right)\propto \left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\left(\frac{e^{\theta +{\eta }_{ik}}}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{il}}}}\right)}^{n_{ik}}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{il}}}}\right)}^{n_{iK}}}}\right]\frac{1}{1+{\theta }^2}\mathrm{.}$$

$$\begin{array}{ll}\pi \left({\eta }_{ik}|\text{others}\right)\hfill & \propto I_{\left(z_i=1\right)}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{il}}}}\right)}^{{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{n}_{ik}}}\text{exp}\left\{{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\eta }_{ik}{n}_{ik}}-\frac{1}{2}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{{\sigma }^2}{\eta }_{ik}^2}\right\}\hfill \\ \hfill & +I_{\left(z_i=0\right)}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{il}}}}\right)}^{{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{n}_{ik}}}\text{exp}\left\{{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\eta }_{ik}{n}_{ik}}-\frac{1}{2}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{{\sigma }_k^2}{\eta }_{ik}^2}\right\}\mathrm{.}\hfill \end{array}$$

$$\pi ({\sigma }^2|\text{others})\propto \frac{1}{{\sigma }^{(K-1){\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^{I}I(z_i=1)}}{(1+{\sigma }^2)}^2}\text{exp}\left\{-\frac{1}{2{\sigma }^2}{\displaystyle \sum _{z_i\mathrm{<mo>=</mo>1}}{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^{K-1}{\eta }_{ik}^2}}\right\}.$$

$$\pi ({\sigma }^2|\text{others})\propto \frac{1}{{\sigma }_k^{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^I{I}_{(z_i=0)}}}{(1+{\sigma }_k^2)}^2}\text{exp}\left\{-\frac{1}{2{\sigma }_k^2}{\displaystyle \sum _{z_i\mathrm{<mo>=</mo>0}}{\eta }_{ik}^2}\right\}.$$

In this model, we suggest using the Metropolis-Hastings algorithm, which is the most commonly used Markov Chain Monte Carlo (MCMC) algorithm used to estimate the value of the location parameter, $\left(\theta \mathrm{,}\eta \right).$ Of course, $\left({\sigma }^2\mathrm{,}{\sigma }_1^2\mathrm{,}\dots ,{\sigma }_{\left(K-1\right)}^2\right)$ is drawn from the above full conditionals, and the Gibbs sampler is performed using the grid method.

B. Computations for nonparameteric models

In order to pool the parameters in nonparametric Bayesian models, we apply the slice sampling mehtod introduced by the Kalli, Griffin and Walker (2011). They proposed an efficient version of the slice sampler for Dirichlet process mixture models constructed by Walker (2007). Suppose that the observations $y_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,I$ are generated in the Dirichlet process mixture model with parameter $\theta .$ That is,

$$\begin{array}{l}{y}_i|\text{G}\stackrel{\text{iid}}{{\displaystyle ~}}\text{G}\\ \text{G}~\text{DP}\left(\alpha \mathrm{,}{\text{G}}_0\right)\end{array}$$

Here, we write $\text{G}~\text{DP}\left(\alpha \mathrm{,}{\text{G}}_0\right)$ to denote that $\text{G}$ follows a Dirichlet process with parameter $\alpha \mathrm{<mo>></mo>}0.$ Then $\text{G}$ has a stick-breaking representation (Sethuraman, 1994) given by

$$\begin{array}{l}P\left(y_i|G\mathrm{,}\alpha \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{\infty }{w}_{j}f\left(y_i|{\theta }_j\right)}\hfill \end{array}$$

where ${\theta }_1\mathrm{,}{\theta }_2\mathrm{,}{\theta }_3\mathrm{,}\dots$ are independent and identically distributed (iid) form ${\text{P}}_0$ and

$$\begin{array}{l}{w}_1\mathrm{<mo>=</mo>}{\nu }_1\mathrm{,}{w}_j\mathrm{<mo>=</mo>}{\nu }_j{\displaystyle \prod _{l<j}\left(1-{\nu }_l\right)}\hfill \end{array}$$

with the ${\nu }_j$ being iid from $\text{Beta}\left(\mathrm{1,}\alpha \right),$ see also Antoniak (1974).

The Slice sampler algorithm proposed by Walker (2007) introduces a latent variable $u\in \left(\mathrm{0,}1\right),$ $d_i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots$ to perform sampling on the joint distribution. First, the latent variable $u_i$ has a joint density as follow

$$\begin{array}{l}P\left(y_i\mathrm{,}{u}_i|G\mathrm{,}\alpha \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{\infty }{I}_{\left(u_i<w_j\right)}f\left(y_i|{\theta }_j\right).}\hfill \end{array}$$

Later, they introduced a latent variable $d_i$ representing the group assignment of the observation $i.$ At this time, the joint density of $\left(y_i\mathrm{,}{u}_i\mathrm{,}{d}_i\right)$ is as follow

$$\begin{array}{l}P\left(y_i\mathrm{,}{u}_i\mathrm{,}{d}_i|G\mathrm{,}\alpha \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{\infty }{I}_{\left(u_i<w_{d_i=j}\right)}f\left(y_i|{\theta }_j\right)}\mathrm{.}\hfill \end{array}$$

Then we need to sample the parameter $\theta ,$ $\alpha ,$ and $\nu$ including latent variables $u$ and $d$ at each iteration of a Gibbs sampler. Kalli et al. (2011) introduces how to perform slice sampling for Dirichlet process mixture models by processing $u$ and $\nu$ as blocks in the basic algorithm described by Walker (2007). The algorithm is as follow.

For this paper, we need to sample the following variables at each iteration of a Gibbs sampler:

$$\left\{\left({\pi }_j\mathrm{,}{\nu }_j\right)\mathrm{,}j\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots ,\mathrm{<mo>;</mo>}\left(d_i\mathrm{,}{u}_i\right)\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}I\right\}\mathrm{.}$$

In general, $\kappa$ is equal to 0.5. However, we use $\kappa$ as a tuning parameter for the hierarchical Bayesian model. Then, the full posterior density for the nonparametric adaptive pooling of he given data is given as follows:

$$\begin{array}{l}P\left(n\mathrm{,}d\mathrm{,}u|\nu \mathrm{,}\pi \right)\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\pi \left({\pi }_i|\mu \mathrm{,}\tau \right)}\right\}\pi \left(\mu \mathrm{,}\tau \right)\pi \left(\nu |\alpha \right)\pi \left(\text{}\alpha \text{}\right)\\ \propto \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I{I}_{\left(u_i<{\xi }_{d_i}\right)}\frac{w_{d_i}}{{\xi }_{d_i}}{n}_{i\mathrm{.}}\mathrm{<mo>!</mo>}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K\frac{{\pi }_{d_{i}k}^{n_{ik}}}{n_{ik}\mathrm{<mo>!</mo>}}}}\right\}\left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{ik}^{{\mu }_k\tau -1}}}\right\}\\ \times \left\{\frac{\left(K-1\right)!}{{\left(1+\tau \right)}^2}\right\}\left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{B\left(\mathrm{1,}\alpha \right)}{\left(1-{\nu }_j\right)}^{\alpha -1}}\right\}\frac{1}{{\left(1+\alpha \right)}^2}\mathrm{,}\end{array}$$

where $d\mathrm{<mo>=</mo>}\left(d_1\mathrm{,}\dots ,d_I\right),$ $u\mathrm{<mo>=</mo>}\left(u_1\mathrm{,}\dots ,u_I\right),$ $\nu \mathrm{<mo>=</mo>}\left({\nu }_1\mathrm{,}\dots ,{\nu }_j\right),$ and the hyperparameters are mutually independent. Then, our Gibbs sampler can be performed in two steps.

The first step is the pooling of the data:

$$\pi \left(\mu \mathrm{<mo>|</mo>}\text{others}\right)\propto {\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\tau -1}\mathrm{.}}}$$

$$\pi \left(\tau \mathrm{<mo>|</mo>}\text{others}\right)\propto \left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\tau -1}}}\right\}\frac{\left(K-1\right)!}{{\left(\tau +1\right)}^2}\mathrm{.}$$

For our Gibbs sampler, we need to transform $\tau$ to $\rho \mathrm{<mo>=</mo>}1/\left(1+\tau \right),$ $0<\rho <1$ because we need to use the grid method for $\tau ,$ which is taken from the noninformative prior with variable support equal to $\left(0,\infty \right).$

$$\pi \left(\rho \mathrm{<mo>|</mo>}\text{others}\right)\propto \left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{D\left(\mu \frac{1-\rho }{\rho }\right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\frac{1-\rho }{\rho }-1}}}\right\}\left(K-1\right)\mathrm{<mo>!</mo>\; .}$$

$$\pi \left(\alpha \mathrm{<mo>|</mo>}\text{others}\right)\propto \left\{{\displaystyle \prod _{j=1}^J{\left(1-{\nu }_j\right)}^{\alpha -1}}\right\}\frac{{\alpha }^J}{{\left(1+\alpha \right)}^2}\mathrm{.}$$

The parameter $\alpha$ is also taken from the noninformative prior with variable support equal to $\left(0,\infty \right),$ as in the case of $\tau .$ Therefore, we need to transform $\alpha$ to $\delta ,$ with Jacobian $1/{\delta }^2.$ Then, the conditional density for $\delta$ is given as follows:

$$\pi \left(\delta \mathrm{<mo>|</mo>}\text{others}\right)\propto \left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J}{\left(1-{\nu }_j\right)}^{\frac{1-\delta }{\delta }-1}\right\}{\left(\frac{1-\delta }{\delta }\right)}^J\mathrm{.}$$

The nonparametric version for the restricted pooling has ${\pi }_j\mathrm{<mo>=</mo>}\left({\pi }_{j1}\mathrm{,}\dots ,{\pi }_{jK}\right),$ $\mu \mathrm{<mo>=</mo>}\left({\mu }_1\mathrm{,}\dots ,{\mu }_K\right),$ $0\le {\mu }_k\le 1,$ ${\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\mu }_k\mathrm{<mo>=</mo>}1},$ $d\mathrm{<mo>=</mo>}\left(d_1\mathrm{,}\dots ,d_I\right),$ and $d_i\mathrm{<mo>=</mo>}j$ for $j\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots ,J.$ Then, we compose the full posterior density for the given data using equation, as follows:

$$P\left(n\mathrm{,}d\mathrm{,}u|\nu \mathrm{,}\pi \right)\left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\pi \left({\pi }_j|\mu \mathrm{,}\tau \mathrm{,}\varphi \right)}\right\}\pi \left(\mu \mathrm{,}\tau \right)\pi \left(\varphi \right)\pi \left(\nu |\alpha \right)\pi \left(\text{}\alpha \text{}\right)\mathrm{.}$$

In our Gibbs sampler, we consider the latent variables $z_j,$ for $j\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,J,$ as the parametric version for restricted pooling, where the subscripts for the parameter are equal to the parameter $\pi .$ At this time, the pooling step for the data is the same as above, and generating the parameter is as follows:

where $p_j\mathrm{<mo>=</mo>}\left(\varphi {\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\tau -1}}/D\left(\mu \tau \right)\right)/\left\{\varphi {\displaystyle {\prod }_{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\tau -1}/D\left(\mu \tau \right)+\left(1-\varphi \right)I_{\left({\displaystyle {\sum }_{k=1}^K{\pi }_{jk}\mathrm{<mo>=</mo>\; 1}}\right)}/K-1}\right\}.$

$$\pi \left(\mu |\text{others}\right)\propto {\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{{\mu }_k\tau -1}}}\mathrm{.}$$

$$\pi \left(\tau |\text{others}\right)\propto \left[{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^J\frac{1}{D\left(\mu \tau \right)}{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^K{\pi }_{jk}^{*{\mu }_k\tau -1}}}\right]\frac{\left(K-1\right)!}{{\left(\tau +1\right)}^2}\mathrm{.}$$

$$\pi \left(\alpha |\text{others}\right)\propto \left\{{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^m{\left(1-{\nu }_j\right)}^{\alpha -1}}\right\}\frac{{\alpha }^J}{{\left(1+\alpha \right)}^2}\mathrm{.}$$

Lastly, the full posterior density is calculated using the joint posterior density and the priors for their parameters in the nonparametric Bayesian global-local pooling model. Here, the data pooling algorithm is the same as above inference. Then, we estimate each parameter as follows:

$$\pi \left(\theta |\text{others}\right)\propto \left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I{\displaystyle \prod _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\left(\frac{e^{\theta +{\eta }_{d_{i}k}}}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{d_{i}l}}}}\right)}^{n_{ik}}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{d_{i}l}}}}\right)}^{n_{iK}}}}\right]\mathrm{.}$$

$$\begin{array}{ll}\pi \left({\eta }_{jk}|\text{others}\right)\hfill & \propto I_{\left(z_j=1\right)}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{jl}}}}\right)}^{{\displaystyle {\sum }_{d_i\mathrm{<mo>=</mo>}j}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{n}_{ik}}}}\text{exp}\left\{{\displaystyle \sum _{d_i\mathrm{<mo>=</mo>}j}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\eta }_{jk}{n}_{ik}}}-\frac{1}{2{\sigma }^2}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\eta }_{jk}^2}\right\}\hfill \\ \hfill & +I_{\left(z_i=0\right)}{\left(\frac{1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^{K-1}{e}^{\theta +{\eta }_{jl}}}}\right)}^{{\displaystyle {\sum }_{d_i\mathrm{<mo>=</mo>}j}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{n}_{ik}}}}\text{exp}\left\{{\displaystyle \sum _{d_i\mathrm{<mo>=</mo>}j}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}{\eta }_{jk}{n}_{ik}}}-\frac{1}{2}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K-1}\frac{1}{{\sigma }_k^2}{\eta }_{jk}^2}\right\}\mathrm{.}\hfill \end{array}$$