Dependence on a collection of Poisson random variables

Abstract

We propose two novel ways of introducing dependence among Poisson counts through the use of latent variables in a three levels hierarchical model. Marginal distributions of the random variables of interest are Poisson with strict stationarity as special case. Order–p dependence is described in detail for a temporal sequence of random variables. A full Bayesian inference of the models is described and performance of the models is illustrated with a numerical analysis of maternal mortality in Mexico. Extensions to seasonal, periodic, spatial or spatio-temporal dependencies, as well as coping with overdispersion, are also discussed.

Appendix

Full conditional distributions for model parameters $\varvec{\theta }$ and latent variables $(\mathbf{Y},\mathbf{W})$ to perform posterior inference for type A and type B models. For simplicity we assume that $Y_t=0$, $W_t=0$ and $\alpha _t=0$ for $t\le 0$. In the sequel, we use $I_{\mathcal {X}}(x)$ to denote the indicator function that takes the value of one if $x\in \mathcal {X}$ and zero otherwise.

For type A model, the required full conditional distributions are:

• (i) For $Y_t$, $t=1,\ldots ,T$

  $$\begin{aligned} f(y_t\mid \text{ rest})\propto \frac{\left[ \alpha _t\mu ^{-p}\left\{ \prod _{j=0}^p\left( 1-\sum _{i=0}^p\alpha _{t+j-i}\right) \right\} ^{-1}\right] ^{y_t}}{y_t!\prod _{j=0}^p\left( x_{t+j}-\sum _{i=0}^p y_{t+j-i}\right) !}I_{\{0,\ldots ,c_t\}}(y_t), \end{aligned}$$

  with $c_t=\min _{j=0,\ldots ,p}\{x_{t+j}-\sum _{i=0,i\ne j}^p y_{t+j-i}\}$

• (ii) For $\alpha _t$, $t=1,\ldots ,T$

  $$\begin{aligned} f(\alpha _t\mid \text{ rest})\propto & {} \alpha _t^{a_\alpha +y_t-1}(1-\alpha _t)^{b_\alpha -1}e^{p\mu \alpha _t}\\&\prod _{j=0}^p\left( 1-\sum _{i=0}^p\alpha _{t+j-i}\right) ^{x_{t+j}-\sum _{i=0}^p y_{t+j-i}}I_{(0,d_t)}(\alpha _t) \end{aligned}$$

  where $d_t=\min _{j=0,\ldots ,p}\left\{ 1-\sum _{i=0,i\ne j}^p\alpha _{t+j-i}\right\}$

• (iii) For $\mu$

  $$\begin{aligned} f(\mu \mid \text{ rest})=\text{ Ga }\left( \mu \left| a_\mu +\sum _{t=1}^T x_t-\sum _{t=1}^T\sum _{i=1}^p y_{t-i},b_\mu +T+\sum _{t=1}^T\sum _{i=1}^p\alpha _{t-i}\right. \right) \end{aligned}$$

Since (i) is a discrete distribution with bounded support, we simply evaluate at all points of the support and normalize to obtain the probability density and sample a new $y_t^{(l)}$ at iteration l. To sample from (ii) we implement a MH step with random walk proposal distribution. If $\alpha _t^{(l)}$ is the current state of the chain, we sample from $\alpha _t^*\mid \alpha _t^{(l)}\sim \text{ Un }(\max (0,\alpha _t^{(l)}-\delta _\alpha ,\min (d_t,\alpha _t^{(l)}+\delta _\alpha )))$, that is a continuous uniform distribution, and accept it with probability $\min \{1,f(\alpha _t^*\mid \text{ rest})/f(\alpha _t^{(l)}\mid \text{ rest})\}$. Sampling from (iii) is direct since it has a standard form.

For type B model, the required full conditional distributions are:

• (iv) For $Y_t$, $t=1,\ldots ,T$

  $$\begin{aligned} f(y_t\mid \text{ rest})\propto \frac{\left\{ \alpha _t\mu ^{-1}(1-\alpha _t)^{-2}\right\} ^{y_t}}{(x_t-y_t)!y_t!\left( \sum _{i=0}^p w_{t-i}-y_t\right) !}I_{\{0,\ldots ,m_t\}}(y_t), \end{aligned}$$

  with $m_t=\min \{x_{t},\sum _{i=0}^p w_{t-i}\}$

• (v) For $W_t$, $t=1,\ldots ,T$

  $$\begin{aligned} f(w_t\mid \text{ rest})\propto & {} \left\{ \prod _{j=0}^p {{\sum _{i=0}^p w_{t+j-i}}\atopwithdelims (){y_{t+j}}}\right\} \left\{ \frac{\mu }{p+1}\prod _{j=0}^p(1-\alpha _{t+j})\right\} ^{w_t}\\&\times \frac{1}{w_t!}I_{\{h_t,h_t+1\ldots ,\}}(w_t), \end{aligned}$$

  where $h_t=\max _{j=0,\ldots ,p}\{y_{t+j}-\sum _{i=0,i\ne j}^p w_{t+j-i}\}$

• (vi) For $\alpha _t$, $t=1,\ldots ,T$

  $$\begin{aligned} f(\alpha _t\mid \text{ rest})\propto \alpha _t^{a_\alpha +y_t-1}(1-\alpha _t)^{b_\alpha +x_t+\sum _{i=0}^p w_{t-i}-2y_t-1}e^{\mu \alpha _t}I_{(0,1)}(\alpha _t) \end{aligned}$$

• (vii) For $\mu$

  $$\begin{aligned} f(\mu \mid \text{ rest})=\text{ Ga }\left( \mu \left| a_\mu +\sum _{t=1}^T (x_t+w_t-y_t),b_\mu +T\left( \frac{p+2}{p+1}\right) -\sum _{t=1}^T\alpha _{t}\right. \right) \end{aligned}$$

Again, since (iv) is a discrete distribution with bounded support, we proceed as for (i). To sample from (v) we note that the support is discrete but unbounded, so we implement a MH step with random walk proposal of the form $W_t^*\mid W_t^{(l)}=w_t^{(l)}\sim \text{ Un }(\max (h_t,w_t^{(l)}-\delta _w),w_t^{(l)}+\delta _w)$ and accept it with probability $\min \{1,f(w_t^*\mid \text{ rest})/f(w_t^{(l)}\mid \text{ rest})\}$. To sample from (vi) we proceed as for (ii) but with proposal $\alpha _t^*\mid \alpha _t^{(l)}\sim \text{ Un }(\max (0,\alpha _t^{(l)}-\delta _\alpha ,\min (1,\alpha _t^{(l)}+\delta _\alpha )))$. Finally, sampling from (vii) is direct. In all cases, $\delta _\alpha$ and $\delta _w$ are tuning parameters that control the acceptance probability.