---
title: "Conjugate Hierarchical Models"
author: "Paul Northrop"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Conjugate Hierarchical Models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: bang.bib
---

```{r, include = FALSE}
knitr::opts_chunk$set(comment = "#>", collapse = TRUE)
set.seed(47)
```

The *bang* package simulates from the posterior distributions involved in certain Bayesian models.  See the vignette [Introducing bang: Bayesian Analysis, No Gibbs](bang-a-vignette.html) for an introduction.  In this vignette we consider the Bayesian analysis of certain conjugate hierarchical models.  We give only a brief outline of the structure of these models.  For a full description see Chapter 5 of @BDA2014.

Suppose that, for $j = 1, \ldots, J$, experiment $j$ of $J$ experiments yields a data vector $Y_j$ and associated parameter vector $\theta_j$.  Conditional on $\theta_j$ the data $Y_j$ are assumed to follow independent response distributions from the exponential family of probability distributions.  A prior distribution $\pi(\theta_j \mid \phi)$ is placed on each of the *population parameters* $\theta_1, \ldots, \theta_J$, where $\phi$ is a vector of hyperparameters.  For mathematical convenience $\pi(\theta_j \mid \phi)$ is selected to be *conditionally conjugate*, that is, conditionally on $\phi$ the posterior distribution of $\theta_j$ of the same type as $\pi(\theta_j \mid \phi)$. 

Use of a conditionally conjugate prior means that it is possible to derive, and simulate from, the marginal posterior density $\pi(\phi \mid \boldsymbol{\mathbf{y}})$.  The `hef` function does this using the generalized ratio-of-uniforms method, implemented by the function `ru` in the *rust* package [@rust]).  By default a model-specific transformation of the parameter vector $\phi$ is used to improve efficiency.  See the documentation of `hef` and the two examples below for details.  Simulation from the full posterior density $\pi(\boldsymbol{\mathbf{\theta}}, \phi \mid \boldsymbol{\mathbf{y}}) = \pi(\boldsymbol{\mathbf{\theta}} \mid \phi, \boldsymbol{\mathbf{y}}) \pi(\phi \mid \boldsymbol{\mathbf{y}})$ follows directly because conditional conjugacy means that it is simple to simulate from $\pi(\boldsymbol{\mathbf{\theta}} \mid \phi, \boldsymbol{\mathbf{y}})$, given the values simulated from $\pi(\phi \mid \boldsymbol{\mathbf{y}})$.  The simulation is performed in the function `hef`.

## Beta-binomial model

We consider the example presented in Section 5.3 of @BDA2014, in which the data [@Tarone1982] in the matrix `rat` are analysed.  These data contain information about an experiment in which, for each of 71 groups of rats, the total number of rats in the group and the numbers of rats who develop a tumor is recorded, so that $J = 71$. Conditional on $\boldsymbol{\mathbf{\theta}} = (\theta_1, \ldots, \theta_J) = (p_1, \ldots, p_J)$ we assume independent binomial distributions for $(Y_1, \ldots, Y_J)$, that is, $Y_j \sim {\rm binomial}(n_j, p_j)$.  We use the conditionally conjugate priors $p_j \sim {\rm Beta}(\alpha, \beta)$, so that $\phi = (\alpha, \beta)$.

The conditional conjagacy of the priors means that the marginal posterior of $(\alpha, \beta)$ given $\boldsymbol{\mathbf{y}} = (y_1, \ldots, y_J)$ can be determined (equation (5.8) of @BDA2014) as 
\[ \pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}}) \propto \pi(\alpha, \beta) 
\prod_{j=1}^J \frac{B(\alpha + y_j, \beta + n_j - y_j)}{B(\alpha, \beta)}, \]
where $\pi(\alpha, \beta)$ is the hyperprior density for $(\alpha, \beta)$. 
By default $\phi = (\alpha, \beta)$ is transformed prior to sampling using $(\rho_1, \rho_2)=(\log (\alpha/\beta), \log (\alpha+\beta))$. The aim of this is to improve efficiency by rotating and scaling the (mode-relocated) conditional posterior density in an attempt to produce near circularity of this density's contours.    

To simulate from the full posterior density $\pi(\boldsymbol{\mathbf{\theta}}, \alpha, \beta \mid \boldsymbol{\mathbf{y}}) = \pi(\boldsymbol{\mathbf{\theta}} \mid \alpha, \beta, \boldsymbol{\mathbf{y}}) \pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}})$ we first sample from $\pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}})$.  Simulation from the conditional posterior distribution of $\boldsymbol{\mathbf{\theta}}$ given $(\alpha, \beta, \boldsymbol{\mathbf{y}})$ is then straightforward on noting that 
\[ \theta_j \mid \alpha, \beta, y_j \sim {\rm beta}(\alpha + y_j, \beta + n_j - y_j) \]
and that $\theta_j, j = 1, \ldots, J$ are conditionally independent.

The hyperprior for $(\alpha, \beta)$ used by default in `hef` is $\pi(\alpha, \beta) \propto (\alpha+\beta)^{-5/2}, \alpha>0, \beta>0$, following Section 5.3 of @BDA2014.  A user-defined prior may be set using `set_user_prior`.

```{r, fig.show='hold', fig.width = 3.45, fig.height = 3.45}
library(bang)
# Default prior, sampling on (rotated) (log(mean), log(alpha + beta)) scale
rat_res <- hef(model = "beta_binom", data = rat, n = 10000)
plot(rat_res)
plot(rat_res, ru_scale = TRUE)
```

The plot on the left shows the values sampled from the posterior distribution of $(\alpha, \beta)$ with superimposed density contours.  On the right is a similar plot displayed on the scale used for sampling, that is, $(\rho_1, \rho_2)$.  

The following summary is of properties of the generalized ratio-of uniforms algorithm, in particular the probability of acceptance, and summary statistics of the posterior sample of $(\alpha, \beta)$.

```{r}
summary(rat_res)
```

## Gamma-Poisson Model

We perform a fully Bayesian analysis of an empirical Bayesian example presented in Section 4.2 of @GS1990, who fix the hyperparameter $\alpha$ described below at a point estimate derived from the data.  The `pump` dataset [@GoM1987] is a matrix in which each row gives information about one of 10 different pump systems.  The first column contains the number of pump failures.  The second column contains the length of operating time, in thousands of hours.

```{r}
pump
```

The general setup is similar to the beta-binomial model described above but now the response distribution is taken to be Poisson, the prior distribution is gamma and $J = 10$.  For $j = 1, \ldots, J$ let $Y_j$ denote the number of failures and $e_j$ the length of operating time for pump system $j$.  Conditional on $\boldsymbol{\mathbf{\lambda}} = (\lambda_1, \ldots, \lambda_J)$ we assume independent Poisson distributions for $(Y_1, \ldots, Y_J)$ with means that are proportional to the *exposure* time $e_j$, that is, $Y_j \sim {\rm Poisson}(e_j \lambda_j)$.  We use the conditionally conjugate priors $\lambda_j \sim {\rm gamma}(\alpha, \beta)$, so that $\phi = (\alpha, \beta)$.  We use the parameterization where $\beta$ is a rate parameter, so that ${\rm E}(\lambda_j) = \alpha/\beta$ *a priori*.

The marginal posterior of $(\alpha, \beta)$ given $\boldsymbol{\mathbf{y}} = (y_1, \ldots, y_J)$ can be determined as 
\[ \pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}}) \propto \pi(\alpha, \beta) 
\prod_{j=1}^J \frac{\beta^\alpha\Gamma(\alpha + y_j)}{(\beta+e_j)^{\alpha+y_j} \Gamma(\alpha)}, 
\]
where $\pi(\alpha, \beta)$ is the hyperprior density for $(\alpha, \beta)$.  The scale used for sampling is $(\rho_1, \rho_2)=(\log (\alpha/\beta), \log \beta))$.

To simulate from the full posterior $\pi(\boldsymbol{\mathbf{\lambda}}, \alpha, \beta \mid \boldsymbol{\mathbf{y}}) = \pi(\boldsymbol{\mathbf{\lambda}} \mid \alpha, \beta, \boldsymbol{\mathbf{y}}) \pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}})$ we first sample from $\pi(\alpha, \beta \mid \boldsymbol{\mathbf{y}})$ and then note that \[ \lambda_j \mid \alpha, \beta, y_j \sim {\rm gamma}(\alpha + y_j, \beta + e_j) \]
and that $\lambda_j, j = 1, \ldots, J$ are conditionally independent.

By default `hef` takes $\alpha$ and $\beta$ to be independent gamma random variables *a priori*.  The parameters of these gamma distributions can be set by the user, using the argument `hpars` or a different prior may be set using `set_user_prior`.

We produce similar output to the beta-binomial example above.  

```{r, fig.show='hold', fig.width = 3.45, fig.height = 3.45}
pump_res <- hef(model = "gamma_pois", data = pump, hpars = c(1, 0.01, 1, 0.01))
plot(pump_res)
plot(pump_res, ru_scale = TRUE)
summary(pump_res)
```

## References

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