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+\documentclass[11pt, a4paper, titlepage]{report}
+\usepackage{amsmath}
+\usepackage{natbib}
+\usepackage{a4wide}
+\usepackage{url}
+\usepackage{multirow}
+\usepackage{amsfonts}
+\newcommand{\release}{2.2.0}
+\newcommand{\JAGS}{\textsf{JAGS}}
+\newcommand{\rjags}{\textsf{rjags}}
+\newcommand{\BUGS}{\textsf{BUGS}}
+\newcommand{\WinBUGS}{\textsf{WinBUGS}}
+\newcommand{\R}{\textsf{R}}
+\newcommand{\CODA}{\textsf{coda}}
+\begin{document}
+
+\title{JAGS Version \release\ user manual}
+\author{Martyn Plummer}
+\maketitle
+
+\tableofcontents
+
+\chapter{Introduction}
+
+JAGS is Just Another Gibbs Sampler.  It is a program for the analysis
+of Bayesian models using Markov Chain Monte Carlo (MCMC) which is not
+wholly unlike
+\WinBUGS\ (\url{http://www.mrc-bsu.cam.ac.uk}). \JAGS\ was written
+with three aims in mind: to have an engine for the \BUGS\ language
+that runs on Unix; to be extensible, allowing users to write their own
+functions, distributions, and samplers; and to be a platform for
+experimentation with ideas in Bayesian modelling.
+
+\JAGS\ is designed to work closely with the \R\ language and
+environment for statistical computation and graphics
+(\url{http://www.r-project.org}).  You will find it useful to install
+the \CODA\ package for \R\ to analyze the output. You can also use the
+\rjags\ package to work directly with \JAGS\ from within R.
+
+\JAGS\ is licensed under the GNU General Public License
+version 2. You may freely modify and redistribute it under certain
+conditions (see the file \texttt{COPYING} for details).
+
+\chapter{What is new in JAGS \release}
+
+\section{New features of JAGS}
+
+Modules can now be unloaded. You can inspect the list of currently
+loaded modules using the \verb+LIST MODULES+ command. You can inspect
+the factory objects that are currently loaded using the
+\verb+LIST FACTORIES+ command and optionally turn on or off factory
+objects using \verb+SET FACTORY+.
+
+The CODA format has been extended to allow more general monitors
+to be written out to file in a rectangular format.  This replaces
+the former S-style dump format which is no longer available.
+
+\section{New modules}
+
+The \textsf{lecuyer} model provides random number generators with with
+multiple independent streams developed by Lecyer {\em et al}
+\cite{lecuyer02}.
+
+The \textsf{glm} module provides samplers to do block updating of the
+parameters in generalized linear models
+\cite{AlbertChib93,Gamerman97,HolmesHeld06,Fruhwirth-Schnatter09}.
+
+\section{New features in existing modules}
+
+\begin{itemize}
+\item The \texttt{dsum} distribution in the \textsf{bugs} module is now more
+flexible. It may take a variable number of arguments, and the
+arguments may be either discrete-valued or real-valued (but you cannot
+mix discrete-valued and real-valued arguments).
+\item The \textsf{bugs} modules now includes density, distribution, and
+quantile functions for many of the univariate distributions. See
+table \ref{table:bugs:dpq}, section \ref{section:functions:bugs}.
+\item The \textsf{mix} module contains an improved sampler for normal mixture
+models defined with the \verb+dnormmix+ distribution using tempered
+transtions \cite{Neal94,Celeux99} that provides much better label
+switching. It also now works when the mixture probabilities are given
+a Dirichlet distribution.
+\end{itemize}
+
+\section{Old features that no longer work}
+
+\begin{itemize}
+\item The model no longer automatically creates a \verb+deviance+
+  node. You can still monitor the model deviance in the same way as
+  before with the command
+\begin{verbatim}
+monitor deviance
+\end{verbatim}
+and you can write the monitored values out to file using the CODA
+command.  However, you must now load the \verb+dic+ module first.
+\item There are no longer any ``default'' monitors. When you set a monitor
+you must explicitly state the node name or other summary statistic
+from your model that you wish to monitor.
+\item The \verb+inverse+ function only works on symmetric positive-definite
+matrices. The previous version, which worked on general square matrices,
+used a QR decomposition, but for medium to large matrices this would create
+a slight asymmetry in the inverse, which caused errors when the inverted
+matrix was used as a precision matrix in a distribution.
+\end{itemize}
+ 
+\chapter{Running a model in \JAGS}
+
+\JAGS\ is designed for inference on Bayesian models using Markov Chain
+Monte Carlo (MCMC) simulation.  Running a model refers to generating
+samples from the posterior distribution of the model parameters.  This
+takes place in five steps:
+\begin{enumerate}
+\item Defining the model
+\item Compiling
+\item Initializing
+\item Adapting and burning-in
+\item Monitoring
+\end{enumerate}
+The next stages of analysis are done outside of \JAGS: convergence
+diagnostics, model criticism, and summarizing the samples must be done
+using other packages more suited to this task. There are several
+\R\ packages designed for analyzing MCMC output, and \JAGS\ can be
+used from within \R\ using the \rjags\ package.
+
+\section{Definition}
+
+There are two parts to the definition of a model in \JAGS: a
+description of the model and the definition of the data.
+
+\subsection{Model definition}
+
+The model is defined in a text file using a dialect of the
+\BUGS\ language.  The model definition consists of a series of
+relations inside a block delimited by curly brackets \verb+{+ and
+\verb+}+ and preceded by the keyword \verb+model+. Here is the standard
+linear regression example:
+
+\begin{verbatim}
+model {
+    for (i in 1:N) {
+          Y[i]   ~ dnorm(mu[i], tau)
+          mu[i] <- alpha + beta * (x[i] - x.bar)
+    }
+    x.bar   <- mean(x)
+    alpha    ~ dnorm(0.0, 1.0E-4)
+    beta     ~ dnorm(0.0, 1.0E-4)
+    sigma   <- 1.0/sqrt(tau)
+    tau      ~ dgamma(1.0E-3, 1.0E-3)
+}
+\end{verbatim}
+
+Each relation defines a node in the model in terms of other nodes that
+appear on the right hand side. These are referred to as the parent
+nodes. Taken together, the nodes in the model (together with the
+parent/child relationships represented as directed edges) form a
+directed acyclic graph. The very top-level nodes in the graph, with no
+parents, are constant nodes, which are defined either in the model
+definition ({\em e.g.}  \verb+1.0E-3+), or in the data file ({\em
+  e.g.}  \verb+x[1]+).
+
+Relations can be of two types. A {\em stochastic relation} (\verb+~+)
+defines a stochastic node, representing a random variable in the
+model. A {\em deterministic relation} (\verb+<-+) defines a
+deterministic node, the value of which is determined exactly by the
+values of its parents.
+
+Nodes defined by a relation are embedded in named arrays. Array names may
+contain letters, numbers, decimal points and underscores, but they must
+start with a letter.  The node array \verb+mu+ is a vector of length
+$N$ containing $N$ nodes (\verb+mu[1]+, $\ldots$, \verb+mu[N]+). The
+node array \verb+alpha+ is a scalar.  \JAGS\ follows the S language
+convention that scalars are considered as vectors of length 1. Hence
+the array \verb+alpha+ contains a single node \verb+alpha[1]+.
+
+Deterministic nodes do not need to be embedded in node arrays. The
+node \verb+Y[i]+ could equivalently be defined as
+\begin{verbatim}
+Y[i] ~ dnorm(alpha + beta * (x[i] - x.bar), tau)
+\end{verbatim}
+In this version of the model definition, the node previously defined
+as \verb+mu[i]+ still exists, but is not accessible to the user as it
+does not have a name.  This ability to hide deterministic nodes by
+embedding them in other expressions underscores an important fact:
+only the stochastic nodes in a model are really
+important. Deterministic nodes are merely a syntactically convenient
+way of describing the relations between, or transformations of, the
+stochastic nodes.
+
+\subsection{Data}
+\label{section:data}
+
+The data are defined in a separate file from the model definition, in
+the format created by the \texttt{dump()} function in \R.  The
+simplest way to prepare your data is to read them into \R\ and then
+dump them.  Only numeric vectors, matrices and arrays are
+allowed. More complex data structures such as factors, lists and data
+frames cannot be parsed by \JAGS\, nor can non-numeric vectors.  Any
+\R\ attributes of the data (such as names and dimnames) are stripped
+when they are read into \JAGS.
+
+The data may contain missing values, but you cannot supply partially
+missing values for a multivariate node. In \JAGS\ a node is either
+completely observed, or completely unobserved.  The unobserved nodes
+are referred to as the {\em parameters} of the model. The data file
+therefore defines the parameters of the model by omission.
+
+Here are the data for the \verb+LINE+ example:
+\begin{verbatim}
+`x` <-
+c(1, 2, 3, 4, 5)
+#R-style comments, like this one, can be embedded in the data file
+`Y` <-
+c(1, 3, 3, 3, 5)
+`N` <-
+5
+\end{verbatim}
+
+It is an error to supply a data value for a deterministic node. (See, however,
+section \ref{section:obfun} on observable functions).
+
+\subsection{Node Array dimensions}
+
+\subsubsection*{Array declarations}
+
+\JAGS\ allows the option of declaring the dimensions of node arrays in
+the model file. The declarations consist of the keyword \texttt{var}
+(for variable) followed by a comma-separated list of array names, with
+their dimensions in square brackets. The dimensions may be given in
+terms of any expression of the data that returns a single integer
+value.
+
+In the linear regression example, the model block could be preceded by
+\begin{verbatim}
+var x[N], Y[N], mu[N], alpha, beta, tau, sigma, x.bar;
+\end{verbatim}
+
+\subsubsection*{Undeclared nodes}
+
+If a node array is not declared then JAGS has three methods of
+determining its size.
+\begin{enumerate}
+\item {\bf Using the data.}  The dimension of an undeclared node array
+  may be inferred if it is supplied in the data file.
+\item {\bf Using the left hand side of the relations.}  The maximal
+  index values on the left hand side of a relation are taken to be the
+  dimensions of the node array.  For example, in this case:
+\begin{verbatim}
+for (i in 1:N) {
+   for (j in 1:M) {
+      Y[i,j] ~ dnorm(mu[i,j], tau)
+   }
+}
+\end{verbatim}
+$Y$ would be inferred to be an $N \times M$ matrix. Using this method,
+empty indices are not allowed on the left hand side of any relation.
+\item {\bf Using the dimensions of the parents} If a whole node array
+  appears on the left hand side of a relation, then its dimensions can
+  be inferred from the dimensions of the nodes on the right hand side.
+  For example, if \verb+A+ is known to be an $N \times N$ matrix
+  and
+\begin{verbatim}
+B <- inverse(A)
+\end{verbatim}
+Then \verb+B+ is also an $N \times N$ matrix.
+\end{enumerate}
+
+\subsubsection*{Querying array dimensions}  
+
+The \JAGS\ compiler has two built-in functions for querying array
+sizes.  The \verb+length()+ function returns the number of elements in
+a node array, and the \verb+dim()+ function returns a vector
+containing the dimensions of an array.  These two functions may be
+used to simplify the data preparation. For example, if \verb+Y+
+represents a vector of observed values, then using the \verb+length()+
+function in a for loop:
+\begin{verbatim}
+for (i in 1:length(Y)) {
+    Y[i] ~ dnorm(mu[i], tau)
+}
+\end{verbatim}
+avoids the need to put a separate data value \verb+N+ in the file
+representing the length of \verb+Y+.  
+
+For multi-dimensional arrays, the \verb+dim+ function serves a similar
+purpose. The \verb+dim+ function returns a vector, which must be stored
+in an array before its elements can be accessed. For this reason, calls
+to the \verb+dim+ function must always be in a data block (see section
+\ref{section:data:tranformations}).
+\begin{verbatim}
+data {
+   D <- dim(Z)
+}
+model {
+   for (i in 1:D[1]) {
+      for (j in 1:D[2]) {
+         Z[i,j] <- dnorm(alpha[i] + beta[j], tau)
+      }
+   }
+   ...
+}
+\end{verbatim}
+Clearly, the \verb+length()+ and \verb+dim()+ functions can only
+work if the size of the node array can be inferred, using one of the
+three methods outlined above.
+
+Note: the \verb+length()+ and \verb+dim()+ functions are different
+from all other functions in \JAGS: they do not act on nodes, but only
+on node {\em arrays}. As a consequence, an expression such as
+\verb+dim(a %*% b)+ is syntactically incorrect.
+
+\section{Compilation}
+
+When a model is compiled, a graph representing the model is created in
+computer memory. Compilation can fail for a number of reasons:
+\begin{enumerate}
+\item The graph contains a directed cycle.  These are forbidden
+in \JAGS.
+\item A top-level parameter is undefined. Any node that is used on
+the right hand side of a relation, but is not defined on the left
+hand side of any relation, is assumed to be a constant node. Its value
+must be supplied in the data file. 
+\item The model uses a function or distribution that has not been
+defined in any of the loaded modules.
+\end{enumerate}
+The number of parallel chains to be run by \JAGS\ is also defined at
+compilation time.  Each parallel chain should produce an independent
+sequence of samples from the posterior distribution. By default,
+\JAGS\ only runs a single chain.
+
+\section{Initialization}
+
+Before a model can be run, it must be initialized. There are three
+steps in the initialization of a model:
+\begin{enumerate}
+\item The initial values of the model parameters are set.
+\item A Random Number Generator (RNG) is chosen for each parallel chain,
+  and its initial value is set.
+\item The Samplers are chosen for each parameter in the model. 
+\end{enumerate}
+
+\subsection{Parameter values}
+
+The user may supply an initial value file containing values for the
+model parameters. The file may not contain values for logical or
+constant nodes. The format is the same as the data file (see section
+\ref{section:data}).
+
+If initial values are not supplied by the user, then each parameter
+chooses its own initial value based on the values of its parents.  The
+initial value is chosen to be a ``typical value'' from the prior
+distribution. The exact meaning of ``typical value'' depends on the
+distribution of the stochastic node, but is usually the mean, median,
+or mode.
+
+If you rely on automatic initial value generation and are running
+multiple parallel chains, then the initial values will be the same in
+all chains.  You may not want this behaviour, especially if you are
+using the Gelman and Rubin convergence diagnostic, which assumes that
+the initial values are over-dispersed with respect to the posterior
+distribution. In this case, you are advised to set the starting values
+manually using the "parameters in" statement.
+
+\subsection{RNGs}
+\label{section:rngs}
+
+Each chain in \JAGS\ has its own random number generator (RNG). RNGs
+are more correctly referred to as {\em pseudo}-random number
+generators. They generate a sequence of numbers that merely looks
+random but is, in fact, entirely determined by the initial state.  You
+may optionally set the name of the RNG and its initial state in the
+initial values file.
+
+The name of the RNG is set as follows. 
+\begin{verbatim}
+.RNG.name <- "name"
+\end{verbatim}
+There are four RNGs supplied by the \texttt{base} module in \JAGS\
+with the following names:
+\begin{verbatim}
+"base::Wichmann-Hill"
+"base::Marsaglia-Multicarry"
+"base::Super-Duper"
+"base::Mersenne-Twister"
+\end{verbatim}
+
+There are two ways to set the starting state of the RNG. The simplest
+is to supply an integer value to \texttt{.RNG.seed}, {\em e.g.}
+\begin{verbatim}
+".RNG.seed" <- 314159
+\end{verbatim}
+The second is way to save the state of the RNG from one JAGS session
+(see the ``PARAMETERS TO'' statement, section \ref{parameters:to}) and
+use this as the initial state of a new chain. The state of any RNG in
+JAGS can be saved and loaded as an integer vector with the name
+\texttt{.RNG.state}. For example,
+\begin{verbatim}
+".RNG.state" <- as.integer(c(20899,10892,29018))
+\end{verbatim}
+is a valid state for the Marsaglia-Multicarry generator.  You cannot
+supply an arbitrary integer to \texttt{.RNG.state}. Both the length of
+the vector and the permitted values of its elements are determined by
+the class of the RNG. The only safe way to use \texttt{.RNG.state} is
+to re-use a previously saved state.
+
+If no RNG names are supplied, then RNGs will be chosen automatically
+so that each chain has its own independent random number stream.  The
+exact behaviour depends on which modules are loaded. The \texttt{base}
+module uses the four generators listed above for the first four
+chains, then recycles them with different seeds for the next four
+chains, and so on.  
+
+By default, \JAGS\ bases the initial state on the time stamp. This
+means that, when a model is re-run, it generates an independent set of
+samples. If you want your model run to be reproducible, you must
+explicitly set the \verb+.RNG.seed+ for each chain.
+
+\subsection{Samplers}
+
+A Sampler is an object that acts on a set of parameters and updates
+them from one iteration to the next. During initialization of the
+model, Samplers are chosen automatically for all parameters. 
+
+The Model holds an internal list of {\em Sampler Factory} objects,
+which inspect the graph, recognize sets of parameters that can be
+updated with specific methods, and generate Sampler objects for
+them. The list of Sampler Factories is traversed in order, starting with
+sampling methods that are efficient, but limited to certain specific
+model structures and ending with the most generic, possibly
+inefficient, methods. If no suitable Sampler can be generated for one
+of the model parameters, an error message is generated.
+
+The user has no direct control over the process of choosing
+Samplers. However, you may indirectly control the process by loading a
+module that defines a new Sampler Factory. The module will insert the
+new Sampler Factory at the beginning of the list, where it will be
+queried before all of the other Sampler Factories. You can also 
+optionally turn on and off sampler factories using the ``SET FACTORY''
+command. See \ref{set:factory}.
+
+A report on the samplers chosen by the model, and the stochastic nodes
+they act on, can be generated using the ``SAMPLERS TO'' command. See 
+section \ref{samplers:to}.
+ 
+\section{Adaptation and burn-in}
+
+In theory, output from an MCMC sampler converges to the target
+distribution ({\em i.e.} the posterior distribution of the model
+parameters) in the limit as the number of iterations tends to
+infinity. In practice, all MCMC runs are finite.  By convention, the
+MCMC output is divided into two parts: an initial ``burn-in'' period,
+which is discarded, and the remainder of the run, in which the output
+is considered to have converged (sufficiently close) to the target
+distribution. Samples from the second part are used to create
+approximate summary statistics for the target distribution.
+
+By default, \JAGS\ keeps only the current value of each node in the
+model, unless a monitor has been defined for that node. The burn-in
+period of a \JAGS\ run is therefore the interval between model
+initialization and the creation of the first monitor.
+
+When a model is initialized, it is in {\em adaptive mode}, meaning
+that the Samplers used by the model may modify their behaviour for
+increased efficiency. Since this adaptation may depend on the entire
+sample history, the sequence generated by an adapting sampler is no
+longer a Markov chain, and is not guaranteed to converge to the target
+distribution. Therefore, adaptive mode must be turned off at some
+point during burn-in, and a sufficient number of iterations must take
+place {\em after} the adaptive phase to ensure convergence.
+
+By default, adaptive mode is turned off half way through first update
+of a \JAGS\ model, although the user may also control the length of
+the adaptive phase directly.  All samplers have a built in test to
+determine whether they have converged to their optimal sampling
+behaviour.  If any sampler fails this validation test, a warning will
+be printed. To ensure optimal sampling behaviour, the model should be
+run again from scratch using a longer adaptation period.
+
+\section{Monitoring}
+\label{section:monitoring}
+
+A {\em monitor} in \JAGS\ is an object that records sampled
+values. The simplest monitor is a {\em trace monitor}, which stores
+the sampled value of a node at each iteration.
+
+\JAGS\ cannot monitor a node unless it has been defined in the model
+file.  For vector- or array-valued nodes, this means that every
+element must be defined. Here is an example of a simple for loop that
+only defines elements $2$ to $N$ of \verb+theta+
+
+\begin{verbatim}
+for (i in 2:N) {
+   theta[i] ~ dnorm(0,1);
+}
+\end{verbatim}
+
+Unless \verb+theta[1]+ is defined somewhere else in the model file,
+the multivariate node \verb+theta+ is undefined and therefore it
+will not be possible to monitor \verb+theta+ as a whole.  In such
+cases you can request each element separately , e.g. \verb+theta[2]+,
+\verb+theta[3]+, {\em etc.}, or request a subset that is fully defined,
+e.g. \verb+theta[2:6]+.
+
+Monitors can be classified according to whether they pool values over
+iterations and whether they pool values over parallel chains (The
+standard trace monitor does neither). When monitor values are written
+out to file using the CODA command, the output files created depend
+on the pooling of the monitor, as shown in table \ref{table:coda}. By
+default, all of these files have the prefix CODA, but this may be changed
+to any other name using the ``stem'' option to the CODA command
+(See \ref{coda}).
+
+\begin{table}[h]
+\begin{tabular}{ccl}
+\hline
+Pool       & Pool   & Output files \\
+iterations & chains &              \\
+\hline
+no         & no     & CODAindex.txt, CODAchain1.txt, ... 
+                      CODAchainN.txt \\
+no         & yes    & CODAindex0.txt, CODAchain0.txt \\
+yes        & no     & CODAtable1.txt, ... CODAtableN.txt \\
+yes        & yes    & CODAtable0.txt \\
+\hline
+\end{tabular}
+\caption{Output files created by the CODA command depending on whether
+a monitor pools its values over chains or over iterations \label{table:coda}}
+\end{table}
+
+The standard CODA format for monitors that do not pool values over
+iterations is to create an index file and one or more output files.
+The index file is has three columns with, one each line,
+\begin{enumerate}
+\item A string giving the name of the (scalar) value being recorded
+\item The first line in the output file(s)
+\item The last line in the output file(s)
+\end{enumerate}
+The output file(s) contain two columns:
+\begin{enumerate}
+\item The iteration number
+\item The value at that iteration
+\end{enumerate}
+
+Some monitors pool values over iterations. For example a mean monitor
+may record only the sample mean of a node, without keeping the
+individual values from each iteration. Such monitors are written out
+to a table file with two columns:
+\begin{enumerate}
+\item A string giving the name of the (scalar) value being recorded
+\item The value (pooled over all iterations)
+\end{enumerate}
+
+\chapter{Running \JAGS}
+
+\JAGS\ has a command line interface. To invoke jags interactively,
+simply type \texttt{jags} at the shell prompt on Unix, or the Windows
+command prompt on Windows. To invoke JAGS with a script file, type
+\begin{verbatim}
+jags <script file>
+\end{verbatim}
+A typical script file has the following commands: 
+\begin{verbatim}
+model in "line.bug"      # Read model file
+data in "line-data.R"    # Read data in from file
+compile, nchains(2)      # Compile a model with two parallel chains
+parameters in "line-inits.R"  # Read initial values from file
+initialize               # Initialize the model
+update 1000              # Adaptation (if necessary) and burnin for 1000 iterations
+monitor alpha            # Set trace monitor for node alpha ...
+monitor beta             # ... and beta
+monitor sigma            # ... and sigma
+update 10000             # Update model for 10000 iterations
+coda *                   # All monitored values are written out to file
+\end{verbatim}
+More examples can be found in the file \verb+classic-bugs.tar.gz+ which is
+available from the \JAGS\ web page.
+
+Output from \JAGS\ is printed to the standard output, even when a
+script file is being used.  The \JAGS\ interface is designed to be
+forgiving. It will print a warning message if you make a mistake, but
+otherwise try to keep going.  This may create a cascade of error messages,
+of which only the first is informative.
+
+\section{Scripting commands}
+\label{section:scripting}
+
+\JAGS\ has a simple set of scripting commands with a syntax loosely
+based on \textsf{Stata}. Commands are shown below preceded by a dot
+(.). This is the \JAGS\ prompt. Do not type the dot in when you are
+entering the commands.
+
+C-style block comments taking the form /* ... */ can be
+embedded anywhere in the script file.  Additionally, you may use
+\R-style single-line comments starting with \#.
+
+If a scripting command takes a file name, then the name may be
+optionally enclosed in quotes. Quotes are required when the file name
+contains space, or any character which is not alphanumeric, or one of
+the following: \verb+_+, \verb+-+, \verb+.+, \verb+/+, \verb+\+.
+
+In the descriptions below, angular brackets \verb+<>+, and the text
+inside them, represents a parameter that should be replaced with the
+correct value by you.  Anything inside square brackets \verb+[]+ is
+optional. Do not type the square brackets if you wish to use an
+option.
+
+\subsection{MODEL IN}
+
+\begin{verbatim}
+. model in <file>
+\end{verbatim}
+Checks the syntactic correctness of the model description in
+\texttt{file} and reads it into memory. The next compilation
+statement will compile this model. 
+
+See also: MODEL CLEAR (\ref{model:clear})
+
+\subsection{DATA IN}
+\label{data:in}
+
+\begin{verbatim}
+. data in <file>
+\end{verbatim}
+JAGS keeps an internal data table containing the values of observed
+nodes inside each node array.  The DATA IN statement reads data from a
+file into this data table.
+
+Several data statements may be used to read in data from more than one
+file. If two data files contain data for the same node array, the second
+set of values will overwrite the first, and a warning will be printed.
+
+See also: DATA TO (\ref{data:to}).
+
+\subsection{COMPILE}
+
+\begin{verbatim}
+. compile [, nchains(<n>)]
+\end{verbatim}
+Compiles the model using the information provided in the preceding
+model and data statements. By default, a single Markov chain is
+created for the model, but if the \texttt{nchains} option is given,
+then \texttt{n} chains are created 
+
+Following the compilation of the model, further DATA IN statements are
+legal, but have no effect.  A new model statement, on the other hand,
+will replace the current model.
+
+\subsection{PARAMETERS IN}
+\label{parameters:in}
+
+\begin{verbatim}
+. parameters in <file> [, chain(<n>)]
+\end{verbatim}
+Reads the values in \texttt{file} and writes them to the corresponding
+parameters in chain \texttt{n}. The file has the same format as the
+one in the DATA IN statement.  The \texttt{chain} option may be
+omitted, in which case the parameter values in all chains are set to
+the same value.
+
+The PARAMETERS IN statement may be used before a model has been
+initialized.  You may only supply the values of unobserved stochastic
+nodes in the parameters file. Logical nodes and constant nodes are
+forbidden.
+
+See also: PARAMETERS TO (\ref{parameters:to})
+
+\subsection{INITIALIZE}
+
+\begin{verbatim}
+. initialize
+\end{verbatim}
+Initializes the model using the previously supplied data and parameter
+values supplied for each chain.
+
+\subsection{UPDATE}
+
+\begin{verbatim}
+. update <n> [,by(<m>)]
+\end{verbatim}
+Updates the model by \texttt{n} iterations. 
+
+The first UPDATE statement turns off adaptive mode for all samplers in
+the model after \texttt{n/2} iterations. A warning is printed if
+adaptation is incomplete. Incomplete adaptation means that the mixing
+of the Markov chain is not optimal. It is still possible to continue
+with a model that has not completely adapted, but it may be preferable
+to run the model again with a longer adaptation phase, starting from
+the MODEL IN statement.
+
+A progress bar is printed on the standard output consisting of 50
+asterisks. If the \texttt{by} option is supplied, a new asterisk is
+printed every \texttt{m} iterations. If this entails more than 50
+asterisks, the progress bar will be wrapped over several lines.  If
+\texttt{m} is zero, the printing of the progress bar is suppressed.
+
+\subsection{ADAPT}
+
+\begin{verbatim}
+. adapt <n> [,by(<m>)]
+\end{verbatim}
+Updates the model by \texttt{n} iterations and then turns of
+adaptive mode.
+
+Use this instead of the first UPDATE statement if you want explicit
+control over the length of the adaptive sampling phase.
+
+\subsection{MONITOR}
+\label{section:monitor}
+
+In \JAGS, a monitor is an object that calculates summary statistics
+from a model.  The most commonly used monitor simply records the value
+of a single node at each iteration.  This is called a ``trace''
+monitor. 
+\begin{verbatim}
+. monitor <varname> [, thin(n)] [type(<montype>)]
+\end{verbatim}
+The \texttt{thin} option sets the thinning interval of the monitor so
+that it will only record every nth value. The \texttt{thin} option 
+selects the type of monitor to create. The default type is \texttt{trace}.
+
+More complex monitors can be defined that do additional calculations.
+For example, the \verb+dic+ module defines a ``deviance'' monitor that
+records the deviance of the model at each iteration, and a ``pD''
+monitor that calculates an estimate of the effective number of
+parameters on the model \cite{spiegelhalter:etal:2002}.  
+
+\begin{verbatim}
+. monitor clear <varname> [type(<montype>)]
+\end{verbatim}
+Clears the monitor of the given type associated with variable
+\texttt{<varname>}. 
+
+\subsection{CODA}
+\label{coda}
+
+\begin{verbatim}
+. coda <varname> [, stem(<filename>)]
+\end{verbatim}
+If the named node has a trace monitor, this dumps the monitored values
+of to files \texttt{CODAindex.txt}, \texttt{CODAindex1.out},
+\texttt{CODAindex2.txt}, \ldots in a form that can be read by the
+\CODA\ package of \R.  The stem option may be used to modify the
+prefix from ``CODA'' to another string.  The wild-card character ``*''
+may be used to dump all monitored nodes
+
+\subsection{EXIT}
+
+\begin{verbatim}
+. exit
+\end{verbatim}
+Exits \JAGS. \JAGS\ will also exit when it reads an end-of-file character.
+
+\subsection{DATA TO}
+\label{data:to}
+\begin{verbatim}
+. data to <filename>
+\end{verbatim}
+Writes the data ({\em i.e.} the values of the observed nodes) to a
+file in the R \texttt{dump} format. The same file can be used in a
+DATA IN statement for a subsequent model.
+
+See also: DATA IN (\ref{data:in})
+
+\subsection{PARAMETERS TO}
+\label{parameters:to}
+\begin{verbatim}
+. parameters to <file> [, chain(<n>)]
+\end{verbatim}
+Writes the current parameter values ({\em i.e.} the values of the
+unobserved stochastic nodes) in chain \texttt{<n>} to a file in R dump
+format. The name and current state of the RNG for chain \texttt{<n>}
+is also dumped to the file.  The same file can be used as input in a
+PARAMETERS IN statement in a subsequent run.
+
+See also: PARAMETERS IN (\ref{parameters:in})
+
+\subsection{SAMPLERS TO}
+\label{samplers:to}
+\begin{verbatim}
+. samplers to <file>
+\end{verbatim}
+Writes out a summary of the samplers to the given file.  The output appears
+in three tab-separated columns, with one row for each sampled node
+\begin{itemize}
+\item The index number of the sampler (starting with 1). The index number 
+gives the order in which Samplers are updated at each iteration.
+\item The name of the sampler, matching the index number
+\item The name of the sampled node. 
+\end{itemize}
+If a Sampler updates multiple nodes then it is represented by multiple rows
+with the same index number.
+
+\subsection{LOAD}
+\label{load}
+\begin{verbatim}
+. load <module>
+\end{verbatim}
+Loads a module into \JAGS\ (see chapter \ref{section:modules}). Loading
+a module does not affect any previously initialized models, but will
+affect the future behaviour of the compiler and model initialization.
+
+\subsection{UNLOAD}
+\label{unload}
+\begin{verbatim}
+. unload <module>
+\end{verbatim}
+Unloads a module. Currently initialized models are unaffected, but
+the functions, distribution, and factory objects in the model will not
+be accessible to future models.
+
+\subsection{LIST MODULES}
+\label{list:modules}
+\begin{verbatim}
+. list modules
+\end{verbatim}
+Prints a list of the currently loaded modules.
+
+\subsection{LIST FACTORIES}
+\label{list:factories}
+\begin{verbatim}
+. list factories, type(<factype>)
+\end{verbatim}
+List the currently loaded factory objects and whether or not they are
+active.  The \verb+type+ option must be given, and the three possible values
+of \verb+<factype>+ are \verb+sampler+, \verb+monitor+, and \verb+rng+.
+
+\subsection{SET FACTORY}
+\label{set:factory}
+\begin{verbatim}
+. set factory "<facname>" <status>, type(<factype>)
+\end{verbatim}
+Sets the status of a factor object. The possible values of \verb+<status>+
+are \verb+on+ and \verb+off+. Possible factory names are given from the
+LIST MODULES command.
+
+\subsection{MODEL CLEAR}
+\label{model:clear}
+\begin{verbatim}
+. model clear
+\end{verbatim}
+Clears the current model.  The data table (see section \ref{data:in})
+remains intact
+
+\subsection{Print Working Directory (PWD)}
+\begin{verbatim}
+. pwd
+\end{verbatim}
+Prints the name of the current working directory. This is where \JAGS\
+will look for files when the file name is given without a full path, 
+{\em e.g.} \verb+"mymodel.bug"+.
+
+\subsection{Change Directory (CD)}
+\begin{verbatim}
+. cd <dirname>
+\end{verbatim}
+Changes the working directory to \texttt{<dirname>}
+
+\subsection{Directory list (DIR)}
+\begin{verbatim}
+. dir
+\end{verbatim}
+Lists the files in the current working directory.
+
+\subsection{RUN}
+\begin{verbatim}
+. run <cmdfile>
+\end{verbatim}
+Opens the file \texttt{<cmdfile>} and reads further scripting commands
+until the end of the file.  Note that if the file contains an EXIT
+statement, then the \JAGS\ session will terminate. 
+
+
+\section{Errors}
+
+There are two kinds of errors in \JAGS: runtime errors, which are due to
+mistakes in the model specification, and logic errors which are internal
+errors in the JAGS program. 
+
+Logic errors are generally created in the lower-level parts of the \JAGS\
+library, where it is not possible to give an informative error message.
+The upper layers of the \JAGS\ program are supposed to catch such errors
+before they occur, and return a useful error message that will help you
+diagnose the problem.  Inevitably, some errors slip through. Hence,
+if you get a logic error, there is probably an error in your input to
+\JAGS, although it may not be obvious what it is. Please send a bug
+report (see ``Feedback'' below) whenever you get a logic error.
+
+Error messages may also be generated when parsing files (model files,
+data files, command files).  The error messages generated in this case
+are created automatically by the program \texttt{bison}. They
+generally take the form ``syntax error, unexpected FOO, expecting BAR''
+and are not always abundantly clear.
+
+If a model compiles and initializes correctly, but an error occurs
+during updating, then the current state of the model will be dumped
+to a file named \verb+jags.dumpN.R+ where $N$ is the chain number.
+You should then load the dumped data into R to inspect the state of
+each chain when the error occurred.
+
+\chapter{Modules}
+\label{section:modules}
+
+The \JAGS\  library is distributed along with certain dynamically
+loadable modules that extend its functionality. A module can define
+new objects of the following classes:
+\begin{enumerate}
+\item {\bf functions} and {\bf distributions}, the basic building
+blocks of the BUGS language.
+\item {\bf samplers}, the objects which update the parameters of the
+model at each iteration, and {\bf sampler factories}, the objects that 
+create new samplers for specific model structures.  If the module
+defines a new distribution, then it will typically also define a new
+sampler for that distribution.
+\item {\bf monitors}, the objects that record sampled values for
+later analysis, and {\bf monitor factories} that create them. 
+\item {\bf random number generators}, the objects that drive the
+MCMC algorithm and {\bf RNG factories} that create them.
+\end{enumerate}
+
+The \verb+base+ module and the \verb+bugs+ module are loaded automatically
+at start time.  Others may be loaded by the user.
+
+\section{The base module}
+
+The base module supply the base functionality for the \JAGS\ library
+to function correctly. It is loaded first by default.  
+
+\subsection{Base Samplers}
+
+The \verb+base+ module defines samplers that use highly generic update
+methods.  These sampling methods only require basic information about
+the stochastic nodes they sample.  Conversely, they may not be fully
+efficient.
+
+Three samplers are currently defined:
+\begin{enumerate}
+\item The Finite sampler can sample a discrete-valued node with
+fixed support of less than 20 possible values. The node must not
+be bounded using the \verb+T(,)+ construct
+\item The Real Slice Sampler can sample any scalar real-valued 
+stochastic node.
+\item The Discrete Slice Sampler can sample any scalar
+discrete-valued stochastic node.
+\end{enumerate}
+
+\subsection{Base RNGs}
+
+The \verb+base+ module defines four RNGs, taken directly from \R,
+with the following names:
+\begin{enumerate}
+\item \verb+"base::Wichmann-Hill"+
+\item \verb+"base::Marsaglia-Multicarry"+
+\item \verb+"base::Super-Duper"+
+\item \verb+"base::Mersenne-Twister"+
+\end{enumerate}
+
+A single RNG factory object is also defined by the \verb+base+
+module which will supply these RNGs for chains 1 to 4 respectively, if
+``RNG.name'' is not specified in the initial values file.  All chains
+generated by the base RNG factory are initialized using the current
+time stamp.
+
+If you have more than four parallel chains, then the base module will
+recycle the same for RNGs, but using different seeds. If you want many
+parallel chains then you may wish to load the \verb+lecuyer+ module.
+
+\subsection{Base Monitors}
+
+The \verb+base+ module defines the TraceMonitor class (type
+``trace''). This is the monitor class that simply records the current
+value of the node at each iteration.
+
+\section{The bugs module}
+
+The \verb+bugs+ module defines some of the functions and distributions
+from \WinBUGS. These are described in more detail in sections
+\ref{section:functions} and \ref{section:distributions}.  The
+\verb+bugs+ module also defines conjugate samplers for efficient Gibbs
+sampling.
+
+\section{The mix module}
+
+The \verb+mix+ module defines a novel distribution
+\verb+dnormmix(mu,tau,pi)+ representing a finite mixture of normal
+distributions. In the parameterization of the \verb+dnormmix+
+distribution, $\mu$, $\tau$, and $\pi$ are vectors of the same length,
+and the density of \verb+y ~ dnormmix(mu, tau, pi)+ is
+\[
+f(y | \mu, \tau, \pi) = \sum_i \pi_i \tau_i^{\frac{1}{2}} \phi( \tau^{\frac{1}{2}}_i (y - \mu_i))
+\]
+where $\phi()$ is the probability density function of a standard
+normal distribution.
+
+The \verb+mix+ module also defines a sampler that is designed to act
+on finite normal mixtures. It uses tempered transitions to jump
+between distant modes of the multi-modal posterior distribution
+generated by such models \cite{Neal94,Celeux99}. The tempered
+transition method is computationally very expensive. If you want to
+use the \verb+dnormmix+ distribution but do not care about label
+switching, then you can disable the tempered transition sampler with
+\begin{verbatim}
+set factory "mix::TemperedMix" off, type(sampler)
+\end{verbatim}
+
+\section{The dic module}
+
+The \verb+dic+ module defines new monitor classes for Bayesian model
+criticism using deviance-based measures. 
+
+\subsection{The deviance monitor}
+
+The deviance monitor records the deviance of the model ({\em i.e.} the
+sum of the deviances of all the observed stochastic nodes) at each
+iteration. The command
+\begin{verbatim}
+monitor deviance
+\end{verbatim}
+will create a deviance monitor {\em unless} you have defined a node
+called ``deviance'' in your model. In this case, you will get a trace
+monitor for your deviance node.
+
+\subsection{The \texttt{pD} monitor}
+
+The \verb+pD+ monitor is used to estimate the effective number of
+parameters ($p_D$) of the model \cite{spiegelhalter:etal:2002}. It
+requires at least two parallel chains in the model, but calculates
+a single estimate of $p_D$ across all chains \cite{plummer:2002}.
+A pD monitor can be created using the command:
+\begin{verbatim}
+monitor pD
+\end{verbatim}
+Like the deviance monitor, however, if you have defined a node called
+``pD'' in your model then this will take precedence, and you will get
+a trace monitor for your \verb+pD+ node.
+
+Since the $p_D$ monitor pools its value across all chains, its values
+will be written out to the index file ``CODAindex0.txt'' and
+output file ``CODAoutput0.txt'' when you use the CODA command.
+
+The effective number of parameters is the sum of separate contributions
+from all observed stochastic nodes: $p_D = \sum_i p_{D_i}$. There is
+also a monitor that stores the sample mean of $p_{D_i}$. These statistics
+may be used as influence diagnostics \cite{spiegelhalter:etal:2002}.
+The mean monitor for $p_{D_i}$ is created with:
+\begin{verbatim}
+monitor pD, type(mean)
+\end{verbatim}
+Its values can be written out to a file ``PDtable0.txt'' with
+\begin{verbatim}
+coda pD, type(mean) stem(PD)
+\end{verbatim}
+
+\subsection{The \texttt{popt} monitor}
+
+The \verb+popt+ monitor works exactly like the mean monitor for $p_D$,
+but records contributions to the optimism of the expected deviance
+($p_{opt_i}$). The total optimism $p_{opt} = \sum_i p_{opt_i}$  can be
+added to the mean deviance to give the penalized expected deviance
+\cite{plummer:2008}.
+
+The mean monitor for $p_{opt_i}$ is created with
+\begin{verbatim}
+monitor popt, type(mean)
+\end{verbatim}
+Its values can be written out to a file ``POPTtable0.txt'' with
+\begin{verbatim}
+coda popt, type(mean) step(POPT)
+\end{verbatim}
+Under asymptotically favourable conditions in which $p_{D_i} \ll 1
+\forall i$,
+\[
+p_{opt} \approx 2 p_D
+\]
+For generalized linear models, a better approximation is
+\[
+p_{opt} \approx \sum_{i=1}^n \frac {p_{D_i}}{1 - p_{D_i}}
+\]
+
+The \verb+popt+ monitor uses importance weights to estimate
+$p_{opt}$. The resulting estimates may be numerically unstable when
+$p_{D_i}$ is not small.  This typically occurs in random-effects
+models, so it is recommended to use caution with the \verb+popt+
+until I can find a better way of estimating $p_{opt_i}$.
+
+\section{The msm module}
+
+The \verb+msm+ module defines the matrix exponential function
+\verb+mexp+ and the multi-state distribution \verb+dmstate+ which
+describes the transitions between observed states in continuous-time
+multi-state Markov transition models. 
+
+\section{The glm module}
+
+The \verb+glm+ module implements samplers for efficient updating of
+generalized linear mixed models.  The fundamental idea is to do block
+updating of the parameters in the linear predictor.  The \verb+glm+
+module is built on top of the \textsf{Csparse} sparse matrix library
+\cite{Davis2006} which allows updating of both fixed and random
+effects in the same block. Currently, the methods only work on
+parameters that have a normal prior distribution.
+
+Some of the samplers are based in the idea of introducing latent
+normal variables that reduce the GLM to a linear model. This idea was
+introduced by Albert and Chib \cite{AlbertChib93} for probit
+regression with a binary outcome, and was later refined and extended
+to logistic regression with binary outcomes by Holmes and Held
+\cite{HolmesHeld06}. Another approach, auxiliary mixture sampling,
+was developed by Fr{\"u}hwirth-Schnatter {\em et al}
+\cite{Fruhwirth-Schnatter09} and is used for more general Poisson
+regression and logistic regression models with binomial outcomes.
+Gamerman \cite{Gamerman97} proposed a stochastic version of the iteratively
+weighted least squares algorithm for GLMs, which is also implemented
+in the \verb+glm+ module. However the IWLS sampler tends to break down
+when there are many random effects in the model. It uses
+Metropolis-Hastings updates, and the acceptance probability may be
+very small under these circumstances.
+
+Block updating in GLMMs frees the user from the need to center
+predictor variables, like this:
+\begin{verbatim}
+y[i] ~ dnorm(mu[i], tau)
+mu[i] <- alpha + beta * (x[i] - mean(x))
+\end{verbatim}
+The second line can simply be written
+\begin{verbatim}
+mu[i] <- alpha + beta * x[i]
+\end{verbatim}
+without affecting the mixing of the Markov chain.  
+
+\chapter{Functions}
+\label{section:functions}
+
+Functions allow deterministic nodes to be defined using the \verb+<-+
+(``gets'') operator.  Most of the functions in \JAGS\ are scalar
+functions taking scalar arguments. However, \JAGS\ also allows
+arbitrary vector- and array-valued functions, such as the matrix
+multiplication operator \verb+%*%+ and the transpose function
+\verb+t()+ defined in the \verb+bugs+ module, and the matrix
+exponential function \verb+mexp()+ defined in the \verb+msm+
+module. \JAGS\ also uses an enriched dialect of the BUGS language with
+a number of operators that are used in the S language.
+
+Scalar functions taking scalar arguments are automatically vectorized.
+They can also be called when the arguments are arrays with conforming
+dimensions, or scalars. So, for example, the scalar $c$ can be added to
+the matrix $A$ using
+\begin{verbatim}
+B <- A + c
+\end{verbatim}
+instead of the more verbose form
+\begin{verbatim}
+D <- dim(A)
+for (i in 1:D[1])
+   for (j in 1:D[2]) {
+      B[i,j] <- A[i,j] + c
+   }
+}
+\end{verbatim}
+
+\section{Base functions}
+\label{section:functions:base}
+
+The functions defined by the \verb+base+ module all appear as infix or
+prefix operators. The syntax of these operators is built into the
+\JAGS\ parser. They are therefore considered part of the modelling
+language.  Table \ref{table:base:functions} lists them in reverse
+order of precedence.
+
+\begin{table}[h]
+\begin{center}
+\begin{tabular}{lll}
+\hline
+Type & Usage & Description\\ 
+\hline
+Logical           & \verb+x || y+ & Or \\
+operators         & \verb+x && y+ & And \\
+                  & \verb+!x+     & Not \\
+\hline
+Comparison  & \verb+x > y+ & Greater than\\
+operators   & \verb+x >= y+ & Greater than or equal to  \\
+            & \verb+x < y+ & Less than \\
+            & \verb+x <= y+ & Less than or equal to \\
+            & \verb+x == y+ & Equal \\
+\hline
+Arithmetic  & \verb-x + y- & Addition \\
+operators   & \verb+x - y+ & Subtraction\\
+            & \verb+x * y+ & Multiplication \\
+            & \verb+x / y+ & Division \\
+            & \verb+x %special% y+ &User-defined operators\\
+            & \verb+-x+ & Unary minus\\
+\hline
+Power function & \verb+x^y+ & \\
+\hline
+\end{tabular}
+\caption{Base functions listed in reverse order of precedence 
+  \label{table:base:functions}}
+\end{center}
+\end{table}
+
+Logical operators convert numerical arguments to logical values: zero
+arguments are converted to FALSE and non-zero arguments to
+TRUE. Logical and comparison operators return the value 1 if the
+result is TRUE and 0 if the result is FALSE.  Comparison operators are
+non-associative: the expression \verb+x < y < z+, for example, is
+syntactically incorrect.
+
+The \verb+%special%+ function is an exception in table
+\ref{table:base:functions}. It is not a function defined by the
+\verb+base+ module, but is a place-holder for any function
+with a name starting and ending with the character ``\verb+%+'' Such
+functions are automatically recognized as infix operators by the
+\JAGS\ model parser, with precedence defined by table
+\ref{table:base:functions}.
+
+\section{Functions in the bugs module}
+\label{section:functions:bugs}
+
+\subsection{Scalar functions}
+
+Table \ref{table:bugs:scalar} lists the scalar-valued functions in the
+\texttt{bugs} module that also have scalar arguments.  These functions
+are automatically vectorized when they are given vector, matrix, or
+array arguments with conforming dimensions.
+
+Table \ref{table:bugs:link} lists the link functions in the
+\texttt{bugs} module.  These are smooth scalar-valued functions that
+may be specified using an S-style replacement function notation. So,
+for example, the log link
+\begin{verbatim}
+log(y) <- x
+\end{verbatim}
+is equivalent to the more direct use of its inverse, the exponential
+function:
+\begin{verbatim}
+y <- exp(x)
+\end{verbatim}
+This usage comes from the use of link functions in generalized linear
+models.
+
+Table \ref{table:bugs:dpq} shows functions to calculate the
+probability density, probability function, and quantiles of some of
+the distributions provided by the \texttt{bugs} module. These
+functions are parameterized in the same way as the corresponding
+distribution.  For example, if $x$ has a normal distribution with mean
+$\mu$ and precision $\tau$
+\begin{verbatim}
+x ~ dnorm(mu, tau)
+\end{verbatim}
+Then the usage of the corresponding density, probability, and quantile
+functions is: 
+\begin{verbatim}
+density.x  <- dnorm(x, mu, tau)     # Density of normal distribution at x
+prob.x     <- pnorm(x, mu, tau)     # P(X <= x)
+quantile90.x <- qnorm(0.9, mu, tau) # 90th percentile
+\end{verbatim}
+For details of the parameterization of the other distributions, see
+tables \ref{table:bugs:distributions:real} and
+\ref{table:bugs:distributions:discrete}.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{llll}
+\hline
+Usage  & Description & Value & Restrictions on arguments \\ 
+\hline
+\verb+abs(x)+       & Absolute value        & Real & \\
+\verb+arccos(x)+      & Arc-cosine            & Real & $-1 < x < 1$\\
+\verb+arccosh(x)+     & Hyperbolic arc-cosine & Real & $1 < x$ \\
+\verb+arcsin(x)+      & Arc-sine              & Real & $-1 < x < 1$\\
+\verb+arcsinh(x)+     & Hyperbolic arc-sine   & Real &\\
+\verb+arctan(x)+      & Arc-tangent           & Real &\\
+\verb+arctanh(x)+     & Hyperbolic arc-tangent & Real & $-1 < x < 1$\\
+\verb+cos(x)+       & Cosine              & Real & \\
+\verb+cosh(x)+      & Hyperbolic Cosine   & Real & \\
+\verb+cloglog(x)+    & Complementary log log & Real & $0 < x < 1$ \\
+\verb+equals(x,y)+   & Test for equality   & Logical & \\
+\verb+exp(x)+       & Exponential         & Real & \\
+\verb+icloglog(x)+  & Inverse complementary & Real & \\
+                    & log log function    & \\
+\verb+ilogit(x)+    & Inverse logit       & Real & \\
+\verb+log(x)+       & Log function        & Real & $x > 0$ \\
+\verb+logfact(x)+   & Log factorial       & Real & $x > -1$ \\
+\verb+loggam(x)+    & Log gamma           & Real & $x > 0$ \\
+\verb+logit(x)+     & Logit               & Real & $0 < x < 1$ \\
+\verb+phi(x)+       & Standard normal cdf & Real & \\
+\verb+pow(x,z)+     & Power function      & Real & If $x < 0$ then $z$ is integer \\ 
+\verb+probit(x)+    & Probit              & Real & $0 < x < 1$ \\
+\verb+round(x)+     & Round to integer    & Integer & \\
+                    & away from zero      &      & \\
+\verb+sin(x)+       & Sine                & Real & \\
+\verb+sinh(x)+      & Hyperbolic Sine     & Real & \\
+\verb+sqrt(x)+      & Square-root         & Real & $x >= 0$ \\
+\verb+step(x)+      & Test for $x \geq 0$ & Logical & \\
+\verb+tan(x)+       & Tangent             & Real & \\
+\verb+tanh(x)+      & Hyperbolic Tangent  & Real & \\
+\verb+trunc(x)+     & Round to integer    & Integer & \\
+                    & towards zero        & \\
+\hline
+\end{tabular}
+\caption{Scalar functions in the \texttt{bugs} module \label{table:bugs:scalar}}
+\end{center}
+\end{table}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{llll}
+\hline
+Distribution & Density & Distribution & Quantile \\
+\hline
+Bernoulli          & dbern     & pbern     & qbern \\
+Beta               & dbeta     & pbeta     & qbeta \\
+Binomial           & dbin      & pbin      & qbin \\
+Chi-square         & dchisqr   & pchisqr   & qchisqr \\
+Double exponential & ddexp     & pdexp     & qdexp \\
+Exponential        & dexp      & pexp      & qexp \\
+F                  & df        & pf        & qf \\
+Gamma              & dgamma    & pgamma    & qgamma \\
+Generalized gamma  & dgengamma & pgengamma & qgengamma \\
+Hypergeometric     & dhyper    & phyper    & qhyper \\
+Log-normal         & dlnorm    & plnorm    & qlnorm \\
+Negative binomial  & dnegbin   & pnegbin   & qnegbin \\
+Normal             & dnorm     & pnorm     & qnorm \\
+Pareto             & dpar      & ppar      & qpar \\
+Poisson            & dpois     & ppois     & qpois \\
+Student t          & dt        & pt        & qt \\
+Weibull            & dweib     & pweib     & qweib \\
+\hline
+\end{tabular}
+\caption{Wrappers for the d-p-q functions from the \texttt{Rmath} library
+  \label{table:bugs:dpq}}
+\end{center}
+\end{table}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{llll}
+\hline
+Link function         & Description & Range & Inverse \\
+\hline
+\verb+cloglog(y) <- x+ & Complementary log log & $0 < y < 1$ & \verb+y <- icloglog(x)+ \\
+\verb+log(y) <- x+    & Log           & $0 < y$ &  \verb+y <- exp(x)+ \\
+\verb+logit(y) <- x+  & Logit         & $0 < y < 1$ &  \verb+y <- ilogit(x)+ \\
+\verb+probit(y) <- x+ & Probit        & $0 < y < 1$ &  \verb+y <- phi(x)+\\
+\hline
+\end{tabular}
+\caption{Link functions in the \texttt{bugs} module \label{table:bugs:link}}
+\end{center}
+\end{table}
+
+\section{Scalar-valued functions with vector arguments}
+
+Table \ref{table:bugs:scalar2} lists the scalar-valued functions in the
+\texttt{bugs} module that take general arguments. Unless otherwise
+stated in table \ref{table:bugs:scalar2}, the arguments to these functions
+may be scalar, vector, or higher-dimensional arrays.
+
+The \verb+max()+ and \verb+min()+ functions work like the
+corresponding \R\ functions. They take a variable number of arguments
+and return the maximum/minimum element over all supplied
+arguments. This usage is  compatible with \WinBUGS, although more general.
+
+\begin{table}
+\begin{tabular}{lll}
+\hline
+Function & Description & Restrictions \\
+\hline
+\verb+inprod(x1,x2)+ & Inner product & Dimensions of $a$, $b$ conform \\
+\verb+interp.lin(e,v1,v2)+ & Linear Interpolation & $e$ scalar, \\
+                          &                     & $v1,v2$ conforming vectors \\
+\verb+logdet(a)+ & Log determinant & $a$ is a square matrix \\
+\verb+max(x1,x2,...)+ & Maximum element among all arguments & \\
+\verb+mean(x)+  & Mean of elements of $a$ & \\
+\verb+min(x1,x2,...)+ & Minimum element among all arguments & \\
+\verb+prod(x)+  & Product of elements of $a$ & \\
+\verb+sum(a)+   & Sum of elements of $a$& \\
+\verb+sd(a)+    & Standard deviation of elements of $a$ & \\
+\hline
+\end{tabular}
+\caption{Scalar-valued functions with general
+  arguments in the \texttt{bugs} module \label{table:bugs:scalar2}}
+\end{table}
+
+\section{Vector- and array-valued functions}
+
+Table \ref{table:bugs:vector} lists vector- or matrix-valued functions
+in the \texttt{bugs} module.
+
+The \texttt{sort} and \texttt{rank} functions behaves like their R
+namesakes: \texttt{sort} accepts a vector and returns the same values
+sorted in ascending order; \texttt{rank} returns a vector of ranks.
+This is distinct from \WinBUGS, which has two scalar-valued functions
+\verb+rank+ and \verb+ranked+.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{lll}
+\hline
+Usage & Description & Restrictions \\
+\hline
+\verb+inverse(a)+ & Matrix inverse & $a$ is a symmetric positive definite matrix  \\
+\verb+mexp(a)+ & Matrix exponential & $a$ is a square matrix \\
+\verb+rank(v)+ & Ranks of elements of $v$ & $v$ is a vector   \\
+\verb+sort(v)+ & Elements of $v$ in order & $v$ is a vector  \\
+\verb+t(a)+    & Transpose                & $a$ is a matrix \\
+\verb+a %*% b+  & Matrix multiplication & $a,b$ conforming vector or matrices\\
+
+\hline
+\end{tabular}
+\caption{Vector- or matrix-valued functions in the \texttt{bugs}
+  module \label{table:bugs:vector}}
+\end{center}
+\end{table}
+
+\chapter{Distributions}
+\label{section:distributions}
+
+Distributions are used to define stochastic nodes using the \verb+~+
+operator. The distributions defined in the bugs module are listed in
+table \ref{table:bugs:distributions:real} (real-valued distributions),
+\ref{table:bugs:distributions:discrete} (discrete-valued
+distributions), and \ref{table:bugs:distributions:multi}
+(multivariate distributions).
+
+Some distributions have restrictions on the valid parameter values,
+and these are indicated in the tables. If a Distribution is
+given invalid parameter values when evaluating the log-likelihood, it
+returns $-\infty$. When a model is initialized, all stochastic nodes
+are checked to ensure that the initial parameter values are valid for
+their distribution.
+
+
+\begin{table}
+  \begin{center}
+    \begin{tabular}{llcll}
+      \hline
+      Name & Usage & Density & Lower & Upper \\
+      \hline
+      Beta & \verb+dbeta(a,b)+ & 
+      \multirow{2}{*}{
+        $\frac{\textstyle x^{a-1}(1-x)^{b-1}}{\textstyle \beta(a,b)}$
+      } & $0$ & $1$ \\
+      & $a > 0$, $b > 0$ \\
+      Chi-square & \verb+dchisqr(k)+ & 
+      \multirow{2}{*}{
+        $\frac{\textstyle x^{\frac{k}{2} - 1} \exp(-x/2)}
+        {\textstyle 2^{\frac{k}{2}} \Gamma({\scriptstyle \frac{k}{2}})}$
+      } & 0 & \\
+      & $k > 0$ \\
+      Double  & \verb+ddexp(mu,tau)+ & 
+      \multirow{2}{*}{$\tau \exp(-\tau | x-\mu |)/2$} & & \\
+      exponential & $\tau > 0$ \\
+      Exponential & \verb+dexp(lambda)+ & 
+      \multirow{2}{*}{$\lambda \exp(-\lambda x)$} & 0 & \\ 
+      & $\lambda > 0$ \\
+      F   & \verb+df(n,m)+ & 
+      \multirow{2}{*}{
+        $\textstyle \frac{\Gamma(\frac{n + m}{2})}
+                         {\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})}
+        \left(\frac{n}{m} \right)^{\frac{n}{2}} x^{\frac{n}{2} - 1} 
+        \left\{1 + \frac{nx}{m} \right\}^{-\frac{(n + m)}{2}}$} & 0 & \\
+      & $n > 0$, $m > 0$ \\
+      Gamma       & \verb+dgamma(r, mu)+ & 
+      \multirow{2}{*}{
+        $\frac{\textstyle \mu^r x^{r - 1} \exp(-\mu x)}
+        {\textstyle \Gamma(r)}$} & 0 & \\
+      & $\mu > 0$, $r > 0$ \\
+      Generalized & \verb+dgen.gamma(r,mu,beta)+ &  
+      \multirow{2}{*}{
+        $\beta \mu^{\beta r} x^{\beta r - 1}  \exp\{-(\mu x)^{\beta}\}$
+      } & $0$ & \\
+      gamma       & $\mu >0$, $\beta > 0$, $r > 0$ \\
+      Log-normal  & \verb+dlnorm(mu,tau)+ & 
+      \multirow{2}{*}{
+        $\tau^{\frac{1}{2}} x^{-1} \exp \left\{-\tau (\log(x) - \mu)^2/2 \right\}$} & 0 \\
+      ~ & $\tau > 0$ \\
+      Normal   & \verb+dnorm(mu,tau)+ & 
+      \multirow{2}{*}{
+        $\left(\frac{\tau}{2\pi}\right)^{\frac{1}{2}} \exp\{-(x - \mu)^2 \tau\}$} & & \\
+      ~ & $\tau > 0$ \\
+      Pareto      & \verb+dpar(alpha, c)+ & 
+      \multirow{2}{*}{
+        $\alpha c^{\alpha} x^{-(\alpha + 1)}$
+      } & $c$ & \\
+      ~ & $\alpha > 0$, $c > 0$ \\
+      Student t   & \verb+dt(mu,tau,k)+ & 
+      \multirow{2}{*}{
+        $\textstyle \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})} 
+        \left(\frac{\tau}{k\pi} \right)^{\frac{1}{2}} 
+        \left\{1 + \frac{\tau (x - \mu)^2}{k} \right\}^{-\frac{(k+1)}{2}}$} & & \\
+      ~ & $\tau > 0$, $k > 0$ \\
+      Uniform     & \verb+dunif(a,b)+ & 
+      \multirow{2}{*}{$\frac{\textstyle 1}{\textstyle b - a}$} & $a$ & $b$ \\
+      ~ & $a < b$ \\ 
+      Weibull     & \verb+dweib(v, lambda)+ & 
+      \multirow{2}{*}{$v  \lambda  x^{v - 1} \exp (- \lambda x^v)$} & 0 & \\
+      ~ & $v > 0$, $\lambda > 0$ \\
+      \hline
+    \end{tabular}
+    \caption{Univariate real-valued distributions in the \texttt{bugs} module
+      \label{table:bugs:distributions:real}}
+  \end{center}
+\end{table}
+
+\begin{table}
+  \begin{center}
+    \begin{tabular}{llllll}
+      \hline
+      Name & Usage & Density & Lower & Upper \\
+      \hline
+      Bernoulli & \verb+dbern(p)+ & 
+      \multirow{2}{*}{$p^x (1 - p)^{1 -x}$} & 
+      $0$ & $1$ \\
+      ~ & $0 < p < 1$ \\
+      Binomial  & \verb+dbin(p,n)+ & 
+      %\multirow{2}{*}{$\begin{array}{c} n \\ x \end{array} p^x (1-p)^{n-x}$}
+      \multirow{2}{*}{${n \choose x}  p^x (1-p)^{n-x}$}
+      ~  & $0$ & $n$ \\
+      ~ & $0 < p < 1$, $n \in \mathbb{N}^+$ \\
+      Categorical & \verb+dcat(p)+ & \multirow{2}{*}{$\frac{\textstyle p_x}{\textstyle \sum_i p_i}$} & $1$ & $N$ \\
+      ~ & $p \in (\mathbb{R}^+)^N$  \\
+      Hypergeometric & \verb+dhyper(n1,n2,m1,psi)+ &
+      \multirow{2}{*}{
+        $\textstyle {n_1 \choose x} {n_2 \choose m_1 - x} \psi^x$
+      } &
+      $\text{max}(0,n_+ - m_1)$ & $\text{min}(n_1,m_1)$ \\
+      ~ & $0 \leq n_i$, $0 < m_1 \leq n_+$  \\
+      Negative & \verb+dnegbin(p, r)+ &
+      \multirow{2}{*}{${x + r -1 \choose x} p^r (1-p)^x$} & 0 & \\
+      binomial & $0 < p < 1$, $r \in \mathbb{N}^+$ \\
+      Poisson & \verb+dpois(lambda)+ & 
+      \multirow{2}{*}{$\frac{\textstyle \exp(-\lambda) \lambda^x}{\textstyle x!}$} & 0 & \\
+      ~ & $\lambda > 0$ \\
+      \hline
+    \end{tabular}
+  \caption{Discrete univariate distributions in the \texttt{bugs} module
+    \label{table:bugs:distributions:discrete}}
+  \end{center}
+\end{table}
+
+
+\begin{table}
+  \begin{center}
+    \begin{tabular}{lll}
+      \hline
+      Name & Usage & Density \\
+      \hline
+      Dirichlet & \verb+p ~ ddirch(alpha)+ & 
+      \multirow{2}{*}{$\Gamma(\sum_i \alpha_i) \prod_j 
+        \frac{\textstyle p_j^{\alpha_j - 1}}{\textstyle \Gamma(\alpha_j)}$} \\
+      ~ & $\alpha_j \geq 0$ \\
+      & \\
+      Multivariate & \verb+x ~ dmnorm(mu,Omega)+ &
+      \multirow{2}{*}{
+        $\left(\frac{|\Omega|}{2\pi}\right)^{\frac{1}{2}} exp\{-(x-\mu)^T \Omega (x-\mu) / 2\}$} \\
+      normal & $\Omega$ positive definite \\
+      Wishart & \verb+Omega ~ dwish(R,k)+ &
+      \multirow{2}{*}{
+        $\frac{\textstyle |\Omega|^{(k-p-1)/2} |R|^{k/2} \exp\{-\text{Tr}(R\Omega/2)\}}
+               {\textstyle 2^{pk/2} \Gamma_p (k/2)}$
+      } \\
+      & $R$ pos. def. \\
+      Multivariate & \verb+x ~ dmt(mu,Omega,k)+ &
+      \multirow{2}{*}{
+        $\frac{\textstyle \Gamma \{(k+p)/2\}}{\textstyle \Gamma(k/2) (n\pi)^{p/2}}
+        |\Omega|^{1/2}
+        \left\{1 + \frac{1}{k} (x - \mu)^T \Omega (x - \mu) \right\}^{-\frac{(k+p)}{2}}$   } \\
+      Student t &  $\Omega$ pos. def. & \\
+      Multinomial  & \verb+x ~ dmulti(p, n)+ & 
+      \multirow{2}{*}{$n! \prod_j 
+        \frac{\textstyle p_j^{x_j}}{\textstyle x_j!}$} \\
+      ~ & $\sum_i x_i = n$ \\
+      & \\
+    \hline
+    \end{tabular}
+    \caption{Multivariate distributions in the \texttt{bugs} module
+      \label{table:bugs:distributions:multi}}
+  \end{center}
+\end{table}
+
+\chapter{Differences between \JAGS\ and \WinBUGS}
+
+Although \JAGS\ aims for the same functionality as \WinBUGS, there are
+a number of important differences.
+
+\subsection{Data format}
+
+There is no need to transpose matrices and arrays when transferring
+data between \R\ and \JAGS, since \JAGS\ stores the values of an array
+in ``column major'' order, like \R\ and FORTRAN ({\em i.e.} filling
+the left-hand index first).
+
+If you have an \textsf{S}-style data file for \WinBUGS\ and you wish
+to convert it for \JAGS, then use the command \texttt{bugs2jags},
+which is supplied with the \CODA\ package.
+
+\subsection{Distributions}
+
+Structural zeros are allowed in the Dirichlet distribution. If
+\begin{verbatim}
+p ~ ddirch(alpha)
+\end{verbatim}
+and some of the elements of alpha are zero, then the corresponding
+elements of p will be fixed to zero.
+
+The Multinomial (\verb+dmulti+) and Categorical (\verb+dcat+)
+distributions, which take a vector of probabilities as a parameter,
+may use unnormalized probabilities. The probability vector is
+normalized internally so that
+\[
+p_i \rightarrow \frac{p_i}{\sum_j p_j}
+\]
+
+\subsection{Observable Functions}
+\label{section:obfun}
+
+Logical nodes in the \BUGS\ language are a convenient way of
+describing the relationships between observables (constant and
+stochastic nodes), but are not themselves observable. You cannot
+supply data values for a logical node.  
+
+This restriction can occasionally be inconvenient, as there are
+important cases where the data are a deterministic function of
+unobserved variables.  Two important examples are
+\begin{enumerate}
+\item Censored data, which commonly occurs in survival analysis. In
+the most general case, we know that unobserved failure time $T$
+lies in the interval $(L,U]$.
+\item Aggregate data when we observe the sum of two or more
+unobserved variables.
+\end{enumerate}
+\JAGS\ contains two novel distributions to handle these situations.  
+\begin{enumerate}
+\item The \texttt{dinterval} distribution represents interval-censored
+data. It has two parameters: $t$ the original continuous variable, and
+$c[]$, a vector of cut points of length $M$, say. If \texttt{X $\sim$
+dinterval(t, c)} then 
+
+\begin{tabular}{lll}
+$X=0$   & if & $t \leq c[1]$\\
+$X=m$   & if & $c[m] < t \leq c[m+1]$ for $1 \leq m < M$\\
+$X = M$ & if & $c[M] < t$.
+\end{tabular}
+
+\item The \texttt{dsum} distribution represents the sum of two or more
+variables.  It takes a variable number of parameters. If \texttt{Y $\sim$
+dsum(x1,x2,x3)} then $Y=x1+x2+x3$.
+\end{enumerate}
+These distributions exist to give a likelihood to data that is, in fact,
+a deterministic function of the parameters.  The relation
+\begin{verbatim}
+Y ~ dsum(x1, x2)
+\end{verbatim}
+is logically equivalent to
+\begin{verbatim}
+Y <- x1 + x2
+\end{verbatim}
+But the latter form does not create a contribution to the likelihood,
+and does not allow you to define $Y$ as data.  The likelihood function
+is trivial: it is 1 if the parameters are consistent with the data and
+0 otherwise.  The \texttt{dsum} distribution also requires a special
+sampler, which can currently only handle the case where the parameters
+of \texttt{dsum} are unobserved stochastic nodes, and where the
+parameters are either all discrete-valued or all continuous-valued. A node
+cannot be subject to more than one \texttt{dsum} constraint.
+
+\subsection{Data transformations}
+\label{section:data:tranformations}
+
+\JAGS\ allows data transformations, but the syntax is different from
+\BUGS.  \BUGS\ allows you to put a stochastic node twice on the left
+hand side of a relation, as in this example taken from the manual
+\begin{verbatim}
+   for (i in 1:N) {
+      z[i] <- sqrt(y[i])
+      z[i] ~ dnorm(mu, tau)
+   }
+\end{verbatim}
+This is forbidden in \JAGS. You must put data transformations in a 
+separate block of relations preceded by the keyword \texttt{data}:
+\begin{verbatim}
+data {
+   for (i in 1:N) {
+      z[i] <- sqrt(y[i])
+   }
+}
+model {
+   for (i in 1:N) {
+      z[i] ~ dnorm(mu, tau)
+   }
+   ...
+}
+\end{verbatim}
+This syntax preserves the declarative nature of the \BUGS\ language.
+In effect, the data block defines a distinct model, which describes
+how the data is generated. Each node in this model is forward-sampled
+once, and then the node values are read back into the data table. The
+data block is not limited to logical relations, but may also include
+stochastic relations. You may therefore use it in simulations,
+generating data from a stochastic model that is different from the one
+used to analyse the data in the \texttt{model} statement.
+
+This example shows a simple location-scale problem in which the ``true''
+values of the parameters \texttt{mu} and \texttt{tau} are generated
+from a given prior in the \texttt{data} block, and the generated
+data is analyzed in the \texttt{model} block.
+\begin{verbatim}
+data {
+   for (i in 1:N) {
+      y[i] ~ dnorm(mu.true, tau.true) 
+   }
+   mu.true ~ dnorm(0,1);
+   tau.true ~ dgamma(1,3);
+}
+model {
+   for (i in 1:N) {
+      y[i] ~ dnorm(mu, tau)
+   }
+   mu ~ dnorm(0, 1.0E-3)
+   tau ~ dgamma(1.0E-3, 1.0E-3)
+}
+\end{verbatim}
+Beware, however, that every node in the \texttt{data} statement will
+be considered as data in the subsequent \texttt{model} statement. This
+example, although superficially similar, has a quite different interpretation.
+\begin{verbatim}
+data {
+   for (i in 1:N) {
+      y[i] ~ dnorm(mu, tau) 
+   }
+   mu ~ dnorm(0,1);
+   tau ~ dgamma(1,3);
+}
+model {
+   for (i in 1:N) {
+      y[i] ~ dnorm(mu, tau)
+   }
+   mu ~ dnorm(0, 1.0E-3)
+   tau ~ dgamma(1.0E-3, 1.0E-3)
+}
+\end{verbatim}
+Since the names \texttt{mu} and \texttt{tau} are used in both
+\texttt{data} and \texttt{model} blocks, these nodes will be
+considered as {\em observed} in the model and their values will be
+fixed at those values generated in the \texttt{data} block.
+
+\subsection{Directed cycles}
+
+Directed cycles are forbidden in \JAGS. There are two important
+instances where directed cycles are used in \BUGS.
+\begin{itemize}
+\item Defining autoregressive priors
+\item Defining ordered priors
+\end{itemize}
+For the first case, the \texttt{GeoBUGS} extension to \WinBUGS\ provides
+some convenient ways of defining autoregressive priors. These should be
+available in a future version of \JAGS.
+
+\subsection{Censoring, truncation and prior ordering}
+\label{section:censoring}
+
+These are three, closely related issues that are all handled using
+the \texttt{I(,)} construct in \BUGS.
+
+Censoring occurs when a variable $X$ is not observed directly,
+but is observed only to lie in the range $(L,U]$.  Censoring is
+an {\em a posteriori} restriction of the data, and is represented
+in WinBUGS by the \texttt{I(,)} construct, {\em e.g.}
+\begin{verbatim}
+X ~ dnorm(theta, tau) I(L,U)
+\end{verbatim}
+where $L$ and $U$ are constant nodes.
+
+Truncation occurs when a variable is known {\em a priori} to lie in
+a certain range.  Although \BUGS\ has no construct for representing
+truncated variables, it turns out that there is no difference between
+censoring and truncation for top-level parameters ({\em i.e.} variables
+with no unobserved parents).  Hence, for example, this
+\begin{verbatim}
+theta ~ dnorm(0, 1.0E-3) I(0, )
+\end{verbatim}
+is a perfectly valid way to describe a parameter $\theta$ with a
+half-normal prior distribution.
+
+Prior ordering occurs when a vector of nodes is known {\em a priori}
+to be strictly increasing or decreasing. It can be represented in
+WinBUGS with symmetric $I(,)$ constructs,  {\em e.g.}
+\begin{verbatim}
+X[1] ~ dnorm(0, 1.0E-3) I(,X[2])
+X[2] ~ dnorm(0, 1.0E-3) I(X[1],)
+\end{verbatim}
+ensures that $X[1] \leq X[2]$.
+
+\JAGS\ makes an attempt to separate these three concepts.
+
+Censoring is handled in \JAGS\ using the new distribution
+\texttt{dinterval} (section \ref{section:obfun}). This can be
+illustrated with a survival analysis example.  A right-censored
+survival time $t_i$ with a Weibull distribution is described in
+\WinBUGS\ as follows:
+\begin{verbatim}
+t[i] ~ dweib(r, mu[i]) I(c[i], )
+\end{verbatim}
+where $t_i$ is unobserved if $t_i > c_i$.  In \JAGS\ this becomes
+\begin{verbatim}
+is.censored[i] ~ dinterval(t[i], c[i])
+t[i] ~ dweib(r, mu[i])
+\end{verbatim}
+where \verb+is.censored[i]+ is an indicator variable that takes the
+value 1 if $t_i$ is censored and 0 otherwise. See the MICE and KIDNEY
+examples in the ``classic bugs'' set of examples.
+
+Truncation is represented in \JAGS\ using the \texttt{T(,)} construct,
+which has the same syntax as the \texttt{I(,)} construct in \WinBUGS,
+but has a different interpretation. If
+\begin{verbatim}
+X ~ dfoo(theta) T(L,U)
+\end{verbatim}
+then {\em a priori} $X$ is known to lie between $L$ and $U$. This
+generates a likelihood
+\[
+\frac{p(x \mid \theta)}{P(L \leq X \leq U \mid \theta)}
+\]
+if $L \leq X \leq U$ and zero otherwise, where $p(x \mid \theta)$ is
+the density of $X$ given $\theta$ according to the distribution
+\texttt{foo}. Note that calculation of the denominator may be
+computationally expensive.
+
+Prior ordering of top-level parameters in the model can be achieved
+using the \texttt{sort} function, which sorts a vector in ascending
+order.
+
+Symmetric truncation relations like this
+\begin{verbatim}
+alpha[1] ~ dnorm(0, 1.0E-3) I(,alpha[2])
+alpha[2] ~ dnorm(0, 1.0E-3) I(alpha[1],alpha[3])
+alpha[3] ~ dnorm(0, 1.0E-3) I(alpha[2],)
+\end{verbatim}
+Should be replaced by this
+\begin{verbatim}
+for (i in 1:3) {
+   alpha0[i] ~ dnorm(0, 1.0E-3)
+}
+alpha[1:3] <- sort(alpha0)
+\end{verbatim}
+
+\chapter{Feedback}
+
+Please send feedback to \url{martyn_plummer@users.sourceforge.net}.
+I am particularly interested in the following problems:
+
+\begin{itemize}
+\item Crashes, including both segmentation faults and uncaught exceptions.
+\item Incomprehensible error messages
+\item Models that should compile, but don't 
+\item Output that cannot be validated against \WinBUGS
+\item Documentation erors
+\end{itemize}
+
+If you want to send a bug report, it must be reproducible. Send the
+model file, the data file, the initial value file and a script file
+that will reproduce the problem. Describe what you think should
+happen, and what did happen.
+
+\chapter{Acknowledgments}
+
+Many thanks to the \BUGS\ development team, without whom \JAGS\ would
+not exist.  Thanks also to Simon Frost for pioneering \JAGS\ on
+Windows and Bill Northcott for getting \JAGS\ on Mac OS X to
+work. Kostas Oikonomou found many bugs while getting \JAGS\ to work on
+Solaris using Sun development tools and libraries.  Bettina Gruen,
+Chris Jackson, Greg Ridgeway and Geoff Evans also provided useful
+feedback.  Special thanks to Jean-Baptiste Denis who has been very
+diligent in providing feedback on JAGS.
+
+Testing of \JAGS\ on IRIX 6.5 was carried out on Helix Systems at the
+National Institutes of Health, Bethesda, MD (\url{http://helix.nih.gov}).
+
+\bibliographystyle{plain}
+\bibliography{jags_user_manual}
+
+\end{document}
+
+
+