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cha-writing-models.html
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<!DOCTYPE html>
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<title>Chapter 5 Writing models in NIMBLE’s dialect of BUGS | NimbleUserManual.knit</title>
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<ul class="summary">
<img src="./nimble-icon.png"
width=100>
<li><a href="./cha-welcome-nimble.html">NIMBLE User Manual, Version 1.0.1</a></li>
<li><a href="https://github.com/nimble-dev/nimble">NIMBLE Development Team</a></li>
<li><a href="https://R-nimble.org">https://R-nimble.org</a></li>
<li class="divider"></li>
<li class="part"><span><b>I Introduction</b></span></li>
<li class="chapter" data-level="1" data-path="cha-welcome-nimble.html"><a href="cha-welcome-nimble.html"><i class="fa fa-check"></i><b>1</b> Welcome to NIMBLE</a>
<ul>
<li class="chapter" data-level="1.1" data-path="cha-welcome-nimble.html"><a href="cha-welcome-nimble.html#sec:what-is-nimble"><i class="fa fa-check"></i><b>1.1</b> What does NIMBLE do?</a></li>
<li class="chapter" data-level="1.2" data-path="cha-welcome-nimble.html"><a href="cha-welcome-nimble.html#how-to-use-this-manual"><i class="fa fa-check"></i><b>1.2</b> How to use this manual</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html"><i class="fa fa-check"></i><b>2</b> Lightning introduction</a>
<ul>
<li class="chapter" data-level="2.1" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:brief-example"><i class="fa fa-check"></i><b>2.1</b> A brief example</a></li>
<li class="chapter" data-level="2.2" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:creating-model"><i class="fa fa-check"></i><b>2.2</b> Creating a model</a></li>
<li class="chapter" data-level="2.3" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:compiling-model"><i class="fa fa-check"></i><b>2.3</b> Compiling the model</a></li>
<li class="chapter" data-level="2.4" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:intro-runMCMC"><i class="fa fa-check"></i><b>2.4</b> One-line invocation of MCMC</a></li>
<li class="chapter" data-level="2.5" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:creating-mcmc"><i class="fa fa-check"></i><b>2.5</b> Creating, compiling and running a basic MCMC configuration</a></li>
<li class="chapter" data-level="2.6" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:customizing-mcmc"><i class="fa fa-check"></i><b>2.6</b> Customizing the MCMC</a></li>
<li class="chapter" data-level="2.7" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:running-mcem"><i class="fa fa-check"></i><b>2.7</b> Running MCEM</a></li>
<li class="chapter" data-level="2.8" data-path="cha-lightning-intro.html"><a href="cha-lightning-intro.html#sec:creating-your-own"><i class="fa fa-check"></i><b>2.8</b> Creating your own functions</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="cha-more-introduction.html"><a href="cha-more-introduction.html"><i class="fa fa-check"></i><b>3</b> More introduction</a>
<ul>
<li class="chapter" data-level="3.1" data-path="cha-more-introduction.html"><a href="cha-more-introduction.html#nimble-adopts-and-extends-the-bugs-language-for-specifying-models"><i class="fa fa-check"></i><b>3.1</b> NIMBLE adopts and extends the BUGS language for specifying models</a></li>
<li class="chapter" data-level="3.2" data-path="cha-more-introduction.html"><a href="cha-more-introduction.html#sec:nimble-lang-writ"><i class="fa fa-check"></i><b>3.2</b> nimbleFunctions for writing algorithms</a></li>
<li class="chapter" data-level="3.3" data-path="cha-more-introduction.html"><a href="cha-more-introduction.html#sec:nimble-algor-libr"><i class="fa fa-check"></i><b>3.3</b> The NIMBLE algorithm library</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html"><i class="fa fa-check"></i><b>4</b> Installing NIMBLE</a>
<ul>
<li class="chapter" data-level="4.1" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#sec:requ-run-nimble"><i class="fa fa-check"></i><b>4.1</b> Requirements to run NIMBLE</a></li>
<li class="chapter" data-level="4.2" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#sec:compiler"><i class="fa fa-check"></i><b>4.2</b> Installing a C++ compiler for NIMBLE to use</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#macos"><i class="fa fa-check"></i><b>4.2.1</b> MacOS</a></li>
<li class="chapter" data-level="4.2.2" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#linux"><i class="fa fa-check"></i><b>4.2.2</b> Linux</a></li>
<li class="chapter" data-level="4.2.3" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#windows"><i class="fa fa-check"></i><b>4.2.3</b> Windows</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#installing-the-nimble-package"><i class="fa fa-check"></i><b>4.3</b> Installing the NIMBLE package</a></li>
<li class="chapter" data-level="4.4" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#troubleshooting-installation-problems"><i class="fa fa-check"></i><b>4.4</b> Troubleshooting installation problems</a></li>
<li class="chapter" data-level="4.5" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#customizing-your-installation"><i class="fa fa-check"></i><b>4.5</b> Customizing your installation</a>
<ul>
<li class="chapter" data-level="4.5.1" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#using-your-own-copy-of-eigen"><i class="fa fa-check"></i><b>4.5.1</b> Using your own copy of Eigen</a></li>
<li class="chapter" data-level="4.5.2" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#using-libnimble"><i class="fa fa-check"></i><b>4.5.2</b> Using libnimble</a></li>
<li class="chapter" data-level="4.5.3" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#sec:blas"><i class="fa fa-check"></i><b>4.5.3</b> BLAS and LAPACK</a></li>
<li class="chapter" data-level="4.5.4" data-path="cha-installing-nimble.html"><a href="cha-installing-nimble.html#customizing-compilation-of-the-nimble-generated-c"><i class="fa fa-check"></i><b>4.5.4</b> Customizing compilation of the NIMBLE-generated C++</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>II Models in NIMBLE</b></span></li>
<li class="chapter" data-level="5" data-path="cha-writing-models.html"><a href="cha-writing-models.html"><i class="fa fa-check"></i><b>5</b> Writing models in NIMBLE’s dialect of BUGS</a>
<ul>
<li class="chapter" data-level="5.1" data-path="cha-writing-models.html"><a href="cha-writing-models.html#sec:supp-feat-bugs"><i class="fa fa-check"></i><b>5.1</b> Comparison to BUGS dialects supported by WinBUGS, OpenBUGS and JAGS</a>
<ul>
<li class="chapter" data-level="5.1.1" data-path="cha-writing-models.html"><a href="cha-writing-models.html#supported-features-of-bugs-and-jags"><i class="fa fa-check"></i><b>5.1.1</b> Supported features of BUGS and JAGS</a></li>
<li class="chapter" data-level="5.1.2" data-path="cha-writing-models.html"><a href="cha-writing-models.html#sec:extensions-bugs"><i class="fa fa-check"></i><b>5.1.2</b> NIMBLE’s Extensions to BUGS and JAGS</a></li>
<li class="chapter" data-level="5.1.3" data-path="cha-writing-models.html"><a href="cha-writing-models.html#sec:not-yet-supported"><i class="fa fa-check"></i><b>5.1.3</b> Not-supported features of BUGS and JAGS</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="cha-writing-models.html"><a href="cha-writing-models.html#writing-models"><i class="fa fa-check"></i><b>5.2</b> Writing models</a>
<ul>
<li class="chapter" data-level="5.2.1" data-path="cha-writing-models.html"><a href="cha-writing-models.html#declaring-stochastic-and-deterministic-nodes"><i class="fa fa-check"></i><b>5.2.1</b> Declaring stochastic and deterministic nodes</a></li>
<li class="chapter" data-level="5.2.2" data-path="cha-writing-models.html"><a href="cha-writing-models.html#sec:more-kinds-bugs"><i class="fa fa-check"></i><b>5.2.2</b> More kinds of BUGS declarations</a></li>
<li class="chapter" data-level="5.2.3" data-path="cha-writing-models.html"><a href="cha-writing-models.html#subsec:vectorized-versus-scalar-declarations"><i class="fa fa-check"></i><b>5.2.3</b> Vectorized versus scalar declarations</a></li>
<li class="chapter" data-level="5.2.4" data-path="cha-writing-models.html"><a href="cha-writing-models.html#subsec:dists-and-functions"><i class="fa fa-check"></i><b>5.2.4</b> Available distributions</a></li>
<li class="chapter" data-level="5.2.5" data-path="cha-writing-models.html"><a href="cha-writing-models.html#subsec:BUGS-lang-fxns"><i class="fa fa-check"></i><b>5.2.5</b> Available BUGS language functions</a></li>
<li class="chapter" data-level="5.2.6" data-path="cha-writing-models.html"><a href="cha-writing-models.html#subsec:BUGS-link"><i class="fa fa-check"></i><b>5.2.6</b> Available link functions</a></li>
<li class="chapter" data-level="5.2.7" data-path="cha-writing-models.html"><a href="cha-writing-models.html#subsec:trunc"><i class="fa fa-check"></i><b>5.2.7</b> Truncation, censoring, and constraints</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="6" data-path="cha-building-models.html"><a href="cha-building-models.html"><i class="fa fa-check"></i><b>6</b> Building and using models</a>
<ul>
<li class="chapter" data-level="6.1" data-path="cha-building-models.html"><a href="cha-building-models.html#creating-model-objects"><i class="fa fa-check"></i><b>6.1</b> Creating model objects</a>
<ul>
<li class="chapter" data-level="6.1.1" data-path="cha-building-models.html"><a href="cha-building-models.html#using-nimblemodel-to-create-a-model"><i class="fa fa-check"></i><b>6.1.1</b> Using <em>nimbleModel</em> to create a model</a></li>
<li class="chapter" data-level="6.1.2" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:readBUGSmodel"><i class="fa fa-check"></i><b>6.1.2</b> Creating a model from standard BUGS and JAGS input files</a></li>
<li class="chapter" data-level="6.1.3" data-path="cha-building-models.html"><a href="cha-building-models.html#sub:multiple-instances"><i class="fa fa-check"></i><b>6.1.3</b> Making multiple instances from the same model definition</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:nodes-and-variables"><i class="fa fa-check"></i><b>6.2</b> NIMBLE models are objects you can query and manipulate</a>
<ul>
<li class="chapter" data-level="6.2.1" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:what-are-nodes-and-variables"><i class="fa fa-check"></i><b>6.2.1</b> What are variables and nodes?</a></li>
<li class="chapter" data-level="6.2.2" data-path="cha-building-models.html"><a href="cha-building-models.html#determining-the-nodes-and-variables-in-a-model"><i class="fa fa-check"></i><b>6.2.2</b> Determining the nodes and variables in a model</a></li>
<li class="chapter" data-level="6.2.3" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:accessing-nodes"><i class="fa fa-check"></i><b>6.2.3</b> Accessing nodes</a></li>
<li class="chapter" data-level="6.2.4" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:how-nodes-are"><i class="fa fa-check"></i><b>6.2.4</b> How nodes are named</a></li>
<li class="chapter" data-level="6.2.5" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:why-use-node"><i class="fa fa-check"></i><b>6.2.5</b> Why use node names?</a></li>
<li class="chapter" data-level="6.2.6" data-path="cha-building-models.html"><a href="cha-building-models.html#sec:cdisdata"><i class="fa fa-check"></i><b>6.2.6</b> Checking if a node holds data</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="cha-building-models.html"><a href="cha-building-models.html#using-models-in-parallel"><i class="fa fa-check"></i><b>6.3</b> Using models in parallel</a></li>
</ul></li>
<li class="part"><span><b>III Algorithms in NIMBLE</b></span></li>
<li class="chapter" data-level="7" data-path="cha-mcmc.html"><a href="cha-mcmc.html"><i class="fa fa-check"></i><b>7</b> MCMC</a>
<ul>
<li class="chapter" data-level="7.1" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:nimbleMCMC"><i class="fa fa-check"></i><b>7.1</b> One-line invocation of MCMC: <em>nimbleMCMC</em></a></li>
<li class="chapter" data-level="7.2" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:mcmc-configuration"><i class="fa fa-check"></i><b>7.2</b> The MCMC configuration</a>
<ul>
<li class="chapter" data-level="7.2.1" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:default-mcmc-conf"><i class="fa fa-check"></i><b>7.2.1</b> Default MCMC configuration</a></li>
<li class="chapter" data-level="7.2.2" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:customizing-mcmc-conf"><i class="fa fa-check"></i><b>7.2.2</b> Customizing the MCMC configuration</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:build-compile-mcmc"><i class="fa fa-check"></i><b>7.3</b> Building and compiling the MCMC</a></li>
<li class="chapter" data-level="7.4" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:initMCMC"><i class="fa fa-check"></i><b>7.4</b> Initializing MCMC</a></li>
<li class="chapter" data-level="7.5" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:runMCMC"><i class="fa fa-check"></i><b>7.5</b> User-friendly execution of MCMC algorithms: <em>runMCMC</em></a></li>
<li class="chapter" data-level="7.6" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:executing-the-mcmc-algorithm"><i class="fa fa-check"></i><b>7.6</b> Running the MCMC</a>
<ul>
<li class="chapter" data-level="7.6.1" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:mcmc-rerun"><i class="fa fa-check"></i><b>7.6.1</b> Rerunning versus restarting an MCMC</a></li>
<li class="chapter" data-level="7.6.2" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:sampler-time"><i class="fa fa-check"></i><b>7.6.2</b> Measuring sampler computation times: <em>getTimes</em></a></li>
<li class="chapter" data-level="7.6.3" data-path="cha-mcmc.html"><a href="cha-mcmc.html#assessing-the-adaption-process-of-rw-and-rw_block-samplers"><i class="fa fa-check"></i><b>7.6.3</b> Assessing the adaption process of <em>RW</em> and <em>RW_block</em> samplers</a></li>
</ul></li>
<li class="chapter" data-level="7.7" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:extracting-samples"><i class="fa fa-check"></i><b>7.7</b> Extracting MCMC samples</a></li>
<li class="chapter" data-level="7.8" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:WAIC"><i class="fa fa-check"></i><b>7.8</b> Calculating WAIC</a></li>
<li class="chapter" data-level="7.9" data-path="cha-mcmc.html"><a href="cha-mcmc.html#k-fold-cross-validation"><i class="fa fa-check"></i><b>7.9</b> k-fold cross-validation</a></li>
<li class="chapter" data-level="7.10" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:rjmcmc"><i class="fa fa-check"></i><b>7.10</b> Variable selection using Reversible Jump MCMC</a>
<ul>
<li class="chapter" data-level="7.10.1" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:rjmcmc-indicator"><i class="fa fa-check"></i><b>7.10.1</b> Using indicator variables</a></li>
<li class="chapter" data-level="7.10.2" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:rjmcmc-no-indicator"><i class="fa fa-check"></i><b>7.10.2</b> Without indicator variables</a></li>
</ul></li>
<li class="chapter" data-level="7.11" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:samplers-provided"><i class="fa fa-check"></i><b>7.11</b> Samplers provided with NIMBLE</a>
<ul>
<li class="chapter" data-level="7.11.1" data-path="cha-mcmc.html"><a href="cha-mcmc.html#conjugate-gibbs-samplers"><i class="fa fa-check"></i><b>7.11.1</b> Conjugate (‘Gibbs’) samplers</a></li>
<li class="chapter" data-level="7.11.2" data-path="cha-mcmc.html"><a href="cha-mcmc.html#subsec:HMC"><i class="fa fa-check"></i><b>7.11.2</b> Hamiltonian Monte Carlo (HMC)</a></li>
<li class="chapter" data-level="7.11.3" data-path="cha-mcmc.html"><a href="cha-mcmc.html#customized-log-likelihood-evaluations-rw_llfunction-sampler"><i class="fa fa-check"></i><b>7.11.3</b> Customized log-likelihood evaluations: <em>RW_llFunction sampler</em></a></li>
<li class="chapter" data-level="7.11.4" data-path="cha-mcmc.html"><a href="cha-mcmc.html#particle-mcmc-sampler"><i class="fa fa-check"></i><b>7.11.4</b> Particle MCMC sampler</a></li>
</ul></li>
<li class="chapter" data-level="7.12" data-path="cha-mcmc.html"><a href="cha-mcmc.html#sec:mcmc-example-litters"><i class="fa fa-check"></i><b>7.12</b> Detailed MCMC example: <em>litters</em></a></li>
<li class="chapter" data-level="7.13" data-path="cha-mcmc.html"><a href="cha-mcmc.html#mcmc-suite-compare-mcmcs"><i class="fa fa-check"></i><b>7.13</b> Comparing different MCMCs with <em>MCMCsuite</em> and <em>compareMCMCs</em></a></li>
<li class="chapter" data-level="7.14" data-path="cha-mcmc.html"><a href="cha-mcmc.html#running-mcmc-chains-in-parallel"><i class="fa fa-check"></i><b>7.14</b> Running MCMC chains in parallel</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html"><i class="fa fa-check"></i><b>8</b> Sequential Monte Carlo, Particle MCMC, Iterated Filtering, and MCEM</a>
<ul>
<li class="chapter" data-level="8.1" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html#particle-filters-sequential-monte-carlo-and-iterated-filtering"><i class="fa fa-check"></i><b>8.1</b> Particle filters / sequential Monte Carlo and iterated filtering</a>
<ul>
<li class="chapter" data-level="8.1.1" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html#filtering-algorithms"><i class="fa fa-check"></i><b>8.1.1</b> Filtering algorithms</a></li>
<li class="chapter" data-level="8.1.2" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html#sec:particle-mcmc"><i class="fa fa-check"></i><b>8.1.2</b> Particle MCMC (PMCMC)</a></li>
</ul></li>
<li class="chapter" data-level="8.2" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html#monte-carlo-expectation-maximization-mcem"><i class="fa fa-check"></i><b>8.2</b> Monte Carlo Expectation Maximization (MCEM)</a>
<ul>
<li class="chapter" data-level="8.2.1" data-path="cha-algos-provided.html"><a href="cha-algos-provided.html#sec:estimate-mcem-cov"><i class="fa fa-check"></i><b>8.2.1</b> Estimating the asymptotic covariance From MCEM</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="9" data-path="cha-spatial.html"><a href="cha-spatial.html"><i class="fa fa-check"></i><b>9</b> Spatial models</a>
<ul>
<li class="chapter" data-level="9.1" data-path="cha-spatial.html"><a href="cha-spatial.html#intrinsic-gaussian-car-model-dcar_normal"><i class="fa fa-check"></i><b>9.1</b> Intrinsic Gaussian CAR model: <em>dcar_normal</em></a>
<ul>
<li class="chapter" data-level="9.1.1" data-path="cha-spatial.html"><a href="cha-spatial.html#specification-and-density"><i class="fa fa-check"></i><b>9.1.1</b> Specification and density</a></li>
<li class="chapter" data-level="9.1.2" data-path="cha-spatial.html"><a href="cha-spatial.html#example"><i class="fa fa-check"></i><b>9.1.2</b> Example</a></li>
</ul></li>
<li class="chapter" data-level="9.2" data-path="cha-spatial.html"><a href="cha-spatial.html#proper-gaussian-car-model-dcar_proper"><i class="fa fa-check"></i><b>9.2</b> Proper Gaussian CAR model: <em>dcar_proper</em></a>
<ul>
<li class="chapter" data-level="9.2.1" data-path="cha-spatial.html"><a href="cha-spatial.html#specification-and-density-1"><i class="fa fa-check"></i><b>9.2.1</b> Specification and density</a></li>
<li class="chapter" data-level="9.2.2" data-path="cha-spatial.html"><a href="cha-spatial.html#example-1"><i class="fa fa-check"></i><b>9.2.2</b> Example</a></li>
</ul></li>
<li class="chapter" data-level="9.3" data-path="cha-spatial.html"><a href="cha-spatial.html#sec:spatial-mcmc-sampling-car"><i class="fa fa-check"></i><b>9.3</b> MCMC Sampling of CAR models</a>
<ul>
<li class="chapter" data-level="9.3.1" data-path="cha-spatial.html"><a href="cha-spatial.html#initial-values"><i class="fa fa-check"></i><b>9.3.1</b> Initial values</a></li>
<li class="chapter" data-level="9.3.2" data-path="cha-spatial.html"><a href="cha-spatial.html#zero-neighbor-regions"><i class="fa fa-check"></i><b>9.3.2</b> Zero-neighbor regions</a></li>
<li class="chapter" data-level="9.3.3" data-path="cha-spatial.html"><a href="cha-spatial.html#zero-mean-constraint"><i class="fa fa-check"></i><b>9.3.3</b> Zero-mean constraint</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="10" data-path="cha-bnp.html"><a href="cha-bnp.html"><i class="fa fa-check"></i><b>10</b> Bayesian nonparametric models</a>
<ul>
<li class="chapter" data-level="10.1" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:bnpmixtures"><i class="fa fa-check"></i><b>10.1</b> Bayesian nonparametric mixture models</a></li>
<li class="chapter" data-level="10.2" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:crp"><i class="fa fa-check"></i><b>10.2</b> Chinese Restaurant Process model</a>
<ul>
<li class="chapter" data-level="10.2.1" data-path="cha-bnp.html"><a href="cha-bnp.html#specification-and-density-2"><i class="fa fa-check"></i><b>10.2.1</b> Specification and density</a></li>
<li class="chapter" data-level="10.2.2" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:excrp"><i class="fa fa-check"></i><b>10.2.2</b> Example</a></li>
<li class="chapter" data-level="10.2.3" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:extensionscrp"><i class="fa fa-check"></i><b>10.2.3</b> Extensions</a></li>
</ul></li>
<li class="chapter" data-level="10.3" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:sb"><i class="fa fa-check"></i><b>10.3</b> Stick-breaking model</a>
<ul>
<li class="chapter" data-level="10.3.1" data-path="cha-bnp.html"><a href="cha-bnp.html#specification-and-function"><i class="fa fa-check"></i><b>10.3.1</b> Specification and function</a></li>
<li class="chapter" data-level="10.3.2" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:exsb"><i class="fa fa-check"></i><b>10.3.2</b> Example</a></li>
</ul></li>
<li class="chapter" data-level="10.4" data-path="cha-bnp.html"><a href="cha-bnp.html#mcmc-sampling-of-bnp-models"><i class="fa fa-check"></i><b>10.4</b> MCMC sampling of BNP models</a>
<ul>
<li class="chapter" data-level="10.4.1" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:mcmcdcrp"><i class="fa fa-check"></i><b>10.4.1</b> Sampling CRP models</a></li>
<li class="chapter" data-level="10.4.2" data-path="cha-bnp.html"><a href="cha-bnp.html#sec:mcmcsb"><i class="fa fa-check"></i><b>10.4.2</b> Sampling stick-breaking models</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>IV Programming with NIMBLE</b></span></li>
<li class="chapter" data-level="" data-path="overview.html"><a href="overview.html"><i class="fa fa-check"></i>Overview</a></li>
<li class="chapter" data-level="11" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html"><i class="fa fa-check"></i><b>11</b> Writing simple nimbleFunctions</a>
<ul>
<li class="chapter" data-level="11.1" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:RC-intro"><i class="fa fa-check"></i><b>11.1</b> Introduction to simple nimbleFunctions</a></li>
<li class="chapter" data-level="11.2" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:r-fiunctions-implemented"><i class="fa fa-check"></i><b>11.2</b> R functions (or variants) implemented in NIMBLE</a>
<ul>
<li class="chapter" data-level="11.2.1" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#finding-help-for-nimbles-versions-of-r-functions"><i class="fa fa-check"></i><b>11.2.1</b> Finding help for NIMBLE’s versions of R functions</a></li>
<li class="chapter" data-level="11.2.2" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#basic-operations"><i class="fa fa-check"></i><b>11.2.2</b> Basic operations</a></li>
<li class="chapter" data-level="11.2.3" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:basic-math-linear"><i class="fa fa-check"></i><b>11.2.3</b> Math and linear algebra</a></li>
<li class="chapter" data-level="11.2.4" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:nimble-dist-funs"><i class="fa fa-check"></i><b>11.2.4</b> Distribution functions</a></li>
<li class="chapter" data-level="11.2.5" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:basic-flow-control"><i class="fa fa-check"></i><b>11.2.5</b> Flow control: <em>if-then-else</em>, <em>for</em>, <em>while</em>, and <em>stop</em></a></li>
<li class="chapter" data-level="11.2.6" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:print"><i class="fa fa-check"></i><b>11.2.6</b> <em>print</em> and <em>cat</em></a></li>
<li class="chapter" data-level="11.2.7" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:check-user-interr"><i class="fa fa-check"></i><b>11.2.7</b> Checking for user interrupts: <em>checkInterrupt</em></a></li>
<li class="chapter" data-level="11.2.8" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#optimization-optim-and-nimoptim"><i class="fa fa-check"></i><b>11.2.8</b> Optimization: <em>optim</em> and <em>nimOptim</em></a></li>
<li class="chapter" data-level="11.2.9" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:altern-keyw-some"><i class="fa fa-check"></i><b>11.2.9</b> ‘nim’ synonyms for some functions</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:how-nimble-handles"><i class="fa fa-check"></i><b>11.3</b> How NIMBLE handles types of variables</a>
<ul>
<li class="chapter" data-level="11.3.1" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:nimbleList-RCFuns"><i class="fa fa-check"></i><b>11.3.1</b> nimbleList data structures</a></li>
<li class="chapter" data-level="11.3.2" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:how-types-work"><i class="fa fa-check"></i><b>11.3.2</b> How numeric types work</a></li>
</ul></li>
<li class="chapter" data-level="11.4" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:decl-argum-return"><i class="fa fa-check"></i><b>11.4</b> Declaring argument and return types</a></li>
<li class="chapter" data-level="11.5" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:comp-nimbl-pass"><i class="fa fa-check"></i><b>11.5</b> Compiled nimbleFunctions pass arguments by reference</a></li>
<li class="chapter" data-level="11.6" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:calling-external-code"><i class="fa fa-check"></i><b>11.6</b> Calling external compiled code</a></li>
<li class="chapter" data-level="11.7" data-path="cha-RCfunctions.html"><a href="cha-RCfunctions.html#sec:calling-R-code"><i class="fa fa-check"></i><b>11.7</b> Calling uncompiled R functions from compiled nimbleFunctions</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="cha-user-defined.html"><a href="cha-user-defined.html"><i class="fa fa-check"></i><b>12</b> Creating user-defined BUGS distributions and functions</a>
<ul>
<li class="chapter" data-level="12.1" data-path="cha-user-defined.html"><a href="cha-user-defined.html#sec:user-functions"><i class="fa fa-check"></i><b>12.1</b> User-defined functions</a></li>
<li class="chapter" data-level="12.2" data-path="cha-user-defined.html"><a href="cha-user-defined.html#sec:user-distributions"><i class="fa fa-check"></i><b>12.2</b> User-defined distributions</a>
<ul>
<li class="chapter" data-level="12.2.1" data-path="cha-user-defined.html"><a href="cha-user-defined.html#sec:registerDistributions"><i class="fa fa-check"></i><b>12.2.1</b> Using <em>registerDistributions</em> for alternative parameterizations and providing other information</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="13" data-path="cha-using-models.html"><a href="cha-using-models.html"><i class="fa fa-check"></i><b>13</b> Working with NIMBLE models</a>
<ul>
<li class="chapter" data-level="13.1" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:accessing-variables"><i class="fa fa-check"></i><b>13.1</b> The variables and nodes in a NIMBLE model</a>
<ul>
<li class="chapter" data-level="13.1.1" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:get-nodes"><i class="fa fa-check"></i><b>13.1.1</b> Determining the nodes in a model</a></li>
<li class="chapter" data-level="13.1.2" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:introduced-nodes"><i class="fa fa-check"></i><b>13.1.2</b> Understanding lifted nodes</a></li>
<li class="chapter" data-level="13.1.3" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:cdgetdependencies"><i class="fa fa-check"></i><b>13.1.3</b> Determining dependencies in a model</a></li>
</ul></li>
<li class="chapter" data-level="13.2" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:nodeInfo"><i class="fa fa-check"></i><b>13.2</b> Accessing information about nodes and variables</a>
<ul>
<li class="chapter" data-level="13.2.1" data-path="cha-using-models.html"><a href="cha-using-models.html#getting-distributional-information-about-a-node"><i class="fa fa-check"></i><b>13.2.1</b> Getting distributional information about a node</a></li>
<li class="chapter" data-level="13.2.2" data-path="cha-using-models.html"><a href="cha-using-models.html#getting-information-about-a-distribution"><i class="fa fa-check"></i><b>13.2.2</b> Getting information about a distribution</a></li>
<li class="chapter" data-level="13.2.3" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:getParam"><i class="fa fa-check"></i><b>13.2.3</b> Getting distribution parameter values for a node</a></li>
<li class="chapter" data-level="13.2.4" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:getBound"><i class="fa fa-check"></i><b>13.2.4</b> Getting distribution bounds for a node</a></li>
</ul></li>
<li class="chapter" data-level="13.3" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:cdcalc-cdsim-cdgetl"><i class="fa fa-check"></i><b>13.3</b> Carrying out model calculations</a>
<ul>
<li class="chapter" data-level="13.3.1" data-path="cha-using-models.html"><a href="cha-using-models.html#core-model-operations-calculation-and-simulation"><i class="fa fa-check"></i><b>13.3.1</b> Core model operations: calculation and simulation</a></li>
<li class="chapter" data-level="13.3.2" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:cdsimn-cdcalcn-cdget"><i class="fa fa-check"></i><b>13.3.2</b> Pre-defined nimbleFunctions for operating on model nodes: <em>simNodes</em>, <em>calcNodes</em>, and <em>getLogProbNodes</em></a></li>
<li class="chapter" data-level="13.3.3" data-path="cha-using-models.html"><a href="cha-using-models.html#sec:access-log-prob"><i class="fa fa-check"></i><b>13.3.3</b> Accessing log probabilities via <em>logProb</em> variables</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="14" data-path="cha-data-structures.html"><a href="cha-data-structures.html"><i class="fa fa-check"></i><b>14</b> Data structures in NIMBLE</a>
<ul>
<li class="chapter" data-level="14.1" data-path="cha-data-structures.html"><a href="cha-data-structures.html#sec:modelValues-struct"><i class="fa fa-check"></i><b>14.1</b> The modelValues data structure</a>
<ul>
<li class="chapter" data-level="14.1.1" data-path="cha-data-structures.html"><a href="cha-data-structures.html#creating-modelvalues-objects"><i class="fa fa-check"></i><b>14.1.1</b> Creating modelValues objects</a></li>
<li class="chapter" data-level="14.1.2" data-path="cha-data-structures.html"><a href="cha-data-structures.html#sec:access-cont-modelv"><i class="fa fa-check"></i><b>14.1.2</b> Accessing contents of modelValues</a></li>
</ul></li>
<li class="chapter" data-level="14.2" data-path="cha-data-structures.html"><a href="cha-data-structures.html#sec:nimbleLists"><i class="fa fa-check"></i><b>14.2</b> The nimbleList data structure</a>
<ul>
<li class="chapter" data-level="14.2.1" data-path="cha-data-structures.html"><a href="cha-data-structures.html#sec:predef-nimbleLists"><i class="fa fa-check"></i><b>14.2.1</b> Pre-defined nimbleList types</a></li>
<li class="chapter" data-level="14.2.2" data-path="cha-data-structures.html"><a href="cha-data-structures.html#sec:eigen-nimFunctions"><i class="fa fa-check"></i><b>14.2.2</b> Using <em>eigen</em> and <em>svd</em> in nimbleFunctions</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="15" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html"><i class="fa fa-check"></i><b>15</b> Writing nimbleFunctions to interact with models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:writ-nimble-funct"><i class="fa fa-check"></i><b>15.1</b> Overview</a></li>
<li class="chapter" data-level="15.2" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:using-comp-nimbl"><i class="fa fa-check"></i><b>15.2</b> Using and compiling nimbleFunctions</a></li>
<li class="chapter" data-level="15.3" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#writing-setup-code"><i class="fa fa-check"></i><b>15.3</b> Writing setup code</a>
<ul>
<li class="chapter" data-level="15.3.1" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#useful-tools-for-setup-functions"><i class="fa fa-check"></i><b>15.3.1</b> Useful tools for setup functions</a></li>
<li class="chapter" data-level="15.3.2" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:access-modify-numer"><i class="fa fa-check"></i><b>15.3.2</b> Accessing and modifying numeric values from setup</a></li>
<li class="chapter" data-level="15.3.3" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#determining-numeric-types-in-nimblefunctions"><i class="fa fa-check"></i><b>15.3.3</b> Determining numeric types in nimbleFunctions</a></li>
<li class="chapter" data-level="15.3.4" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:determ-pers-texttts"><i class="fa fa-check"></i><b>15.3.4</b> Control of setup outputs</a></li>
</ul></li>
<li class="chapter" data-level="15.4" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:nimble-lang-comp"><i class="fa fa-check"></i><b>15.4</b> Writing run code</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:driv-models:-calc"><i class="fa fa-check"></i><b>15.4.1</b> Driving models: <em>calculate</em>, <em>calculateDiff</em>, <em>simulate</em>, <em>getLogProb</em></a></li>
<li class="chapter" data-level="15.4.2" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#getting-and-setting-variable-and-node-values"><i class="fa fa-check"></i><b>15.4.2</b> Getting and setting variable and node values</a></li>
<li class="chapter" data-level="15.4.3" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#getting-parameter-values-and-node-bounds"><i class="fa fa-check"></i><b>15.4.3</b> Getting parameter values and node bounds</a></li>
<li class="chapter" data-level="15.4.4" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:access-model-modelv"><i class="fa fa-check"></i><b>15.4.4</b> Using modelValues objects</a></li>
<li class="chapter" data-level="15.4.5" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:using-model-variable"><i class="fa fa-check"></i><b>15.4.5</b> Using model variables and modelValues in expressions</a></li>
<li class="chapter" data-level="15.4.6" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:incl-other-meth"><i class="fa fa-check"></i><b>15.4.6</b> Including other methods in a nimbleFunction</a></li>
<li class="chapter" data-level="15.4.7" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:using-other-nimbl"><i class="fa fa-check"></i><b>15.4.7</b> Using other nimbleFunctions</a></li>
<li class="chapter" data-level="15.4.8" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:virt-nimbl-nimbl"><i class="fa fa-check"></i><b>15.4.8</b> Virtual nimbleFunctions and nimbleFunctionLists</a></li>
<li class="chapter" data-level="15.4.9" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#character-objects"><i class="fa fa-check"></i><b>15.4.9</b> Character objects</a></li>
<li class="chapter" data-level="15.4.10" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:user-defined-data"><i class="fa fa-check"></i><b>15.4.10</b> User-defined data structures</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:user-samplers"><i class="fa fa-check"></i><b>15.5</b> Example: writing user-defined samplers to extend NIMBLE’s MCMC engine</a>
<ul>
<li class="chapter" data-level="15.5.1" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#user-defined-samplers-and-posterior-predictive-nodes"><i class="fa fa-check"></i><b>15.5.1</b> User-defined samplers and posterior predictive nodes</a></li>
</ul></li>
<li class="chapter" data-level="15.6" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#copying-nimblefunctions-and-nimble-models"><i class="fa fa-check"></i><b>15.6</b> Copying nimbleFunctions (and NIMBLE models)</a></li>
<li class="chapter" data-level="15.7" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#sec:debugging"><i class="fa fa-check"></i><b>15.7</b> Debugging nimbleFunctions</a></li>
<li class="chapter" data-level="15.8" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#timing-nimblefunctions-with-run.time"><i class="fa fa-check"></i><b>15.8</b> Timing nimbleFunctions with <em>run.time</em></a></li>
<li class="chapter" data-level="15.9" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#clearing-and-unloading-compiled-objects"><i class="fa fa-check"></i><b>15.9</b> Clearing and unloading compiled objects</a></li>
<li class="chapter" data-level="15.10" data-path="cha-progr-with-models.html"><a href="cha-progr-with-models.html#reducing-memory-usage"><i class="fa fa-check"></i><b>15.10</b> Reducing memory usage</a></li>
</ul></li>
<li class="part"><span><b>V Automatic Derivatives in NIMBLE</b></span></li>
<li class="chapter" data-level="16" data-path="cha-AD.html"><a href="cha-AD.html"><i class="fa fa-check"></i><b>16</b> Automatic Derivatives</a>
<ul>
<li class="chapter" data-level="16.1" data-path="cha-AD.html"><a href="cha-AD.html#sec:use-derivs"><i class="fa fa-check"></i><b>16.1</b> How to turn on derivatives in a model</a>
<ul>
<li class="chapter" data-level="16.1.1" data-path="cha-AD.html"><a href="cha-AD.html#finish-setting-up-the-glmm-example"><i class="fa fa-check"></i><b>16.1.1</b> Finish setting up the GLMM example</a></li>
</ul></li>
<li class="chapter" data-level="16.2" data-path="cha-AD.html"><a href="cha-AD.html#how-to-use-laplace-approximation"><i class="fa fa-check"></i><b>16.2</b> How to use Laplace approximation</a></li>
<li class="chapter" data-level="16.3" data-path="cha-AD.html"><a href="cha-AD.html#how-to-support-derivatives-in-user-defined-functions-and-distributions"><i class="fa fa-check"></i><b>16.3</b> How to support derivatives in user-defined functions and distributions</a></li>
<li class="chapter" data-level="16.4" data-path="cha-AD.html"><a href="cha-AD.html#what-operations-are-and-arent-supported-for-ad"><i class="fa fa-check"></i><b>16.4</b> What operations are and aren’t supported for AD</a></li>
<li class="chapter" data-level="16.5" data-path="cha-AD.html"><a href="cha-AD.html#basics-of-obtaining-derivatives-in-nimblefunctions"><i class="fa fa-check"></i><b>16.5</b> Basics of obtaining derivatives in <code>nimbleFunctions</code></a>
<ul>
<li class="chapter" data-level="16.5.1" data-path="cha-AD.html"><a href="cha-AD.html#checking-derivatives-with-uncompiled-execution"><i class="fa fa-check"></i><b>16.5.1</b> Checking derivatives with uncompiled execution</a></li>
<li class="chapter" data-level="16.5.2" data-path="cha-AD.html"><a href="cha-AD.html#holding-some-local-variables-out-of-derivative-tracking"><i class="fa fa-check"></i><b>16.5.2</b> Holding some local variables out of derivative tracking</a></li>
<li class="chapter" data-level="16.5.3" data-path="cha-AD.html"><a href="cha-AD.html#using-ad-with-multiple-nimblefunctions"><i class="fa fa-check"></i><b>16.5.3</b> Using AD with multiple nimbleFunctions</a></li>
<li class="chapter" data-level="16.5.4" data-path="cha-AD.html"><a href="cha-AD.html#understanding-more-about-how-ad-works-taping-of-operations"><i class="fa fa-check"></i><b>16.5.4</b> Understanding more about how AD works: <em>taping</em> of operations</a></li>
<li class="chapter" data-level="16.5.5" data-path="cha-AD.html"><a href="cha-AD.html#resetting-a-nimderivs-call"><i class="fa fa-check"></i><b>16.5.5</b> Resetting a <code>nimDerivs</code> call</a></li>
<li class="chapter" data-level="16.5.6" data-path="cha-AD.html"><a href="cha-AD.html#a-note-on-performance-benchmarking"><i class="fa fa-check"></i><b>16.5.6</b> A note on performance benchmarking</a></li>
</ul></li>
<li class="chapter" data-level="16.6" data-path="cha-AD.html"><a href="cha-AD.html#advanced-uses-double-taping"><i class="fa fa-check"></i><b>16.6</b> Advanced uses: double taping</a></li>
<li class="chapter" data-level="16.7" data-path="cha-AD.html"><a href="cha-AD.html#derivatives-involving-model-calculations"><i class="fa fa-check"></i><b>16.7</b> Derivatives involving model calculations</a>
<ul>
<li class="chapter" data-level="16.7.1" data-path="cha-AD.html"><a href="cha-AD.html#method-1-nimderivs-of-modelcalculate"><i class="fa fa-check"></i><b>16.7.1</b> Method 1: <code>nimDerivs</code> of <code>model$calculate</code></a></li>
<li class="chapter" data-level="16.7.2" data-path="cha-AD.html"><a href="cha-AD.html#method-2-nimderivs-of-a-method-that-calls-modelcalculate"><i class="fa fa-check"></i><b>16.7.2</b> Method 2: <code>nimDerivs</code> of a method that calls <code>model$calculate</code></a></li>
</ul></li>
<li class="chapter" data-level="16.8" data-path="cha-AD.html"><a href="cha-AD.html#sec:parameter-transform"><i class="fa fa-check"></i><b>16.8</b> Parameter transformations</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="example-maximum-likelihood-estimation-using-optim-with-gradients-from-nimderivs..html"><a href="example-maximum-likelihood-estimation-using-optim-with-gradients-from-nimderivs..html"><i class="fa fa-check"></i><b>17</b> Example: maximum likelihood estimation using <code>optim</code> with gradients from <code>nimDerivs</code>.</a></li>
<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
</ul>
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<div id="cha-writing-models" class="section level1 hasAnchor" number="5">
<h1><span class="header-section-number">Chapter 5</span> Writing models in NIMBLE’s dialect of BUGS<a href="cha-writing-models.html#cha-writing-models" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Models in NIMBLE are written using a variation on the BUGS language.
From BUGS code, NIMBLE creates a model object. This chapter
describes NIMBLE’s version of BUGS. The next chapter explains how to
build and manipulate model objects.
<!--- With NIMBLE you can also define your own distributions and functions for use in BUGS code; discussion of this functionality is deferred to Chapter \@ref(cha-user-defined) as it requires the use of nimbleFunctions. --></p>
<div id="sec:supp-feat-bugs" class="section level2 hasAnchor" number="5.1">
<h2><span class="header-section-number">5.1</span> Comparison to BUGS dialects supported by WinBUGS, OpenBUGS and JAGS<a href="cha-writing-models.html#sec:supp-feat-bugs" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Many users will come to NIMBLE with some familiarity with WinBUGS,
OpenBUGS, or JAGS, so we start by summarizing how NIMBLE is similar to
and different from those before documenting NIMBLE’s version of BUGS
more completely. In general, NIMBLE aims to be compatible with the
original BUGS language and also JAGS’ version. However, at this
point, there are some features not supported by NIMBLE, and there are
some extensions that are planned but not implemented.</p>
<div id="supported-features-of-bugs-and-jags" class="section level3 hasAnchor" number="5.1.1">
<h3><span class="header-section-number">5.1.1</span> Supported features of BUGS and JAGS<a href="cha-writing-models.html#supported-features-of-bugs-and-jags" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ol style="list-style-type: decimal">
<li>Stochastic and deterministic<a href="#fn10" class="footnote-ref" id="fnref10"><sup>10</sup></a> node declarations.</li>
<li>Most univariate and multivariate distributions.</li>
<li>Link functions.</li>
<li>Most mathematical functions.</li>
<li>‘for’ loops for iterative declarations.</li>
<li>Arrays of nodes up to 4 dimensions.</li>
<li>Truncation and censoring as in JAGS using the <code>T()</code>
notation and <code>dinterval</code>.</li>
</ol>
</div>
<div id="sec:extensions-bugs" class="section level3 hasAnchor" number="5.1.2">
<h3><span class="header-section-number">5.1.2</span> NIMBLE’s Extensions to BUGS and JAGS<a href="cha-writing-models.html#sec:extensions-bugs" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>NIMBLE extends the BUGS language in the following ways:</p>
<ol style="list-style-type: decimal">
<li>User-defined functions and distributions – written as nimbleFunctions – can be used in model code. See Chapter <a href="cha-user-defined.html#cha-user-defined">12</a>.</li>
<li>Multiple parameterizations for distributions, similar to those in R, can be used.</li>
<li>Named parameters for distributions and functions, similar to R function calls, can be used.</li>
<li>Linear algebra, including for vectorized
calculations of simple algebra, can be used in deterministic declarations.</li>
<li>Distribution parameters can be expressions, as in JAGS but not in WinBUGS. Caveat: parameters to <em>multivariate</em>
distributions (e.g., <code>dmnorm</code>) cannot be expressions (but an expression can be defined in a separate deterministic expression and the resulting variable then used). <!--- still true. -Perry --></li>
<li>Alternative models can be defined from the same model code by using if-then-else statements that are evaluated when the model is defined.</li>
<li>More flexible indexing of vector nodes within larger variables is allowed. For example one can place a multivariate normal vector arbitrarily within a higher-dimensional object, not just in the last index.</li>
<li>More general constraints can be declared using <code>dconstraint</code>, which extends the concept of JAGS’ <code>dinterval</code>.</li>
<li>Link functions can be used in stochastic, as well as deterministic, declarations.<a href="#fn11" class="footnote-ref" id="fnref11"><sup>11</sup></a></li>
<li>Data values can be reset, and which parts of a model are flagged as data can be changed, allowing one model to be used for different data sets without rebuilding the model each time.</li>
<li>One can use stochastic/dynamic indexes, i.e., indexes can be other nodes or functions of other nodes. For a given dimension of a node being indexed, if the index is not constant, it must be a scalar value. So expressions such as <code>mu[k[i], 3]</code> or <code>mu[k[i], 1:3]</code> or <code>mu[k[i], j[i]]</code> are allowed, but not <code>mu[k[i]:(k[i]+1)]</code><a href="#fn12" class="footnote-ref" id="fnref12"><sup>12</sup></a> Nested dynamic indexes such as <code>mu[k[j[i]]]</code> are also allowed.</li>
</ol>
</div>
<div id="sec:not-yet-supported" class="section level3 hasAnchor" number="5.1.3">
<h3><span class="header-section-number">5.1.3</span> Not-supported features of BUGS and JAGS<a href="cha-writing-models.html#sec:not-yet-supported" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>The following are not supported.</p>
<ol style="list-style-type: decimal">
<li>The appearance of the same node on the left-hand side of both a
<code><-</code> and a <span class="math inline">\(\sim\)</span> declaration (used in WinBUGS for data
assignment for the value of a stochastic node).</li>
<li>Multivariate nodes must appear with brackets, even if they are
empty. E.g., <code>x</code> cannot be multivariate but <code>x[]</code> or
<code>x[2:5]</code> can be.</li>
<li>NIMBLE generally determines the dimensionality and
sizes of variables from the BUGS code. However, when a variable
appears with blank indices, such as in <code>x.sum <- sum(x[])</code>,
and if the dimensions of the variable are not clearly defined in
other declarations, NIMBLE currently requires that the dimensions of
x be provided when the model object is created (via <code>nimbleModel</code>).</li>
<li>Use of non-sequential indexes in the definition of a for loop. JAGS allows syntax such as <code>for(i in expr)</code> when <code>expr</code> evaluates to an integer vector. In NIMBLE one can use the work-around of defining a constant vector, say <code>k</code>, and using <code>k[i]</code> in the body of the for loop.</li>
<li>Use of non-sequential indexes to subset variables. JAGS allows syntax such as <code>sum(mu[expr])</code> when <code>expr</code> evaluates to an integer vector. In NIMBLE one can use the work-around of defining a nimbleFunction that takes <code>mu</code> and the index vector as arguments and does the calculation of interest.</li>
</ol>
</div>
</div>
<div id="writing-models" class="section level2 hasAnchor" number="5.2">
<h2><span class="header-section-number">5.2</span> Writing models<a href="cha-writing-models.html#writing-models" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Here we introduce NIMBLE’s version of BUGS. The WinBUGS, OpenBUGS and
JAGS manuals are also useful resources for writing BUGS models,
including many examples.</p>
<div id="declaring-stochastic-and-deterministic-nodes" class="section level3 hasAnchor" number="5.2.1">
<h3><span class="header-section-number">5.2.1</span> Declaring stochastic and deterministic nodes<a href="cha-writing-models.html#declaring-stochastic-and-deterministic-nodes" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>BUGS is a declarative language for graphical (or hierarchical) models.
Most programming languages are imperative, which means a series of
commands will be executed in the order they are written. A
declarative language like BUGS is more like building a machine before
using it. Each line declares that a component should be plugged into
the machine, but it doesn’t matter in what order they are declared as
long as all the right components are plugged in by the end of the code.</p>
<p>The machine in this case is a graphical model<a href="#fn13" class="footnote-ref" id="fnref13"><sup>13</sup></a>. A <em>node</em> (sometimes called
a <em>vertex</em>) holds one value, which may be a scalar or a vector.
<em>Edges</em> define the relationships between nodes. A huge variety
of statistical models can be thought of as graphs.</p>
<p>Here is the code to define and create a simple linear regression model with four observations.</p>
<div class="sourceCode" id="cb76"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb76-1"><a href="cha-writing-models.html#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(nimble)</span>
<span id="cb76-2"><a href="cha-writing-models.html#cb76-2" aria-hidden="true" tabindex="-1"></a>mc <span class="ot"><-</span> <span class="fu">nimbleCode</span>({</span>
<span id="cb76-3"><a href="cha-writing-models.html#cb76-3" aria-hidden="true" tabindex="-1"></a> intercept <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb76-4"><a href="cha-writing-models.html#cb76-4" aria-hidden="true" tabindex="-1"></a> slope <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb76-5"><a href="cha-writing-models.html#cb76-5" aria-hidden="true" tabindex="-1"></a> sigma <span class="sc">~</span> <span class="fu">dunif</span>(<span class="dv">0</span>, <span class="dv">100</span>)</span>
<span id="cb76-6"><a href="cha-writing-models.html#cb76-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>) {</span>
<span id="cb76-7"><a href="cha-writing-models.html#cb76-7" aria-hidden="true" tabindex="-1"></a> predicted.y[i] <span class="ot"><-</span> intercept <span class="sc">+</span> slope <span class="sc">*</span> x[i]</span>
<span id="cb76-8"><a href="cha-writing-models.html#cb76-8" aria-hidden="true" tabindex="-1"></a> y[i] <span class="sc">~</span> <span class="fu">dnorm</span>(predicted.y[i], <span class="at">sd =</span> sigma)</span>
<span id="cb76-9"><a href="cha-writing-models.html#cb76-9" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb76-10"><a href="cha-writing-models.html#cb76-10" aria-hidden="true" tabindex="-1"></a>})</span>
<span id="cb76-11"><a href="cha-writing-models.html#cb76-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-12"><a href="cha-writing-models.html#cb76-12" aria-hidden="true" tabindex="-1"></a>model <span class="ot"><-</span> <span class="fu">nimbleModel</span>(mc, <span class="at">data =</span> <span class="fu">list</span>(<span class="at">y =</span> <span class="fu">rnorm</span>(<span class="dv">4</span>)))</span>
<span id="cb76-13"><a href="cha-writing-models.html#cb76-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-14"><a href="cha-writing-models.html#cb76-14" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(igraph)</span>
<span id="cb76-15"><a href="cha-writing-models.html#cb76-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-16"><a href="cha-writing-models.html#cb76-16" aria-hidden="true" tabindex="-1"></a>layout <span class="ot"><-</span> <span class="fu">matrix</span>(<span class="at">ncol =</span> <span class="dv">2</span>, <span class="at">byrow =</span> <span class="cn">TRUE</span>,</span>
<span id="cb76-17"><a href="cha-writing-models.html#cb76-17" aria-hidden="true" tabindex="-1"></a> <span class="co"># These seem to be rescaled to fit in the plot area,</span></span>
<span id="cb76-18"><a href="cha-writing-models.html#cb76-18" aria-hidden="true" tabindex="-1"></a> <span class="co"># so I'll just use 0-100 as the scale</span></span>
<span id="cb76-19"><a href="cha-writing-models.html#cb76-19" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> <span class="fu">c</span>(<span class="dv">33</span>, <span class="dv">100</span>,</span>
<span id="cb76-20"><a href="cha-writing-models.html#cb76-20" aria-hidden="true" tabindex="-1"></a> <span class="dv">66</span>, <span class="dv">100</span>,</span>
<span id="cb76-21"><a href="cha-writing-models.html#cb76-21" aria-hidden="true" tabindex="-1"></a> <span class="dv">50</span>, <span class="dv">0</span>, <span class="co"># first three are parameters</span></span>
<span id="cb76-22"><a href="cha-writing-models.html#cb76-22" aria-hidden="true" tabindex="-1"></a> <span class="dv">15</span>, <span class="dv">50</span>, <span class="dv">35</span>, <span class="dv">50</span>, <span class="dv">55</span>, <span class="dv">50</span>, <span class="dv">75</span>, <span class="dv">50</span>, <span class="co"># x's</span></span>
<span id="cb76-23"><a href="cha-writing-models.html#cb76-23" aria-hidden="true" tabindex="-1"></a> <span class="dv">20</span>, <span class="dv">75</span>, <span class="dv">40</span>, <span class="dv">75</span>, <span class="dv">60</span>, <span class="dv">75</span>, <span class="dv">80</span>, <span class="dv">75</span>, <span class="co"># predicted.y's</span></span>
<span id="cb76-24"><a href="cha-writing-models.html#cb76-24" aria-hidden="true" tabindex="-1"></a> <span class="dv">25</span>, <span class="dv">25</span>, <span class="dv">45</span>, <span class="dv">25</span>, <span class="dv">65</span>, <span class="dv">25</span>, <span class="dv">85</span>, <span class="dv">25</span>) <span class="co"># y's</span></span>
<span id="cb76-25"><a href="cha-writing-models.html#cb76-25" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb76-26"><a href="cha-writing-models.html#cb76-26" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-27"><a href="cha-writing-models.html#cb76-27" aria-hidden="true" tabindex="-1"></a>sizes <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">45</span>, <span class="dv">30</span>, <span class="dv">30</span>,</span>
<span id="cb76-28"><a href="cha-writing-models.html#cb76-28" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(<span class="dv">20</span>, <span class="dv">4</span>),</span>
<span id="cb76-29"><a href="cha-writing-models.html#cb76-29" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(<span class="dv">50</span>, <span class="dv">4</span>),</span>
<span id="cb76-30"><a href="cha-writing-models.html#cb76-30" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(<span class="dv">20</span>, <span class="dv">4</span>))</span>
<span id="cb76-31"><a href="cha-writing-models.html#cb76-31" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-32"><a href="cha-writing-models.html#cb76-32" aria-hidden="true" tabindex="-1"></a>edge.color <span class="ot"><-</span> <span class="st">"black"</span></span>
<span id="cb76-33"><a href="cha-writing-models.html#cb76-33" aria-hidden="true" tabindex="-1"></a> <span class="co"># c(</span></span>
<span id="cb76-34"><a href="cha-writing-models.html#cb76-34" aria-hidden="true" tabindex="-1"></a> <span class="co"># rep("green", 8),</span></span>
<span id="cb76-35"><a href="cha-writing-models.html#cb76-35" aria-hidden="true" tabindex="-1"></a> <span class="co"># rep("red", 4),</span></span>
<span id="cb76-36"><a href="cha-writing-models.html#cb76-36" aria-hidden="true" tabindex="-1"></a> <span class="co"># rep("blue", 4),</span></span>
<span id="cb76-37"><a href="cha-writing-models.html#cb76-37" aria-hidden="true" tabindex="-1"></a> <span class="co"># rep("purple", 4))</span></span>
<span id="cb76-38"><a href="cha-writing-models.html#cb76-38" aria-hidden="true" tabindex="-1"></a>stoch.color <span class="ot"><-</span> <span class="st">"deepskyblue2"</span></span>
<span id="cb76-39"><a href="cha-writing-models.html#cb76-39" aria-hidden="true" tabindex="-1"></a>det.color <span class="ot"><-</span> <span class="st">"orchid3"</span></span>
<span id="cb76-40"><a href="cha-writing-models.html#cb76-40" aria-hidden="true" tabindex="-1"></a>rhs.color <span class="ot"><-</span> <span class="st">"gray73"</span></span>
<span id="cb76-41"><a href="cha-writing-models.html#cb76-41" aria-hidden="true" tabindex="-1"></a>fill.color <span class="ot"><-</span> <span class="fu">c</span>(</span>
<span id="cb76-42"><a href="cha-writing-models.html#cb76-42" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(stoch.color, <span class="dv">3</span>),</span>
<span id="cb76-43"><a href="cha-writing-models.html#cb76-43" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(rhs.color, <span class="dv">4</span>),</span>
<span id="cb76-44"><a href="cha-writing-models.html#cb76-44" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(det.color, <span class="dv">4</span>),</span>
<span id="cb76-45"><a href="cha-writing-models.html#cb76-45" aria-hidden="true" tabindex="-1"></a> <span class="fu">rep</span>(stoch.color, <span class="dv">4</span>)</span>
<span id="cb76-46"><a href="cha-writing-models.html#cb76-46" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb76-47"><a href="cha-writing-models.html#cb76-47" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-48"><a href="cha-writing-models.html#cb76-48" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb76-49"><a href="cha-writing-models.html#cb76-49" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(model<span class="sc">$</span>graph, <span class="at">vertex.shape =</span> <span class="st">"crectangle"</span>,</span>
<span id="cb76-50"><a href="cha-writing-models.html#cb76-50" aria-hidden="true" tabindex="-1"></a> <span class="at">vertex.size =</span> sizes,</span>
<span id="cb76-51"><a href="cha-writing-models.html#cb76-51" aria-hidden="true" tabindex="-1"></a> <span class="at">vertex.size2 =</span> <span class="dv">20</span>,</span>
<span id="cb76-52"><a href="cha-writing-models.html#cb76-52" aria-hidden="true" tabindex="-1"></a> <span class="at">layout =</span> layout,</span>
<span id="cb76-53"><a href="cha-writing-models.html#cb76-53" aria-hidden="true" tabindex="-1"></a> <span class="at">vertex.label.cex =</span> <span class="fl">3.0</span>,</span>
<span id="cb76-54"><a href="cha-writing-models.html#cb76-54" aria-hidden="true" tabindex="-1"></a> <span class="at">vertex.color =</span> fill.color,</span>
<span id="cb76-55"><a href="cha-writing-models.html#cb76-55" aria-hidden="true" tabindex="-1"></a> <span class="at">edge.width =</span> <span class="dv">3</span>,</span>
<span id="cb76-56"><a href="cha-writing-models.html#cb76-56" aria-hidden="true" tabindex="-1"></a> <span class="at">asp =</span> <span class="fl">0.5</span>,</span>
<span id="cb76-57"><a href="cha-writing-models.html#cb76-57" aria-hidden="true" tabindex="-1"></a> <span class="at">edge.color =</span> edge.color)</span></code></pre></div>
<div class="figure"><span style="display:block;" id="fig:linearRegressionGraph"></span>
<img src="NimbleUserManual_files/figure-html/linearRegressionGraph-1.png" alt="Graph of a linear regression model" width="1920" />
<p class="caption">
Figure 5.1: Graph of a linear regression model
</p>
</div>
<p>The graph representing the model is
shown in Figure <a href="cha-writing-models.html#fig:linearRegressionGraph">5.1</a>. Each observation,
<code>y[i]</code>, is a node whose edges say that it follows a normal
distribution depending on a predicted value, <code>predicted.y[i]</code>, and
standard deviation, <code>sigma</code>, which are each nodes. Each predicted
value is a node whose edges say how it is calculated from <code>slope</code>,
<code>intercept</code>, and one value of an explanatory variable, <code>x[i]</code>,
which are each nodes.</p>
<p>This graph is created from the following BUGS code:</p>
<div class="sourceCode" id="cb77"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb77-1"><a href="cha-writing-models.html#cb77-1" aria-hidden="true" tabindex="-1"></a>{</span>
<span id="cb77-2"><a href="cha-writing-models.html#cb77-2" aria-hidden="true" tabindex="-1"></a> intercept <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb77-3"><a href="cha-writing-models.html#cb77-3" aria-hidden="true" tabindex="-1"></a> slope <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb77-4"><a href="cha-writing-models.html#cb77-4" aria-hidden="true" tabindex="-1"></a> sigma <span class="sc">~</span> <span class="fu">dunif</span>(<span class="dv">0</span>, <span class="dv">100</span>)</span>
<span id="cb77-5"><a href="cha-writing-models.html#cb77-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>) {</span>
<span id="cb77-6"><a href="cha-writing-models.html#cb77-6" aria-hidden="true" tabindex="-1"></a> predicted.y[i] <span class="ot"><-</span> intercept <span class="sc">+</span> slope <span class="sc">*</span> x[i]</span>
<span id="cb77-7"><a href="cha-writing-models.html#cb77-7" aria-hidden="true" tabindex="-1"></a> y[i] <span class="sc">~</span> <span class="fu">dnorm</span>(predicted.y[i], <span class="at">sd =</span> sigma)</span>
<span id="cb77-8"><a href="cha-writing-models.html#cb77-8" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb77-9"><a href="cha-writing-models.html#cb77-9" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>In this code, stochastic relationships are declared with ‘<span class="math inline">\(\sim\)</span>’
and deterministic relationships are declared with ‘<code><-</code>’. For
example, each <code>y[i]</code> follows a normal distribution with mean
<code>predicted.y[i]</code> and standard deviation
<code>sigma</code>. Each
<code>predicted.y[i]</code> is the result of <code>intercept + slope * x[i]</code>.
The for-loop yields the equivalent of writing four lines of code, each
with a different value of <code>i</code>. It does not matter in what order
the nodes are declared. Imagine that each line of code draws part of
Figure <a href="cha-writing-models.html#fig:linearRegressionGraph">5.1</a>, and all that matters is that
the everything gets drawn in the end. Available distributions, default and alternative
parameterizations, and functions are listed in Section <a href="cha-writing-models.html#subsec:dists-and-functions">5.2.4</a>.</p>
<p>An equivalent graph can be created by this BUGS code:</p>
<div class="sourceCode" id="cb78"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb78-1"><a href="cha-writing-models.html#cb78-1" aria-hidden="true" tabindex="-1"></a>{</span>
<span id="cb78-2"><a href="cha-writing-models.html#cb78-2" aria-hidden="true" tabindex="-1"></a> intercept <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb78-3"><a href="cha-writing-models.html#cb78-3" aria-hidden="true" tabindex="-1"></a> slope <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1000</span>)</span>
<span id="cb78-4"><a href="cha-writing-models.html#cb78-4" aria-hidden="true" tabindex="-1"></a> sigma <span class="sc">~</span> <span class="fu">dunif</span>(<span class="dv">0</span>, <span class="dv">100</span>)</span>
<span id="cb78-5"><a href="cha-writing-models.html#cb78-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>) {</span>
<span id="cb78-6"><a href="cha-writing-models.html#cb78-6" aria-hidden="true" tabindex="-1"></a> y[i] <span class="sc">~</span> <span class="fu">dnorm</span>(intercept <span class="sc">+</span> slope <span class="sc">*</span> x[i], <span class="at">sd =</span> sigma)</span>
<span id="cb78-7"><a href="cha-writing-models.html#cb78-7" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb78-8"><a href="cha-writing-models.html#cb78-8" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>In this case, the <code>predicted.y[i]</code> nodes in Figure
<a href="cha-writing-models.html#fig:linearRegressionGraph">5.1</a> will be created automatically by
NIMBLE and will have a different name, generated by NIMBLE.</p>
</div>
<div id="sec:more-kinds-bugs" class="section level3 hasAnchor" number="5.2.2">
<h3><span class="header-section-number">5.2.2</span> More kinds of BUGS declarations<a href="cha-writing-models.html#sec:more-kinds-bugs" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Here are some examples of valid lines of BUGS code. This code does
not describe a sensible or complete model, and it includes some
arbitrary indices (e.g. <code>mvx[8:10, i]</code>) to illustrate flexibility.
Instead the purpose of each line is to illustrate a feature of
NIMBLE’s version of BUGS.</p>
<div class="sourceCode" id="cb79"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb79-1"><a href="cha-writing-models.html#cb79-1" aria-hidden="true" tabindex="-1"></a>{</span>
<span id="cb79-2"><a href="cha-writing-models.html#cb79-2" aria-hidden="true" tabindex="-1"></a> <span class="co"># 1. normal distribution with BUGS parameter order</span></span>
<span id="cb79-3"><a href="cha-writing-models.html#cb79-3" aria-hidden="true" tabindex="-1"></a> x <span class="sc">~</span> <span class="fu">dnorm</span>(a <span class="sc">+</span> b <span class="sc">*</span> c, tau) </span>
<span id="cb79-4"><a href="cha-writing-models.html#cb79-4" aria-hidden="true" tabindex="-1"></a> <span class="co"># 2. normal distribution with a named parameter</span></span>
<span id="cb79-5"><a href="cha-writing-models.html#cb79-5" aria-hidden="true" tabindex="-1"></a> y <span class="sc">~</span> <span class="fu">dnorm</span>(a <span class="sc">+</span> b <span class="sc">*</span> c, <span class="at">sd =</span> sigma) </span>
<span id="cb79-6"><a href="cha-writing-models.html#cb79-6" aria-hidden="true" tabindex="-1"></a> <span class="co"># 3. For-loop and nested indexing</span></span>
<span id="cb79-7"><a href="cha-writing-models.html#cb79-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N) {</span>
<span id="cb79-8"><a href="cha-writing-models.html#cb79-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(j <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>M[i]) {</span>
<span id="cb79-9"><a href="cha-writing-models.html#cb79-9" aria-hidden="true" tabindex="-1"></a> z[i,j] <span class="sc">~</span> <span class="fu">dexp</span>(r[ blockID[i] ]) </span>
<span id="cb79-10"><a href="cha-writing-models.html#cb79-10" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb79-11"><a href="cha-writing-models.html#cb79-11" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb79-12"><a href="cha-writing-models.html#cb79-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># 4. multivariate distribution with arbitrary indexing</span></span>
<span id="cb79-13"><a href="cha-writing-models.html#cb79-13" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>) </span>
<span id="cb79-14"><a href="cha-writing-models.html#cb79-14" aria-hidden="true" tabindex="-1"></a> mvx[<span class="dv">8</span><span class="sc">:</span><span class="dv">10</span>, i] <span class="sc">~</span> <span class="fu">dmnorm</span>(mvMean[<span class="dv">3</span><span class="sc">:</span><span class="dv">5</span>], <span class="at">cov =</span> mvCov[<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>, <span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>, i])</span>
<span id="cb79-15"><a href="cha-writing-models.html#cb79-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># 5. User-provided distribution</span></span>
<span id="cb79-16"><a href="cha-writing-models.html#cb79-16" aria-hidden="true" tabindex="-1"></a> w <span class="sc">~</span> <span class="fu">dMyDistribution</span>(<span class="at">hello =</span> x, <span class="at">world =</span> y) </span>
<span id="cb79-17"><a href="cha-writing-models.html#cb79-17" aria-hidden="true" tabindex="-1"></a> <span class="co"># 6. Simple deterministic node</span></span>
<span id="cb79-18"><a href="cha-writing-models.html#cb79-18" aria-hidden="true" tabindex="-1"></a> d1 <span class="ot"><-</span> a <span class="sc">+</span> b</span>
<span id="cb79-19"><a href="cha-writing-models.html#cb79-19" aria-hidden="true" tabindex="-1"></a> <span class="co"># 7. Vector deterministic node with matrix multiplication</span></span>
<span id="cb79-20"><a href="cha-writing-models.html#cb79-20" aria-hidden="true" tabindex="-1"></a> d2[] <span class="ot"><-</span> A[ , ] <span class="sc">%*%</span> mvMean[<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>] </span>
<span id="cb79-21"><a href="cha-writing-models.html#cb79-21" aria-hidden="true" tabindex="-1"></a> <span class="co"># 8. Deterministic node with user-provided function</span></span>
<span id="cb79-22"><a href="cha-writing-models.html#cb79-22" aria-hidden="true" tabindex="-1"></a> d3 <span class="ot"><-</span> <span class="fu">foo</span>(x, <span class="at">hooray =</span> y) </span>
<span id="cb79-23"><a href="cha-writing-models.html#cb79-23" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>When a variable appears only on the right-hand side, it can be provided via <code>constants</code> (in which case it can never be changed) or via <code>data</code> or <code>inits</code>, as discussed in Chapter <a href="cha-building-models.html#cha-building-models">6</a>.</p>
<p>Notes on the comment-numbered lines are:</p>
<ol style="list-style-type: decimal">
<li><code>x</code> follows a normal distribution with mean <code>a + b*c</code> and precision <code>tau</code> (default BUGS second parameter for <code>dnorm</code>).</li>
<li><code>y</code> follows a normal distribution with the same mean as <code>x</code> but a named standard deviation parameter instead of a precision parameter (sd = 1/sqrt(precision)).</li>
<li><code>z[i, j]</code> follows an exponential distribution with parameter
<code>r[ blockID[i] ]</code>. This shows how for-loops can be used for indexing of variables containing
multiple nodes. Variables that define for-loop indices (<code>N</code> and <code>M</code>) must also be provided as constants.<br />
</li>
<li>The arbitrary block <code>mvx[8:10, i]</code> follows a multivariate
normal distribution, with a named covariance matrix instead of BUGS’
default of a precision matrix. As in R, curly braces for for-loop
contents are only needed if there is more than one line.</li>
<li><code>w</code> follows a user-defined distribution. See Chapter <a href="cha-user-defined.html#cha-user-defined">12</a>.</li>
<li><code>d1</code> is a scalar deterministic node that, when calculated, will be
set to <code>a + b</code>.</li>
<li><code>d2</code> is a vector deterministic node using matrix
multiplication in R’s syntax.</li>
<li><code>d3</code> is a deterministic node using a user-provided
function. See Chapter <a href="cha-user-defined.html#cha-user-defined">12</a>.</li>
</ol>
<div id="sec:indexing" class="section level4 hasAnchor" number="5.2.2.1">
<h4><span class="header-section-number">5.2.2.1</span> More about indexing<a href="cha-writing-models.html#sec:indexing" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p>Examples of allowed indexing include:</p>
<ul>
<li><code>x[i]</code> # a single index</li>
<li><code>x[i:j]</code> # a range of indices</li>
<li><code>x[i:j,k:l]</code> # multiple single indices or ranges for higher-dimensional arrays</li>
<li><code>x[i:j, ]</code> # blank indices indicating the full range</li>
<li><code>x[3*i+7]</code> # computed indices</li>
<li><code>x[(3*i):(5*i+1)]</code> # computed lower and upper ends of an index range</li>
<li><code>x[k[i]+1]</code> # a dynamic (and computed) index</li>
<li><code>x[k[j[i]]]</code> # nested dynamic indexes</li>
<li><code>x[k[i], 1:3]</code> # nested indexing of rows or columns</li>
</ul>
<p>NIMBLE does not allow multivariate nodes to be used without
square brackets, which is an incompatibility with JAGS. Therefore a statement like <code>xbar <- mean(x)</code> in JAGS must be converted to
<code>xbar <- mean(x[])</code> (if <code>x</code> is a vector) or <code>xbar <- mean(x[,])</code> (if <code>x</code> is a matrix) for NIMBLE<a href="#fn14" class="footnote-ref" id="fnref14"><sup>14</sup></a>. Section <a href="cha-building-models.html#sec:provide-dimensions">6.1.1.5</a> discusses how to provide NIMBLE with dimensions of <code>x</code> when needed.</p>
<p>One cannot provide a vector of indices that are not constant. For example, <code>x[start[i]:end[i]]</code> is allowed only if <code>start</code> and <code>end</code> are provided as constants. Also, one cannot provide a vector, or expression evaluating to a vector (apart from use of <code>:</code>) as an index. For example, <code>x[inds]</code> is not allowed. Often one can write a nimbleFunction to achieve the desired result, such as definining a nimbleFunction that takes <code>x</code>, <code>start[i]</code>, and <code>end[i]</code> or takes <code>x</code> and <code>inds</code> as arguments and does the subsetting (possibly in combination with some other calculation). Note that <code>x[L:U]</code> and <code>x[inds]</code> are allowed in nimbleFunction code.</p>
<p>Generally NIMBLE supports R-like linear algebra expressions and attempts to follow the same rules as R about
dimensions (although in some cases this is not possible). For
example, <code>x[1:3] %*% y[1:3]</code> converts <code>x[1:3]</code> into a row
vector and thus computes the inner product, which is returned as a <span class="math inline">\(1 \times 1\)</span>
matrix (use <code>inprod</code> to get it as a scalar, which it typically easier). Like in R,
a scalar index will result in dropping a dimension unless the argument
<code>drop=FALSE</code> is provided. For example, <code>mymatrix[i, 1:3]</code> will
be a vector of length 3, but <code>mymatrix[i, 1:3, drop=FALSE]</code> will be
a <span class="math inline">\(1 \times 3\)</span> matrix. More about indexing and dimensions is
discussed in Section <a href="cha-RCfunctions.html#sec:manag-dimens-sizes">11.3.2.7</a>.</p>
</div>
</div>
<div id="subsec:vectorized-versus-scalar-declarations" class="section level3 hasAnchor" number="5.2.3">
<h3><span class="header-section-number">5.2.3</span> Vectorized versus scalar declarations<a href="cha-writing-models.html#subsec:vectorized-versus-scalar-declarations" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Suppose you need nodes <code>logY[i]</code> that should be the log of the
corresponding <code>Y[i]</code>, say for <code>i</code> from 1 to 10. Conventionally
this would be created with a for loop:</p>
<div class="sourceCode" id="cb80"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb80-1"><a href="cha-writing-models.html#cb80-1" aria-hidden="true" tabindex="-1"></a>{</span>
<span id="cb80-2"><a href="cha-writing-models.html#cb80-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>) {</span>
<span id="cb80-3"><a href="cha-writing-models.html#cb80-3" aria-hidden="true" tabindex="-1"></a> logY[i] <span class="ot"><-</span> <span class="fu">log</span>(Y[i])</span>
<span id="cb80-4"><a href="cha-writing-models.html#cb80-4" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb80-5"><a href="cha-writing-models.html#cb80-5" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>Since NIMBLE supports R-like algebraic expressions, an alternative in
NIMBLE’s dialect of BUGS is to use a vectorized declaration like this:</p>
<div class="sourceCode" id="cb81"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb81-1"><a href="cha-writing-models.html#cb81-1" aria-hidden="true" tabindex="-1"></a>{</span>
<span id="cb81-2"><a href="cha-writing-models.html#cb81-2" aria-hidden="true" tabindex="-1"></a> logY[<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>] <span class="ot"><-</span> <span class="fu">log</span>(Y[<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>])</span>
<span id="cb81-3"><a href="cha-writing-models.html#cb81-3" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>There is an important difference between the models that are created by the
above two methods. The first creates 10 scalar nodes, <code>logY[1]</code>
<span class="math inline">\(,\ldots,\)</span> <code>logY[10]</code>. The second creates one vector node,
<code>logY[1:10]</code>. If each <code>logY[i]</code> is used separately by an algorithm, it may be more efficient computationally if they are declared as scalars. If they are all used together,
it will often make sense to declare them as a vector.</p>
</div>
<div id="subsec:dists-and-functions" class="section level3 hasAnchor" number="5.2.4">
<h3><span class="header-section-number">5.2.4</span> Available distributions<a href="cha-writing-models.html#subsec:dists-and-functions" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="subsec:distributions" class="section level4 hasAnchor" number="5.2.4.1">
<h4><span class="header-section-number">5.2.4.1</span> Distributions<a href="cha-writing-models.html#subsec:distributions" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p>NIMBLE supports most of the distributions allowed in BUGS and
JAGS. Table <a href="cha-writing-models.html#tab:distributions">5.1</a> lists the distributions that are
currently supported, with their default parameterizations, which match
those of BUGS<a href="#fn15" class="footnote-ref" id="fnref15"><sup>15</sup></a>. NIMBLE also allows one to use alternative
parameterizations for a variety of distributions as described next.
See Section <a href="cha-user-defined.html#sec:user-distributions">12.2</a> to learn how to write new distributions using nimbleFunctions.</p>
<table>
<caption><span id="tab:distributions">Table 5.1: </span> Distributions with their default order of parameters. The value of the random variable
is denoted by <span class="math inline">\(x\)</span>.</caption>
<thead>
<tr class="header">
<th align="left">Name</th>
<th align="left">Usage</th>
<th align="left">Density</th>
<th>Lower</th>
<th>Upper</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">Bernoulli</td>
<td align="left"><code>dbern(prob = p)</code></td>
<td align="left"><span class="math inline">\(p^x (1 - p)^{1 -x}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td><span class="math inline">\(1\)</span></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><span class="math inline">\(0 < p < 1\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Beta</td>
<td align="left"><code>dbeta(shape1 = a,</code></td>
<td align="left"><span class="math inline">\(\frac{x^{a-1}(1-x)^{b-1}}{\beta(a,b)}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td><span class="math inline">\(1\)</span></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><code>shape2 = b)</code>, <span class="math inline">\(a > 0\)</span>, <span class="math inline">\(b > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Binomial</td>
<td align="left"><code>dbin(prob = p, size = n)</code></td>
<td align="left"><span class="math inline">\({n \choose x} p^x (1-p)^{n-x}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td><span class="math inline">\(n\)</span></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><span class="math inline">\(0 < p < 1\)</span>, <span class="math inline">\(n \in \mathbb{N}^*\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">CAR</td>
<td align="left"><code>dcar_normal(adj, weights,</code></td>
<td align="left">see chapter <a href="cha-spatial.html#cha-spatial">9</a> for details</td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">(intrinsic)</td>
<td align="left"><code>num, tau, c, zero_mean)</code></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">CAR</td>
<td align="left"><code>dcar_proper(mu, C, adj,</code></td>
<td align="left">see chapter <a href="cha-spatial.html#cha-spatial">9</a> for details</td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">(proper)</td>
<td align="left"><code>num, M, tau, gamma)</code></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Categorical</td>
<td align="left"><code>dcat(prob = p)</code></td>
<td align="left"><span class="math inline">\(\frac{p_x}{\sum_i p_i}\)</span></td>
<td><span class="math inline">\(1\)</span></td>
<td><span class="math inline">\(N\)</span></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><span class="math inline">\(p \in (\mathbb{R}^+)^N\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Chi-square</td>
<td align="left"><code>dchisq(df = k)</code>, <span class="math inline">\(k > 0\)</span></td>
<td align="left"><span class="math inline">\(\frac{x^{\frac{k}{2}-1} \exp(-x/2)}{2^{\frac{k}{2}} \Gamma(\frac{k}{2})}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Double</td>
<td align="left"><code>ddexp(location =</code> <span class="math inline">\(\mu\)</span>,</td>
<td align="left"><span class="math inline">\(\frac{\tau}{2} \exp(-\tau |x - \mu|)\)</span></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">exponential</td>
<td align="left"><code>rate =</code> <span class="math inline">\(\tau\)</span><code>),</code><span class="math inline">\(\tau > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Dirichlet</td>
<td align="left"><code>ddirch(alpha =</code> <span class="math inline">\(\alpha\)</span><code>)</code></td>
<td align="left"><span class="math inline">\(\Gamma(\sum_j \alpha_j) \prod_j \frac{ x_j^{\alpha_j - 1}}{ \Gamma(\alpha_j)}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><span class="math inline">\(\alpha_j \geq 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Exponential</td>
<td align="left"><code>dexp(rate =</code> <span class="math inline">\(\lambda\)</span><code>)</code>, <span class="math inline">\(\lambda > 0\)</span></td>
<td align="left"><span class="math inline">\(\lambda \exp(-\lambda x)\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Flat</td>
<td align="left"><code>dflat()</code></td>
<td align="left"><span class="math inline">\(\propto 1\)</span> (improper)</td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Gamma</td>
<td align="left"><code>dgamma(shape = r, rate =</code> <span class="math inline">\(\lambda\)</span><code>)</code></td>
<td align="left"><span class="math inline">\(\frac{ \lambda^r x^{r - 1} \exp(-\lambda x)}{ \Gamma(r)}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><span class="math inline">\(\lambda > 0\)</span>, <span class="math inline">\(r > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Half flat</td>
<td align="left"><code>dhalfflat()</code></td>
<td align="left"><span class="math inline">\(\propto 1\)</span> (improper)</td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Inverse</td>
<td align="left"><code>dinvgamma(shape = r, scale =</code> <span class="math inline">\(\lambda\)</span><code>)</code></td>
<td align="left"><span class="math inline">\(\frac{ \lambda^r x^{-(r + 1)} \exp(-\lambda / x)}{ \Gamma(r)}\)</span></td>
<td><span class="math inline">\(0\)</span></td>
<td></td>
</tr>
<tr class="even">
<td align="left">gamma</td>
<td align="left"><span class="math inline">\(\lambda > 0\)</span>, <span class="math inline">\(r > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">LKJ Correl’n</td>
<td align="left"><code>dlkj_corr_cholesky(eta =</code> <span class="math inline">\(\eta\)</span><code>,</code></td>
<td align="left"><span class="math inline">\(\prod_{i=2}^{p} x_{ii}^{p - i + 2\eta -2}\)</span></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left">Cholesky</td>
<td align="left"><code>size = p),</code> <span class="math inline">\(\eta > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td align="left">Logistic</td>
<td align="left"><code>dlogis(location =</code> <span class="math inline">\(\mu\)</span>,</td>
<td align="left"><span class="math inline">\(\frac{ \tau \exp\{(x - \mu) \tau\}}{\left[1 + \exp\{(x - \mu) \tau\}\right]^2}\)</span></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><code>rate =</code> <span class="math inline">\(\tau\)</span><code>),</code><span class="math inline">\(\tau > 0\)</span></td>
<td align="left"></td>
<td></td>
<td></td>
</tr>