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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
  28         Chi Squared Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">inverse_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">inverse_chi_squared_distribution</span>
  36 <span class="special">{</span>
  37 <span class="keyword">public</span><span class="special">:</span>
  38    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  39    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  40
  41    <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Not explicitly scaled, default 1/df.</span>
  42    <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">df</span><span class="special">);</span>  <span class="comment">// Scaled.</span>
  43
  44    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Default 1.</span>
  45    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale [xi] (variance), default 1/degrees_of_freedom.</span>
  46 <span class="special">};</span>
  47
  48 <span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
  49 </pre>
  50 <p>
  51           The inverse chi squared distribution is a continuous probability distribution
  52           of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed
  53           according to the chi squared distribution.
  54         </p>
  55 <p>
  56           The sources below give confusingly different formulae using different symbols
  57           for the distribution pdf, but they are all the same, or related by a change
  58           of variable, or choice of scale.
  59         </p>
  60 <p>
  61           Two constructors are available to implement both the scaled and (implicitly)
  62           unscaled versions.
  63         </p>
  64 <p>
  65           The main version has an explicit scale parameter which implements the
  66           <a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
  67           inverse chi_squared distribution</a>.
  68         </p>
  69 <p>
  70           A second version has an implicit scale = 1/degrees of freedom and gives
  71           the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
  72           inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
  73           distribution definition can be implemented by explicitly specifying a scale
  74           = 1.
  75         </p>
  76 <p>
  77           Both definitions are also available in <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
  78           Mathematica</a> and in <a href="http://www.r-project.org/" target="_top">The R
  79           Project for Statistical Computing</a> (geoR) with default scale = 1/degrees
  80           of freedom.
  81         </p>
  82 <p>
  83           See
  84         </p>
  85 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  86 <li class="listitem">
  87               Inverse chi_squared distribution <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Inverse-chi-square_distribution</a>
  88             </li>
  89 <li class="listitem">
  90               Scaled inverse chi_squared distribution<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution</a>
  91             </li>
  92 <li class="listitem">
  93               R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
  94               </a>
  95             </li>
  96 <li class="listitem">
  97               Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
  98               Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
  99               Wolfram Web Resource.</a>
 100             </li>
 101 <li class="listitem">
 102               Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
 103               Eric W. "Inverse Chi-Squared Distribution reference." From
 104               Wolfram Mathematica.</a>
 105             </li>
 106 </ul></div>
 107 <p>
 108           The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
 109           statistics</a>: the scaled inverse chi-square is conjugate prior for
 110           the normal distribution with known mean, model parameter &#963;&#178; (variance).
 111         </p>
 112 <p>
 113           See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
 114           priors including a table of distributions and their priors.</a>
 115         </p>
 116 <p>
 117           See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
 118           Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
 119           Squared Distribution</a>.
 120         </p>
 121 <p>
 122           The inverse_chi_squared distribution is a special case of a inverse_gamma
 123           distribution with &#957; (degrees_of_freedom) shape (&#945;) and scale (&#946;) where
 124         </p>
 125 <div class="blockquote"><blockquote class="blockquote"><p>
 126             <span class="serif_italic">&#945;= &#957; /2 and &#946; = &#189;</span>
 127           </p></blockquote></div>
 128 <div class="note"><table border="0" summary="Note">
 129 <tr>
 130 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 131 <th align="left">Note</th>
 132 </tr>
 133 <tr><td align="left" valign="top">
 134 <p>
 135             This distribution <span class="bold"><strong>does</strong></span> provide the typedef:
 136           </p>
 137 <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">inverse_chi_squared</span><span class="special">;</span></pre>
 138 <p>
 139             If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
 140             inverse_chi_squared distribution you can use
 141           </p>
 142 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;&gt;</span></pre>
 143 <p>
 144             or you can write <code class="computeroutput"><span class="identifier">inverse_chi_squared</span>
 145             <span class="identifier">my_invchisqr</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code>
 146           </p>
 147 </td></tr>
 148 </table></div>
 149 <p>
 150           For degrees of freedom parameter &#957;, the (<span class="bold"><strong>unscaled</strong></span>)
 151           inverse chi_squared distribution is defined by the probability density
 152           function (PDF):
 153         </p>
 154 <div class="blockquote"><blockquote class="blockquote"><p>
 155             <span class="serif_italic">f(x;&#957;) = 2<sup>-&#957;/2</sup> x<sup>-&#957;/2-1</sup> e<sup>-1/2x</sup> / &#915;(&#957;/2)</span>
 156           </p></blockquote></div>
 157 <p>
 158           and Cumulative Density Function (CDF)
 159         </p>
 160 <div class="blockquote"><blockquote class="blockquote"><p>
 161             <span class="serif_italic">F(x;&#957;) = &#915;(&#957;/2, 1/2x) / &#915;(&#957;/2)</span>
 162           </p></blockquote></div>
 163 <p>
 164           For degrees of freedom parameter &#957; and scale parameter &#958;, the <span class="bold"><strong>scaled</strong></span>
 165           inverse chi_squared distribution is defined by the probability density
 166           function (PDF):
 167         </p>
 168 <div class="blockquote"><blockquote class="blockquote"><p>
 169             <span class="serif_italic">f(x;&#957;, &#958;) = (&#958;&#957;/2)<sup>&#957;/2</sup> e<sup>-&#957;&#958;/2x</sup> x<sup>-1-&#957;/2</sup> / &#915;(&#957;/2)</span>
 170           </p></blockquote></div>
 171 <p>
 172           and Cumulative Density Function (CDF)
 173         </p>
 174 <div class="blockquote"><blockquote class="blockquote"><p>
 175             <span class="serif_italic">F(x;&#957;, &#958;) = &#915;(&#957;/2, &#957;&#958;/2x) / &#915;(&#957;/2)</span>
 176           </p></blockquote></div>
 177 <p>
 178           The following graphs illustrate how the PDF and CDF of the inverse chi_squared
 179           distribution varies for a few values of parameters &#957; and &#958;:
 180         </p>
 181 <div class="blockquote"><blockquote class="blockquote"><p>
 182             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.svg" align="middle"></span>
 183
 184           </p></blockquote></div>
 185 <div class="blockquote"><blockquote class="blockquote"><p>
 186             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.svg" align="middle"></span>
 187
 188           </p></blockquote></div>
 189 <h5>
 190 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h0"></a>
 191           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
 192           Functions</a>
 193         </h5>
 194 <pre class="programlisting"><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Implicitly scaled 1/df.</span>
 195 <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">);</span> <span class="comment">// Explicitly scaled.</span>
 196 </pre>
 197 <p>
 198           Constructs an inverse chi_squared distribution with &#957; degrees of freedom
 199           <span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
 200           value 1/df.
 201         </p>
 202 <p>
 203           Requires that the degrees of freedom &#957; parameter is greater than zero, otherwise
 204           calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 205         </p>
 206 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 207 </pre>
 208 <p>
 209           Returns the degrees_of_freedom &#957; parameter of this distribution.
 210         </p>
 211 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 212 </pre>
 213 <p>
 214           Returns the scale &#958; parameter of this distribution.
 215         </p>
 216 <h5>
 217 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h1"></a>
 218           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors">Non-member
 219           Accessors</a>
 220         </h5>
 221 <p>
 222           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 223           functions</a> that are generic to all distributions are supported:
 224           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 225           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 226           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 227           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 228           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 229           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 230           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 231           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 232         </p>
 233 <p>
 234           The domain of the random variate is [0,+&#8734;].
 235         </p>
 236 <div class="note"><table border="0" summary="Note">
 237 <tr>
 238 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 239 <th align="left">Note</th>
 240 </tr>
 241 <tr><td align="left" valign="top"><p>
 242             Unlike some definitions, this implementation supports a random variate
 243             equal to zero as a special case, returning zero for both pdf and cdf.
 244           </p></td></tr>
 245 </table></div>
 246 <h5>
 247 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h2"></a>
 248           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
 249         </h5>
 250 <p>
 251           The inverse gamma distribution is implemented in terms of the incomplete
 252           gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
 253           Gamma Distribution</a> that use <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
 254           and <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their
 255           inverses <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
 256           and <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
 257           refer to the accuracy data for those functions for more information. But
 258           in general, gamma (and thus inverse gamma) results are often accurate to
 259           a few epsilon, &gt;14 decimal digits accuracy for 64-bit double. unless
 260           iteration is involved, as for the estimation of degrees of freedom.
 261         </p>
 262 <h5>
 263 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h3"></a>
 264           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
 265         </h5>
 266 <p>
 267           In the following table &#957; is the degrees of freedom parameter and &#958; is the scale
 268           parameter of the distribution, <span class="emphasis"><em>x</em></span> is the random variate,
 269           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
 270           its complement. Parameters &#945; for shape and &#946; for scale are used for the inverse
 271           gamma function: &#945; = &#957;/2 and &#946; = &#957; * &#958;/2.
 272         </p>
 273 <div class="informaltable"><table class="table">
 274 <colgroup>
 275 <col>
 276 <col>
 277 </colgroup>
 278 <thead><tr>
 279 <th>
 280                   <p>
 281                     Function
 282                   </p>
 283                 </th>
 284 <th>
 285                   <p>
 286                     Implementation Notes
 287                   </p>
 288                 </th>
 289 </tr></thead>
 290 <tbody>
 291 <tr>
 292 <td>
 293                   <p>
 294                     pdf
 295                   </p>
 296                 </td>
 297 <td>
 298                   <p>
 299                     Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(&#945;,
 300                     &#946;/ x, &#946;) / x * x
 301                   </p>
 302                 </td>
 303 </tr>
 304 <tr>
 305 <td>
 306                   <p>
 307                     cdf
 308                   </p>
 309                 </td>
 310 <td>
 311                   <p>
 312                     Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(&#945;,
 313                     &#946; / x)
 314                   </p>
 315                 </td>
 316 </tr>
 317 <tr>
 318 <td>
 319                   <p>
 320                     cdf complement
 321                   </p>
 322                 </td>
 323 <td>
 324                   <p>
 325                     Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(&#945;,
 326                     &#946; / x)
 327                   </p>
 328                 </td>
 329 </tr>
 330 <tr>
 331 <td>
 332                   <p>
 333                     quantile
 334                   </p>
 335                 </td>
 336 <td>
 337                   <p>
 338                     Using the relation: x = &#946;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(&#945;,
 339                     p)
 340                   </p>
 341                 </td>
 342 </tr>
 343 <tr>
 344 <td>
 345                   <p>
 346                     quantile from the complement
 347                   </p>
 348                 </td>
 349 <td>
 350                   <p>
 351                     Using the relation: x = &#945;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(&#945;,
 352                     q)
 353                   </p>
 354                 </td>
 355 </tr>
 356 <tr>
 357 <td>
 358                   <p>
 359                     mode
 360                   </p>
 361                 </td>
 362 <td>
 363                   <p>
 364                     &#957; * &#958; / (&#957; + 2)
 365                   </p>
 366                 </td>
 367 </tr>
 368 <tr>
 369 <td>
 370                   <p>
 371                     median
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     no closed form analytic equation is known, but is evaluated as
 377                     quantile(0.5)
 378                   </p>
 379                 </td>
 380 </tr>
 381 <tr>
 382 <td>
 383                   <p>
 384                     mean
 385                   </p>
 386                 </td>
 387 <td>
 388                   <p>
 389                     &#957;&#958; / (&#957; - 2) for &#957; &gt; 2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 390                   </p>
 391                 </td>
 392 </tr>
 393 <tr>
 394 <td>
 395                   <p>
 396                     variance
 397                   </p>
 398                 </td>
 399 <td>
 400                   <p>
 401                     2 &#957;&#178; &#958;&#178; / ((&#957; -2)&#178; (&#957; -4)) for &#957; &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 402                   </p>
 403                 </td>
 404 </tr>
 405 <tr>
 406 <td>
 407                   <p>
 408                     skewness
 409                   </p>
 410                 </td>
 411 <td>
 412                   <p>
 413                     4 &#8730;2 &#8730;(&#957;-4) /(&#957;-6) for &#957; &gt;6, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 414                   </p>
 415                 </td>
 416 </tr>
 417 <tr>
 418 <td>
 419                   <p>
 420                     kurtosis_excess
 421                   </p>
 422                 </td>
 423 <td>
 424                   <p>
 425                     12 * (5&#957; - 22) / ((&#957; - 6) * (&#957; - 8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 426                   </p>
 427                 </td>
 428 </tr>
 429 <tr>
 430 <td>
 431                   <p>
 432                     kurtosis
 433                   </p>
 434                 </td>
 435 <td>
 436                   <p>
 437                     3 + 12 * (5&#957; - 22) / ((&#957; - 6) * (&#957;-8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 438                   </p>
 439                 </td>
 440 </tr>
 441 </tbody>
 442 </table></div>
 443 <h5>
 444 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h4"></a>
 445           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
 446         </h5>
 447 <div class="orderedlist"><ol class="orderedlist" type="1">
 448 <li class="listitem">
 449               Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
 450               Donald B. Rubin, ISBN-13: 978-1584883883, Chapman &amp; Hall; 2 edition
 451               (29 July 2003).
 452             </li>
 453 <li class="listitem">
 454               Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer;
 455               2nd ed. edition (10 Jun 2009)
 456             </li>
 457 </ol></div>
 458 </div>
 459 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 460 <td align="left"></td>
 461 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 462       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 463       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 464       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 465       Daryle Walker and Xiaogang Zhang<p>
 466         Distributed under the Boost Software License, Version 1.0. (See accompanying
 467         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 468       </p>
 469 </div></td>
 470 </tr></table>
 471 <hr>
 472 <div class="spirit-nav">
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