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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg"></a><a class="link" href="inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example">Inverse |
| Chi-Squared Distribution Bayes Example</a> |
| </h4></div></div></div> |
| <p> |
| The scaled-inversed-chi-squared distribution is the conjugate prior distribution |
| for the variance (σ<sup>2</sup>) parameter of a normal distribution with known expectation |
| (μ). As such it has widespread application in Bayesian statistics: |
| </p> |
| <p> |
| In <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian |
| inference</a>, the strength of belief into certain parameter values |
| is itself described through a distribution. Parameters hence become themselves |
| modelled and interpreted as random variables. |
| </p> |
| <p> |
| In this worked example, we perform such a Bayesian analysis by using the |
| scaled-inverse-chi-squared distribution as prior and posterior distribution |
| for the variance parameter of a normal distribution. |
| </p> |
| <p> |
| For more general information on Bayesian type of analyses, see: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Andrew Gelman, John B. Carlin, Hal E. Stern, Donald B. Rubin, Bayesian |
| Data Analysis, 2003, ISBN 978-1439840955. |
| </li> |
| <li class="listitem"> |
| Jim Albert, Bayesian Compution with R, Springer, 2009, ISBN 978-0387922973. |
| </li> |
| </ul></div> |
| <p> |
| (As the scaled-inversed-chi-squared is another parameterization of the |
| inverse-gamma distribution, this example could also have used the inverse-gamma |
| distribution). |
| </p> |
| <p> |
| Consider precision machines which produce balls for a high-quality ball |
| bearing. Ideally each ball should have a diameter of precisely 3000 μm (3 |
| mm). Assume that machines generally produce balls of that size on average |
| (mean), but individual balls can vary slightly in either direction following |
| (approximately) a normal distribution. Depending on various production |
| conditions (e.g. raw material used for balls, workplace temperature and |
| humidity, maintenance frequency and quality) some machines produce balls |
| tighter distributed around the target of 3000 μm, while others produce balls |
| with a wider distribution. Therefore the variance parameter of the normal |
| distribution of the ball sizes varies from machine to machine. An extensive |
| survey by the precision machinery manufacturer, however, has shown that |
| most machines operate with a variance between 15 and 50, and near 25 μm<sup>2</sup> on |
| average. |
| </p> |
| <p> |
| Using this information, we want to model the variance of the machines. |
| The variance is strictly positive, and therefore we look for a statistical |
| distribution with support in the positive domain of the real numbers. Given |
| the expectation of the normal distribution of the balls is known (3000 |
| μm), for reasons of conjugacy, it is customary practice in Bayesian statistics |
| to model the variance to be scaled-inverse-chi-squared distributed. |
| </p> |
| <p> |
| In a first step, we will try to use the survey information to model the |
| general knowledge about the variance parameter of machines measured by |
| the manufacturer. This will provide us with a generic prior distribution |
| that is applicable if nothing more specific is known about a particular |
| machine. |
| </p> |
| <p> |
| In a second step, we will then combine the prior-distribution information |
| in a Bayesian analysis with data on a specific single machine to derive |
| a posterior distribution for that machine. |
| </p> |
| <h6> |
| <a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.h0"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_one_using_the_survey_inform"></a></span><a class="link" href="inverse_chi_squared_eg.html#math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_one_using_the_survey_inform">Step |
| one: Using the survey information.</a> |
| </h6> |
| <p> |
| Using the survey results, we try to find the parameter set of a scaled-inverse-chi-squared |
| distribution so that the properties of this distribution match the results. |
| Using the mathematical properties of the scaled-inverse-chi-squared distribution |
| as guideline, we see that that both the mean and mode of the scaled-inverse-chi-squared |
| distribution are approximately given by the scale parameter (s) of the |
| distribution. As the survey machines operated at a variance of 25 μm<sup>2</sup> on |
| average, we hence set the scale parameter (s<sub>prior</sub>) of our prior distribution |
| equal to this value. Using some trial-and-error and calls to the global |
| quantile function, we also find that a value of 20 for the degrees-of-freedom |
| (ν<sub>prior</sub>) parameter is adequate so that most of the prior distribution mass |
| is located between 15 and 50 (see figure below). |
| </p> |
| <p> |
| We first construct our prior distribution using these values, and then |
| list out a few quantiles: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">priorDF</span> <span class="special">=</span> <span class="number">20.0</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">priorScale</span> <span class="special">=</span> <span class="number">25.0</span><span class="special">;</span> |
| |
| <span class="identifier">inverse_chi_squared</span> <span class="identifier">prior</span><span class="special">(</span><span class="identifier">priorDF</span><span class="special">,</span> <span class="identifier">priorScale</span><span class="special">);</span> |
| <span class="comment">// Using an inverse_gamma distribution instead, we could equivalently write</span> |
| <span class="comment">// inverse_gamma prior(priorDF / 2.0, priorScale * priorDF / 2.0);</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Prior distribution:"</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 2.5% quantile: "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 50% quantile: "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 97.5% quantile: "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| This produces this output: |
| </p> |
| <pre class="programlisting"><span class="identifier">Prior</span> <span class="identifier">distribution</span><span class="special">:</span> |
| |
| <span class="number">2.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">14.6</span> |
| <span class="number">50</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">25.9</span> |
| <span class="number">97.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">52.1</span> |
| </pre> |
| <p> |
| Based on this distribution, we can now calculate the probability of having |
| a machine working with an unusual work precision (variance) at <= 15 |
| or > 50. For this task, we use calls to the <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span></code> functions <code class="computeroutput"><span class="identifier">cdf</span></code> |
| and <code class="computeroutput"><span class="identifier">complement</span></code>, respectively, |
| and find a probability of about 0.031 (3.1%) for each case. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance <= 15: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">15.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance <= 25: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">25.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance > 50: "</span> |
| <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">prior</span><span class="special">,</span> <span class="number">50.0</span><span class="special">))</span> |
| <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| This produces this output: |
| </p> |
| <pre class="programlisting"><span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special"><=</span> <span class="number">15</span><span class="special">:</span> <span class="number">0.031</span> |
| <span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special"><=</span> <span class="number">25</span><span class="special">:</span> <span class="number">0.458</span> |
| <span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">></span> <span class="number">50</span><span class="special">:</span> <span class="number">0.0318</span> |
| </pre> |
| <p> |
| Therefore, only 3.1% of all precision machines produce balls with a variance |
| of 15 or less (particularly precise machines), but also only 3.2% of all |
| machines produce balls with a variance of as high as 50 or more (particularly |
| imprecise machines). Moreover, slightly more than one-half (1 - 0.458 = |
| 54.2%) of the machines work at a variance greater than 25. |
| </p> |
| <p> |
| Notice here the distinction between a <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian</a> |
| analysis and a <a href="http://en.wikipedia.org/wiki/Frequentist_inference" target="_top">frequentist</a> |
| analysis: because we model the variance as random variable itself, we can |
| calculate and straightforwardly interpret probabilities for given parameter |
| values directly, while such an approach is not possible (and interpretationally |
| a strict <span class="emphasis"><em>must-not</em></span>) in the frequentist world. |
| </p> |
| <h6> |
| <a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.h1"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_2_investigate_a_single_mach"></a></span><a class="link" href="inverse_chi_squared_eg.html#math_toolkit.stat_tut.weg.inverse_chi_squared_eg.step_2_investigate_a_single_mach">Step |
| 2: Investigate a single machine</a> |
| </h6> |
| <p> |
| In the second step, we investigate a single machine, which is suspected |
| to suffer from a major fault as the produced balls show fairly high size |
| variability. Based on the prior distribution of generic machinery performance |
| (derived above) and data on balls produced by the suspect machine, we calculate |
| the posterior distribution for that machine and use its properties for |
| guidance regarding continued machine operation or suspension. |
| </p> |
| <p> |
| It can be shown that if the prior distribution was chosen to be scaled-inverse-chi-square |
| distributed, then the posterior distribution is also scaled-inverse-chi-squared-distributed |
| (prior and posterior distributions are hence conjugate). For more details |
| regarding conjugacy and formula to derive the parameters set for the posterior |
| distribution see <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">Conjugate |
| prior</a>. |
| </p> |
| <p> |
| Given the prior distribution parameters and sample data (of size n), the |
| posterior distribution parameters are given by the two expressions: |
| </p> |
| <p> |
|    ν<sub>posterior</sub> = ν<sub>prior</sub> + n |
| </p> |
| <p> |
| which gives the posteriorDF below, and |
| </p> |
| <p> |
|    s<sub>posterior</sub> = (ν<sub>prior</sub>s<sub>prior</sub> + Σ<sup>n</sup><sub>i=1</sub>(x<sub>i</sub> - μ)<sup>2</sup>) / (ν<sub>prior</sub> + n) |
| </p> |
| <p> |
| which after some rearrangement gives the formula for the posteriorScale |
| below. |
| </p> |
| <p> |
| Machine-specific data consist of 100 balls which were accurately measured |
| and show the expected mean of 3000 μm and a sample variance of 55 (calculated |
| for a sample mean defined to be 3000 exactly). From these data, the prior |
| parameterization, and noting that the term Σ<sup>n</sup><sub>i=1</sub>(x<sub>i</sub> - μ)<sup>2</sup> equals the sample |
| variance multiplied by n - 1, it follows that the posterior distribution |
| of the variance parameter is scaled-inverse-chi-squared distribution with |
| degrees-of-freedom (ν<sub>posterior</sub>) = 120 and scale (s<sub>posterior</sub>) = 49.54. |
| </p> |
| <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">ballsSampleSize</span> <span class="special">=</span> <span class="number">100</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span><span class="string">"balls sample size: "</span> <span class="special"><<</span> <span class="identifier">ballsSampleSize</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">ballsSampleVariance</span> <span class="special">=</span> <span class="number">55.0</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span><span class="string">"balls sample variance: "</span> <span class="special"><<</span> <span class="identifier">ballsSampleVariance</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="keyword">double</span> <span class="identifier">posteriorDF</span> <span class="special">=</span> <span class="identifier">priorDF</span> <span class="special">+</span> <span class="identifier">ballsSampleSize</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"prior degrees-of-freedom: "</span> <span class="special"><<</span> <span class="identifier">priorDF</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"posterior degrees-of-freedom: "</span> <span class="special"><<</span> <span class="identifier">posteriorDF</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="keyword">double</span> <span class="identifier">posteriorScale</span> <span class="special">=</span> |
| <span class="special">(</span><span class="identifier">priorDF</span> <span class="special">*</span> <span class="identifier">priorScale</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">ballsSampleVariance</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">ballsSampleSize</span> <span class="special">-</span> <span class="number">1</span><span class="special">)))</span> <span class="special">/</span> <span class="identifier">posteriorDF</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"prior scale: "</span> <span class="special"><<</span> <span class="identifier">priorScale</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"posterior scale: "</span> <span class="special"><<</span> <span class="identifier">posteriorScale</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| An interesting feature here is that one needs only to know a summary statistics |
| of the sample to parameterize the posterior distribution: the 100 individual |
| ball measurements are irrelevant, just knowledge of the sample variance |
| and number of measurements is sufficient. |
| </p> |
| <p> |
| That produces this output: |
| </p> |
| <pre class="programlisting"><span class="identifier">balls</span> <span class="identifier">sample</span> <span class="identifier">size</span><span class="special">:</span> <span class="number">100</span> |
| <span class="identifier">balls</span> <span class="identifier">sample</span> <span class="identifier">variance</span><span class="special">:</span> <span class="number">55</span> |
| <span class="identifier">prior</span> <span class="identifier">degrees</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">freedom</span><span class="special">:</span> <span class="number">20</span> |
| <span class="identifier">posterior</span> <span class="identifier">degrees</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">freedom</span><span class="special">:</span> <span class="number">120</span> |
| <span class="identifier">prior</span> <span class="identifier">scale</span><span class="special">:</span> <span class="number">25</span> |
| <span class="identifier">posterior</span> <span class="identifier">scale</span><span class="special">:</span> <span class="number">49.5</span> |
| |
| </pre> |
| <p> |
| To compare the generic machinery performance with our suspect machine, |
| we calculate again the same quantiles and probabilities as above, and find |
| a distribution clearly shifted to greater values (see figure). |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/prior_posterior_plot.svg" align="middle"></span> |
| </p> |
| <pre class="programlisting"><span class="identifier">inverse_chi_squared</span> <span class="identifier">posterior</span><span class="special">(</span><span class="identifier">posteriorDF</span><span class="special">,</span> <span class="identifier">posteriorScale</span><span class="special">);</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Posterior distribution:"</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 2.5% quantile: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 50% quantile: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" 97.5% quantile: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance <= 15: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">15.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance <= 25: "</span> <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">25.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" probability variance > 50: "</span> |
| <span class="special"><<</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">posterior</span><span class="special">,</span> <span class="number">50.0</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| This produces this output: |
| </p> |
| <pre class="programlisting"><span class="identifier">Posterior</span> <span class="identifier">distribution</span><span class="special">:</span> |
| |
| <span class="number">2.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">39.1</span> |
| <span class="number">50</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">49.8</span> |
| <span class="number">97.5</span><span class="special">%</span> <span class="identifier">quantile</span><span class="special">:</span> <span class="number">64.9</span> |
| |
| <span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special"><=</span> <span class="number">15</span><span class="special">:</span> <span class="number">2.97e-031</span> |
| <span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special"><=</span> <span class="number">25</span><span class="special">:</span> <span class="number">8.85e-010</span> |
| <span class="identifier">probability</span> <span class="identifier">variance</span> <span class="special">></span> <span class="number">50</span><span class="special">:</span> <span class="number">0.489</span> |
| </pre> |
| <p> |
| Indeed, the probability that the machine works at a low variance (<= |
| 15) is almost zero, and even the probability of working at average or better |
| performance is negligibly small (less than one-millionth of a permille). |
| On the other hand, with an almost near-half probability (49%), the machine |
| operates in the extreme high variance range of > 50 characteristic for |
| poorly performing machines. |
| </p> |
| <p> |
| Based on this information the operation of the machine is taken out of |
| use and serviced. |
| </p> |
| <p> |
| In summary, the Bayesian analysis allowed us to make exact probabilistic |
| statements about a parameter of interest, and hence provided us results |
| with straightforward interpretation. |
| </p> |
| <p> |
| A full sample output is: |
| </p> |
| <pre class="programlisting"> Inverse_chi_squared_distribution Bayes example: |
| |
| Prior distribution: |
| |
| 2.5% quantile: 14.6 |
| 50% quantile: 25.9 |
| 97.5% quantile: 52.1 |
| |
| probability variance <= 15: 0.031 |
| probability variance <= 25: 0.458 |
| probability variance > 50: 0.0318 |
| |
| balls sample size: 100 |
| balls sample variance: 55 |
| prior degrees-of-freedom: 20 |
| posterior degrees-of-freedom: 120 |
| prior scale: 25 |
| posterior scale: 49.5 |
| Posterior distribution: |
| |
| 2.5% quantile: 39.1 |
| 50% quantile: 49.8 |
| 97.5% quantile: 64.9 |
| |
| probability variance <= 15: 2.97e-031 |
| probability variance <= 25: 8.85e-010 |
| probability variance > 50: 0.489 |
| |
| </pre> |
| <p> |
| (See also the reference documentation for the <a class="link" href="../../dist_ref/dists/inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse |
| chi squared Distribution</a>.) |
| </p> |
| <p> |
| See the full source C++ of this example at <a href="../../../../../example/inverse_chi_squared_bayes_eg.cpp" target="_top">../../example/inverse_chi_squared_bayes_eg.cpp</a> |
| </p> |
| </div> |
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| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
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