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mcmc01
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mcmc01
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Certification Method and Definitions

Model:

The joint density for data  
\vec y = \{y_1, \ldots, y_n\} 
, given the parameters  
\mu
and  
\sigma
is

 
(\sqrt{2\pi}\sigma)^{-n} \exp\{-\frac{(n-1)s^2 + n(\overline{y} - \mu)^2}{2\sigma^2}\},

with sample mean,  
\overline y = \frac{\sum_{j=1}^n y_j}{n}
, and sample variance,  
s^2 = \frac{\sum_{j=1}^n (y_j - \overline y)^2}{n-1}
.

The joint prior density for  
\mu
and  
\sigma
is proportional to  
\frac{1}{\sigma}d\mu d\sigma
(a non-informative prior specification).

Methodology:

For all datasets, extended precision calculations (accurate to 500 digits) were made. Data were read in exactly as extended precision numbers and all calculations were made with this very high precision. The results were output in extended precision, and only then rounded to fifteen decimal places. These extended precision results are an idealization. They represent what would be achieved if calculations were made without roundoff or other numerical errors. Any typical numerical algorithm (i.e., not implemented in extended precision) will introduce computational inaccuracies, and will produce results which differ slightly from these certified values.

Definitions:

Using Bayes' Theorem, the joint posterior probability density of  
\mu
and  
\sigma
, given  
\bar{y}
and  
s^2
is

 
\pi (\mu, \sigma |\overline y, s^2) =
 C\sigma^{-(n+1)} \exp [-\frac{n(\mu - \overline y)^2}{2\sigma^2}
  - \frac{(n-1)s^2}{2\sigma^2}],

where the normalizing constant

 
C = \sqrt\frac{2n}{\pi} [\Gamma((n-1)/2)]^{-1}\left[\frac{(n-1) s^2}{2}\right]^{(n-1)/2},

and  
\Gamma(\cdot)
is the gamma function defined by

 
\Gamma(\alpha)= \int_0^{\infty} x^{\alpha -1}e^{-x}dx, \quad\mbox{for $\alpha > 0.$}



It is convenient to denote the marginal posterior distribution of  
\mu
by

 
\mu \stackrel{\it D}= \overline y + \frac{s}{\sqrt{n}}t_{n-1}, \ \mbox{for } \
 n \geq 4,

where  
t_{n-1}
denotes a random variable with a Student's t distribution with  
n-1
degrees of freedom, and the symbol  
\stackrel{\it D}=
denotes equal in distribution.

Similarly the marginal posterior distribution of  
\sigma
can be denoted by

 
\sigma \stackrel{\it D}= s\sqrt{\frac{n-1}{W}}, \ \mbox{for } \
 n \geq 4,

where W is a chi-square random variable with  
n-1
degrees of freedom.


The posterior mean of  
\mu

The certified value of the posterior mean of  
\mu
is defined by

 
\mbox{E}(\mu |\overline y, s) =\bar{y}.


The posterior standard deviation of  
\mu

The certified value of the posterior standard deviation of  
\mu
is defined by

 
\sqrt{\mbox{Var}(\mu |\overline y, s)} =s\sqrt{ \frac{n-1}{n(n-3)}}.


The  
\alpha
th posterior quantile of  
\mu

The certified value of the  
\alpha
th posterior quantile of  
\mu
is defined by

 
q_\alpha(\mu)= \overline y+\frac{s}{\sqrt{n}}t_{\alpha,n-1},
where  
t_{\alpha,n-1}
is the  
\alpha
th quantile of a t distribution with  
n-1
degrees of freedom.


The posterior mean of  
\sigma

The certified value of the posterior mean of  
\sigma
is defined by

 
\mbox{E}(\sigma |\overline y, s) =
s\sqrt{\frac{n-1}{2}} \frac{\Gamma(\frac{n}{2} - 1)}{\Gamma(\frac{n}{2}
                                                    - \frac{1}{2})}.


The posterior standard deviation of  
\sigma

The certified value of the posterior standard deviation of  
\sigma
is defined by

 
\sqrt{\mbox{Var}(\sigma |\overline y, s)} = s\sqrt{ (n-1)}\sqrt{  \frac{1}{n-3}
                      - \frac{1}{2}\left(\frac{\Gamma (\frac{n}{2}-1)}
                             {\Gamma (\frac{n}{2}- \frac{1}{2})}\right)^2}.


The  
\alpha
th posterior quantile of  
\sigma

The certified value of the  
\alpha
th posterior quantile of  
\sigma
is defined by

 
q_\alpha(\sigma)=s\sqrt{\frac{n-1}{\chi^2_{1-\alpha,n-1}}},
where  
\chi^2_{\alpha,n-1}
is the  
\alpha
th quantile of a chi-square distribution with  
n-1
degrees of freedom.