

Certification Method and Definitions


Model: 


The joint density for data , given the parameters
and is
with sample mean, ,
and sample variance, .
The joint prior density for and is
proportional to (a noninformative 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
and , given
and is
where the normalizing constant
and is the gamma function defined by
It is convenient to denote the marginal posterior distribution of
by
where denotes a random variable with
a Student's t distribution with degrees of freedom, and the symbol
denotes equal in distribution.
Similarly the marginal posterior distribution of
can be denoted by
where W is a chisquare random variable with
degrees of freedom.
 The posterior mean of
The certified value of the posterior mean of
is defined by
 The posterior standard deviation of
The certified value of the posterior standard deviation of is defined by
 The th posterior quantile of
The certified value of the th posterior
quantile of is defined by
where
is the th quantile of a t distribution
with degrees of freedom.
 The posterior mean of
The certified value of the posterior mean of
is defined by
 The posterior standard deviation of
The certified value of the posterior standard deviation of
is defined by
 The th posterior quantile of
The certified value of the th posterior
quantile of is defined by
where
is the th quantile of a chisquare distribution
with degrees of freedom.
