Certification Method and Definitions  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 non-informative prior specification). 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. 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 chi-square 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 chi-square distribution with degrees of freedom.