• Tolerance interval is very similar to posterior predictive interval.
  • Confidence/reliability vs. …
Variable Description
\(y\) Observed data
\(n\) Number of observations
\(y_i\) \(i\)th observation
\(\tilde{y}\) New data
\(\bar{y}\) Sample average of observed data

Under assumptions of normally distributed data with a noninformative prior, predicted new data given observed data has a t distribution. (see p. X of Gelman et al. Bayesian Data Analysis, 3rd Edition)

\[p(\tilde{y} \vert y) \sim t_\nu (\mu, \sigma^2)\]

where

\[\mu = \bar{y}\] \[\sigma^2 = \left(1 + \frac{1}{n}\right) s\] \[\nu = n - 1\] \[s^2 = \frac{1}{n-1} \sum_{i=1}^n \left( y_i - \bar{y} \right)^2\]

For the bounds of a two-sided credible interval, we want the values at which the cumulative distribution function (CDF) equals 0.025 and 0.975. The CDF is of the t distribution is

\[\text{CDF}(x) = \frac{1}{2} + \left(\frac{x - \mu}{\sigma}\right) \Gamma \left( \frac{\nu + 1}{2} \right) \times \frac{_2F_1 \left( \frac{1}{2} , \frac{\nu + 1}{2} ; \frac{3}{2} ; - \frac{\left( \frac{x-\mu}{\sigma} \right)^2}{\nu} \right)}{\sqrt{\pi \nu} \Gamma \left( \frac{\nu}{2} \right)}\]

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The joint posterior distribution for normally distributed data with a noninformative prior distribution is

\[p(\mu, \sigma^2 \vert y) \propto \sigma^{-n -2} \exp \left( - \frac{1}{2 \sigma^2} \left[ \left( n - 1 \right) s^2 + n \left( \bar{y} - \mu \right)^2 \right] \right)\]

where

\[s^2 = \frac{1}{n-1} \sum_{i=1}^n \left( y_i - \bar{y} \right)^2\]