knitr::opts_chunk$set(
eval = identical(Sys.getenv("NOT_CRAN"), "true")
)This is a testing vignette to make sure the conditional evaluation works.
print("Starting expensive computation")## [1] "Starting expensive computation"## [1] "Done!"Above should print two lines.
“Likelihood” from rstanarm
The likelihood for one observation under a linear model can be written as a conditionally normal PDF \[\frac{1}{\sigma_{\epsilon} \sqrt{2 \pi}} e^{-\frac{1}{2} \left(\frac{y - \mu}{\sigma_{\epsilon}}\right)^2},\] where \(\mu = \alpha + \mathbf{x}^\top \boldsymbol{\beta}\) is a linear predictor and \(\sigma_{\epsilon}\) is the standard deviation of the error in predicting the outcome, \(y\). The likelihood of the entire sample is the product of \(N\) individual likelihood contributions.
It is well-known that the likelihood of the sample is maximized when the sum-of-squared residuals is minimized, which occurs when \[ \widehat{\boldsymbol{\beta}} = \left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{y}, \] \[ \widehat{\alpha} = \overline{y} - \overline{\mathbf{x}}^\top \widehat{\boldsymbol{\beta}}, \] \[ \widehat{\sigma}_{\epsilon}^2 = \frac{\left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)^\top \left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)}{N},\] where \(\overline{\mathbf{x}}\) is a vector that contains the sample means of the \(K\) predictors, \(\mathbf{X}\) is a \(N \times K\) matrix of centered predictors, \(\mathbf{y}\) is a \(N\)-vector of outcomes and \(\overline{y}\) is the sample mean of the outcome.
Bold symbols should render, i.e. this should render properly: \(\boldsymbol{\beta}\) vs \(\beta\) and not error out.