Bayesian Linear Regression Model

The unnormalized posterior distribution was given for a Bayesian Linear Regression Model in the tutorial. Additional forms of that posterior are given in the following section.

Log-Transformed Distribution and Gradient

Let \mathcal{L} denote the logarithm of a density of interest up to a normalizing constant, and \nabla \mathcal{L} its gradient. Then, the following are obtained for the regression example parameters \bm{\beta} and \theta = \log(\sigma^2), for samplers, like NUTS, that can utilize both.

\mathcal{L}(\bm{\beta}, \theta | \bm{y}) &= \log(p(\bm{y} | \bm{\beta}, \theta) p(\bm{\beta}) p(\theta)) \\
  &= (-n/2 -\alpha_\pi) \theta - \frac{1}{\exp\{\theta\}} \left(\frac{1}{2} (\bm{y} - \bm{X} \bm{\beta})^\top (\bm{y} - \bm{X} \bm{\beta}) + \beta_\pi \right) \\
  &\quad - \frac{1}{2} (\bm{\beta} - \bm{\mu}_\pi)^\top \bm{\Sigma}_\pi^{-1} (\bm{\beta} - \bm{\mu}_\pi) \\
\nabla \mathcal{L}(\bm{\beta}, \theta | \bm{y}) &= \begin{bmatrix}
  \frac{1}{\exp\{\theta\}} \bm{X}^\top (\bm{y} - \bm{X} \bm{\beta}) - \bm{\Sigma}_\pi^{-1} (\bm{\beta} - \bm{\mu}_\pi) \\
  -n/2 -\alpha_\pi + \frac{1}{\exp\{\theta\}} \left(\frac{1}{2} (\bm{y} - \bm{X} \bm{\beta})^\top (\bm{y} - \bm{X} \bm{\beta}) + \beta_\pi \right)