This vignette demonstrates fitting a Poisson regression model via Hamiltonian Monte Carlo (HMC) using the hmclearn package.
For a count response, we let
\[ p(y | \mu) = \frac{e^{-\mu}\mu^y}{y!}, \]
with a log-link function
\[ \begin{aligned} \boldsymbol\mu &:= E(\mathbf{y} | \mathbf{X}) = e^{\mathbf{X}\boldsymbol\beta}, \\ \log \boldsymbol\mu &= \mathbf{X}\boldsymbol\beta. \end{aligned} \] The vector of responses is \(\mathbf{y} = (y_1, ..., y_n)^T\). The covariate values for subject \(i\) are \(\mathbf{x}_i^T = (x_{i0}, ..., x_{iq})\) for \(q\) covariates plus an intercept. We write the full design matrix as \(\mathbf{X} = (\mathbf{x}_1^T, ..., \mathbf{x}_n^T)^T \in \mathbb{R}^{n\times(q+1)}\) for \(n\) observations. The regression coefficients are a vector of length \(q + 1\), \(\boldsymbol\beta = (\beta_0, ..., \beta_q)^T\).
We develop the likelihood
\[ \begin{aligned} f(\mathbf{y} | \boldsymbol\mu) &= \prod_{i=1}^n \frac{e^{-\mu_i}\mu_i^{y_i}}{y_i!}, \\ f(\mathbf{y} | \mathbf{X}, \boldsymbol\beta) &= \prod_{i=1}^n \frac{e^{-e^{\mathbf{x}_i^T\boldsymbol\beta}}e^{y_i\mathbf{x}_i^T\boldsymbol\beta}}{y_i!}, \\ \end{aligned} \]
and log-likelihood
\[ \begin{aligned} f(\mathbf{y} | \mathbf{X}, \boldsymbol\beta) &= \sum_{i=1}^n -e^{\mathbf{x}_i^T\boldsymbol\beta} + y_i \mathbf{x}_i^T \boldsymbol\beta - \log y_i!, \\ &\propto \sum_{i=1}^n -e^{\mathbf{x}_i^T\boldsymbol\beta} + y_i \mathbf{x}_i^T \boldsymbol\beta. \end{aligned} \]
We set a multivariate normal prior for \(\boldsymbol\beta\)
\[ \begin{aligned} \boldsymbol\beta &\sim N(0, \sigma_\beta^2 \mathbf{I}), \end{aligned} \]
with pdf, omitting constants
\[ \begin{aligned} \pi(\boldsymbol\beta | \sigma_\beta^2) &= \frac{1}{\sqrt{\lvert 2\pi \sigma_\beta^2 \rvert }}e^{-\frac{1}{2}\boldsymbol\beta^T \boldsymbol\beta / \sigma_\beta^2}, \\ \log \pi(\boldsymbol\beta | \sigma_\beta^2) &= -\frac{1}{2}\log(2\pi \sigma_\beta^2) - \frac{1}{2}\boldsymbol\beta^T \boldsymbol\beta / \sigma_\beta^2, \\ &\propto -\frac{1}{2}\log \sigma_\beta^2 - \frac{\boldsymbol\beta^T\boldsymbol\beta}{2\sigma_\beta^2}. \end{aligned} \]
Next, we derive the log posterior, omitting constants,
\[ \begin{aligned} f(\boldsymbol\beta | \mathbf{X}, \mathbf{y}, \sigma_\beta^2) &\propto f(\mathbf{y} | \mathbf{X}, \boldsymbol\beta) \pi(\boldsymbol\beta | \sigma_\beta^2), \\ \log f(\boldsymbol\beta | \mathbf{X}, \mathbf{y}, \sigma_\beta^2) & \propto \log f(\mathbf{y} | \mathbf{X}, \boldsymbol\beta) + \log \pi(\beta | \sigma_\beta^2), \\ &\propto \sum_{i=1}^n \left( -e^{\mathbf{x}_i^T\boldsymbol\beta} + y_i \mathbf{x}_i^T \boldsymbol\beta\right) - \frac{1}{2}\beta^T\b