#poisson-regression

Question 1: Single choice

Bayesian Poisson regression: $\alpha, \beta \sim \mathcal{N}(0, 100)$ independently and $y_n \sim \mathrm{Poisson}(\exp(\alpha x_n + \beta))$ for $n = 1, \ldots, N$ . Which function can serve as the target $p^*(\alpha, \beta)$ for Metropolis–Hastings sampling of the posterior?

$\prod_{n=1}^N \mathrm{Poisson}(y_n; e^{\alpha x_n + \beta})$ 1
$\prod_{n=1}^N \mathrm{Poisson}(y_n; e^{\alpha x_n + \beta}) / p(\mathcal{D})$ , with the evidence $p(\mathcal{D})$ computed first2
$\prod_{n=1}^N \mathcal{N}(\alpha; 0, 100)\,\mathcal{N}(\beta; 0, 100)\, \mathrm{Poisson}(y_n; e^{\alpha x_n + \beta})$ 3
$\mathcal{N}(\alpha; 0, 100)\,\mathcal{N}(\beta; 0, 100) \prod_{n=1}^N \mathrm{Poisson}(y_n; e^{\alpha x_n + \beta})$ 4

Timed test