#score-matching

Question 1: Single choice

Score matching fits a model $p(\mathbf{x}; \theta) = \tilde p(\mathbf{x}; \theta)/Z(\theta)$ by minimising $J(\theta) = \frac{1}{n}\sum_i \sum_j \left[\partial_j \psi_j(\mathbf{x}_i; \theta) + \tfrac{1}{2}\psi_j(\mathbf{x}_i; \theta)^2\right]$ , where $\psi_j = \partial \log p / \partial x_j$ . Why does it not need the partition function $Z(\theta)$ ?

$\log Z(\theta)$ is constant in $\mathbf{x}$ , so its derivatives in $x_j$ vanish1
$Z(\theta)$ cancels because the objective only involves ratios of model densities2
$Z(\theta)$ is estimated on the fly with Monte Carlo samples3
The exponential family always has $Z(\theta) = 1$ 4

Timed test