Which density formula is correct for a normalizing flow?

Question: Let z∼p_Z and x=f(z), where f is an invertible differentiable transformation. Which expression correctly gives the data-space log density? A) log p_X(x)=log p_Z(f⁻¹(x))+log|det J_f(f⁻¹(x))| B) log p_X(x)=log p_Z(z)−log|det J_f(z)| C) log p_X(x)=log p_Z(z)+log|det J_f(z)|⁻¹ without taking the logarithm of the determinant D) log p_X(x)=log p_Z(f(x))−log|det J_f(x)| Correct: B Explanation: The forward transformation expands or contracts volume. The data density is therefore the base density minus the forward log-volume change. Equivalently, it is the base log density plus the log absolute determinant of the inverse transformation's Jacobian. Topic: advanced ML / generative modelling / normalizing flows