logJacobians

Log Jacobians refer to the natural logarithm of the absolute value of the determinant of the Jacobian matrix that arises in a change of variables for multivariate functions. When a vector-valued transformation y = g(x) is differentiable and locally invertible, its Jacobian matrix J_g(x) = ∂y/∂x captures how volumes change under the transformation. The determinant det J_g(x) measures the local volume scaling. In probability and statistics, this leads to the change-of-variables formula for densities, where the density of the transformed variable y is p_Y(y) = p_X(x) / |det J_g(x)| with x = g^{-1}(y). Equivalently, log p_Y(y) = log p_X(x) − log |det J_g(x)|. Here the term log |det J_g(x)| is the log Jacobian.

In practice, the log Jacobian is often computed directly for numerical stability, using log det |det J_g(x)| or logdet when available. For special transformations, such as triangular or diagonal Jacobians, the log determinant simplifies to the sum of logarithms of diagonal entries. The log Jacobian plays a central role in composing transformations: for y = g(f(x)), log det J_{g∘f}(x) = log det J_g(f(x)) + log det J_f(x).

Applications include probabilistic modeling, Bayesian inference, variational methods, and normalizing flows, where transforming densities through sequences of simple mappings requires tracking the accumulated log Jacobian. A noninvertible transformation (det J_g = 0) creates singularities, making the log Jacobian undefined.