logdetJ

LogdetJ refers to the logarithm of the absolute value of the determinant of the Jacobian matrix of a differentiable transformation. For a function f: R^n → R^n, the Jacobian J_f(x) = ∂f/∂x is the n×n matrix of first-order partial derivatives. The quantity logdetJ(x) is defined as log |det J_f(x)| and encodes how volumes are locally scaled by f near x.

In probability and statistics, logdetJ appears in the change-of-variables formula. If Y = f(X) and f is differentiable with invertible Jacobian, then the density transforms as p_Y(y) = p_X(x) / |det J_f(x)| with x = f^{-1}(y). Equivalently, log p_Y(y) = log p_X(x) − log |det J_f(x)|. In practice, logdetJ is often denoted as log |det J_f(x)| and is used to adjust log-likelihoods when applying a nonlinear transformation to random variables.

When a sequence of transformations is applied, the total logdetJ is the sum of the log determinants from each step, reflecting the cumulative volume change. This additive property is particularly important in normalizing flows and other probabilistic models that transform simple base distributions into complex ones while tracking log-determinants to preserve tractable likelihoods.

Computational aspects vary. Computing det J_f(x) directly can be expensive for high dimensions. If J_f(x) is triangular (for example in autoregressive models), logdetJ(x) equals the sum of the logs of the diagonal entries. More generally, numerical linear algebra techniques such as LU decomposition enable stable computation of log|det J_f(x)|. The Jacobian must be invertible (non-singular) on the region of interest for logdetJ to be well defined.