Jacobian
Topic of Functions of random variables and limit theorems, part of Probability
Jacobian
| Type | Topic |
| Taught in | Functions of random variables and limit theorems |
| Part | Probability |
| Questions using it | 8 |
The determinant of the matrix of partial derivatives of a change of variables. In the transformation rule it rescales a density: if \mathbf{Y}=g(\mathbf{X}) with g one-to-one and differentiable, f_{\mathbf{Y}}(\mathbf{y})=f_{\mathbf{X}}(g^{-1}(\mathbf{y}))\,|\det J_{g^{-1}}(\mathbf{y})|.
In Functions of random variables and limit theorems
Theorem 1 (Transformation rule) Let \mathbf{X} take values in an open set of \mathbb{R}^n with density f_{\mathbf{X}}, and let \mathbf{Z}=g(\mathbf{X}) where g is a smooth bijection onto its image, with inverse g^{-1} and Jacobian determinant J that does not vanish. The density of \mathbf{Z} is
f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}\bigl(g^{-1}(\mathbf{z})\bigr)\,\bigl|\det Dg^{-1}(\mathbf{z})\bigr|.
Equivalently, if the Jacobian is computed for g rather than for the inverse, divide by |\det Dg| evaluated at the preimage. For a linear map \mathbf{Z}=A\mathbf{X} with A invertible, f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}(A^{-1}\mathbf{z})/|\det A|.
The absolute value of the Jacobian is the local change in volume. Probability in a small region is density times volume, so the density must be divided by that factor.
Box–Muller is this rule in the plane. Let U,V be independent and uniform on (0,1), and set
Z_1=\sqrt{-2\log U}\,\cos(2\pi V), \qquad Z_2=\sqrt{-2\log U}\,\sin(2\pi V).
Passing to polar-type coordinates and computing the Jacobian shows that (Z_1,Z_2) has joint density (2\pi)^{-1}\exp(-(z_1^2+z_2^2)/2) on \mathbb{R}^2, the product of two standard normal densities. The construction is the one used in Simulation when a normal generator has to be built from uniforms.
A second standard example splits a pair of independent exponential variables. Let X and Y be independent and exponential with rate 1, and set U=X+Y and V=X/(X+Y). The map (u,v)\mapsto(uv,\,u(1-v)) has Jacobian determinant of absolute value u. The joint density of (U,V) is u e^{-u} on u>0, 0<v<1, which factors into a \operatorname{Gamma}(2,1) density for U and a uniform density for V. So U and V are independent.
Used in
- Taught in Functions of random variables and limit theorems
- Also used in Simulation · Hierarchical Bayes · Hierarchical clustering and kernel density estimation · Networks in practice
- Questions that use it (8) — Sum and proportion of two exponentials · A logit-normal prior for a coin (answer) · A kernel is a density (answer) · Q42 · Q58 (answer) · Question 5 (answer) · Question 7 (answer) · Question 15 (answer)
- All 21 questions on Functions of random variables and limit theorems
See also
Transformation rule · Grid Monte Carlo approximation · Uniformity depends on a domain · Multivariate normal distribution · Affine images of normal vectors · Normal linear model · Law of large numbers · Central limit theorem