Jacobian and covariance matrix

Question

Anyone know the Bishop's book in 2.53 they use Jacobian to convert covariance matrix x to y.

$$J_{ij}=\dfrac{\partial x_i}{\partial y_i}=U_{ji} \qquad{(2.53)}$$ $$\int_{\bf x} f({\bf x})d{\bf x} = \int_{\bf y} f({\bf y})|{\bf J}|d{\bf y}$$ $${\vert}{\bf J}{\vert}^2 = {\vert}{\bf U}^T{\vert}^2 = {\vert}{\bf U}^T{\vert}\;{\vert}{\bf U}{\vert} = {\vert}{\bf U}^T{\bf U}{\vert} = {\vert}{\bf I}{\vert} = 1 \qquad{(2.54)}$$ $$\left|\Sigma\right|^{\frac{1}{2}}=\prod_{j=1}^{D}\lambda_j^{\frac{1}{2}} \qquad{(2.55)}$$

$$p({\bf y}) = p(x)|{\bf J}| = \prod_{j=1}^{D}\dfrac{1}{(2\pi\lambda_j)^{1/2}}\exp\left\{-\dfrac{y_j^2}{2\lambda_j}\right\} \qquad{(2.56)}$$

1) why they need to change it with Jacobian ?(what is the reason) 2) what is the p(y) represent in 2.56 3) why they use jacobian for in machine learning covariance matrix?

Answer 1

1) why they need to change it with Jacobian: The Jacobian is used when you transform functions of multiple variables and integrate. The process is similar to u-substitution in single variable calculus but for multiple variables you have to include the Jacobian. Wikipedia will explain why better then me : wikipedia

2) what is the p(y) represent: If we perform the following linear transformation: $$y=U^{T}(x -u)$$

Where U is the eigenvectors of the covariance matrix. The new variable y will be zero mean and uncorrelated i.e: $$p(y) = N(0,diag(λ_1,..,λ_N)) $$

As the data is Gaussian uncorrelated means independent:

$$p(y_1,..,y_N)=p(y_1)p(y_2)..p(y_{N-1})p(y_N) $$