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?