proof of Box–Muller transform (polar form)

Question

There're proofs of Box–Muller transform available online but my book (pattern recognition and machine learning) seems to have put it in a different form.

I didn't follow the derivation of equation 11.12, can anyone please help? Thanks!

EDIT
As mentioned in Nadiels's answer, there's a mistake in formula 11.10 and 11.11, as logarithm has to take in a positive number (PRML errata).

Answer 1

So first thing there is an error in equations $(11.10)$ and $(11.11)$ and in fact you should have the transformations $$ y_i = z_i \left( \frac{-2 \ln r^2 }{r^2 } \right)^{1/2} $$ and in particular we have \begin{align*} \exp\left( -\frac{1}{2} \left(y_1^2 + y_2^2 \right) \right) &=\exp\left( \left( z_1^2 +z_2^2\right)\frac{\ln(r^2)}{r^2} \right) = r^2, \end{align*} which using the inverse function theorem tells us that if $$ \mathbf{J} =\begin{bmatrix} \frac{\partial y_1}{\partial z_1} & \frac{\partial y_1}{\partial z_2} \\ \frac{\partial y_2}{\partial z_1} & \frac{\partial y_2}{\partial z_2}\end{bmatrix}, $$ then to get the desired result we want to show that $\left| \operatorname{det}(\mathbf{J}) \right| = 2/r^2$. Or $$ \left| \left(\frac{\partial y_1}{\partial z_1}\right)\left(\frac{\partial y_2}{\partial z_2}\right) - \left(\frac{\partial y_1}{\partial z_2}\right)^2 \right|= \frac{2}{r^2}. $$ Let $$ y_i = z_i h(r^2), \qquad \mbox{where } h(r^2) = \left(-\frac{2\ln r^2}{r^2} \right)^{1/2} $$ then \begin{align*} \left(\frac{\partial y_1}{\partial z_1}\right)\left(\frac{\partial y_2}{\partial z_2}\right) - \left(\frac{\partial y_1}{\partial z_2}\right)^2 &= h(r^2)^2+2r^2h'(r^2)h(r^2) \\ &=h(r^2)^2 + \frac{2}{r^2}\left( \ln(r^2) - 1 \right)\\ &=\frac{-2\ln(r^2)}{r^2} + \frac{2 \ln r^2}{r^2} - \frac{2}{r^2} \\ &= -\frac{2}{r^2}. \end{align*} as desired.

Answer 2

As a general rule, a pdf in one set of variables is given by the pdf in another set of variables multiplied by the determinant of the Jacobian matrix.

In you case, you must calculate: $$\left|\left(\begin{array}(\partial z_1 / \partial y_1 & \partial z_2 / \partial y_1 \\ \partial z_1 / \partial y_2 & \partial z_2 / \partial y_2\end{array}\right)\right|$$ Now, this looks a bit complicated, since you are given $y_1,y_2$ in terms of $z_1,z_2$ and not the other way round. Luckily, we can use the Inverse function theorem to get that: $$\left|\frac{\partial (z_1, z_2)}{\partial (y_1, y_2)}\right|= \left|\frac{\partial (y_1, y_2)}{\partial (z_1, z_2)}^{-1}\right|=\left|\frac{\partial (y_1, y_2)}{\partial (z_1, z_2)}\right|^{-1}$$

So you should be able to calculate the derivatives and then the inverse determinant.