Change of variables by doing a transformation with a Jacobian versus finding an inverse

Question

I have been solving one problem and there is something unclear to me in the solutions.

Namely, let's consider a probability density $p_x(x)$ defined over a continuous variable $x$, and suppose that we make a nonlinear change of variable using $x = g(y)$, so that the density transforms according to (1.27). $$\begin{align} p_y(y) &= p_x(x) \left| \dfrac{dx}{dy} \right| \\ &= p_x(g(y)) |g'(y)| \tag{1.27} \end{align}$$

An example of this: Let's take a Gaussian distribution $p_x(x)$ over $x$ with mean $\mu = 6$ and standard deviation $\sigma = 1$. Next we draw a sample of $N = 50,000$ points from this distribution.

Now let's consider a non-linear change of variables from $x$ to $y$ gives by

$$x = g(y) = \ln(y) - \ln(1 - y) + 5 \tag{5}$$

The inverse of this function is given by

$$y = g^{-1}(x) = \dfrac{1}{1 + \exp(-x + 5)} \ \tag{6}$$

which is a logistic sigmoid function.

Now, in order to calculate $p_y(y)$ we can use the (1.27) defined above, by basically taking into account (5) and computing the Jacobian, i.e. $|g'(y)|$.

The unclear part: The author claims that we can confirm whether we did the non-linear transformation correctly by taking out sample of 50,000 values of $x$, evaluate the corresponding values of $y$ using (6) and plot this, and this would match the transformation given by 1.27.

So my question now: how did we avoid computing the Jacobian by using the inverse i.e (6) instead, and why don't we always do this instead, i.e. just compute the inverse, and not do a full transformation according to 1.27?

Thanks in advance!

Answer 1

The Jacobian is avoided via simulation since the density for $y$ is defined as \begin{eqnarray*} p_y(t) &=& \underset{\substack{\Delta t \rightarrow 0 \\ n \rightarrow \infty}}{\mbox{lim}}\frac{\frac{1}{n} \sum_{i=1}^n \mbox{I} \left[t \lt g^{-1}(x_i) \lt t + \Delta t \right]}{\Delta t} \\ &=& \underset{\substack{\Delta t \rightarrow 0 \\ n \rightarrow \infty}}{\mbox{lim}}\frac{\frac{1}{n} \sum_{i=1}^n \mbox{I} \left[t \lt y_i \lt t + \Delta t \right]}{\Delta t}\\ &=& \underset{\Delta t \rightarrow 0}{\mbox{lim}} \frac{\mbox{Pr} \left[ t \lt y \lt t + \Delta t\right]}{\Delta t} \\ &=& \underset{\Delta t \rightarrow 0}{\mbox{lim}} \frac{F_y(t+\Delta t) - F_y(t)}{\Delta t}, \end{eqnarray*} where $\mbox{I} \left[\cdot\right]$ denotes the indicator function and $F_y (\cdot)$ denotes the CDF of $y$. Since we have 50,000 simulations, it is safe to assume that $n$ is large enough for the limiting argument to hold. In other words, \begin{eqnarray*} p_y(t) & \approx & \underset{\substack{\Delta t \rightarrow 0}}{\mbox{lim}}\frac{\frac{1}{50,000} \sum_{i=1}^{50,000} \mbox{I} \left[t \lt g^{-1}(x_i) \lt t + \Delta t \right]}{\Delta t}. \end{eqnarray*}

Using simulation is undesirable in the sense that it does not give a closed-form solution for $p_y(t)$.