# Transformation of probability distribution

I have a question about a snippet on page 526 in the PRML book of Bishop.

Can someone explain to me why the right-hand side of equation (11.6) equals $$z$$?

It's unclear to me where this derivation comes from. Thanks for your help! Plugging $$p(z)=1$$, in equation (11.5), we get $$p(y)=|dz/dy|$$.

Taking the integral of a derivative of $$z$$ (right-hand side), should give back the original $$z$$, leading to equation (11.6). However, where do the bounds $$-\infty$$ and $$y$$ of the integral come from?

It is mentioned that it is an indefinite integral, why do we have these bounds?

I also didn't take into account that there is an absolute value around $$dz/dy$$.

Can we just discard this as if it isn't there?

• This looks wrong to me. Consider the transformation $z=1-y$ for which $|dz/dy|=1,$ whence (by $(11.5)$) $p(y)=p(z)(1)=1$ when $0\le y\le 1$ (and is $0$ otherwise). For $0\le y\le 1$ and $y\ne 1/2$ equation $(11.6)$ gives $$1-y=z=\int_{-\infty}^y I(0\le \hat y\le 1) p(\hat y) d\hat y = \int_0^y d\hat y = y,$$ a clear contradiction.
– whuber
Oct 2 '18 at 14:15

The bounds come from the Cumulative Distribution Function (CDF) of $$p(y)$$. I.e.,

$$p(Y\leq y)=\int\limits_{-\infty}^yp(Y=\hat y)d\hat y$$

The absolute value $$\left | \frac{dz}{dy}\right |$$ come from the change of variable. Check this for more details on how to proceed with a change of variable over probability distributions.

For a more detailed answer, check below:

We have that $$z$$ is uniformly distributed over (0,1), thus

$$p(z)=\frac{1}{1-0}=1$$

Then, by defining $$z=h(y)$$ we have that

$$p(y)=p(z)\left | \frac{d}{dy}z\right |=1\left | \frac{d}{dy}h(y) \right |$$

We want to determine what the change of variable function should look like, thus we isolate it by integrating $$p(y)$$. We proceed by noticing that the function $$h(y)$$ is either monotonically increasing or decreasing. Thus, if it is increasingly monotonic we have that the right-hand side of the above equation is equal to

$$\int\limits_{-\infty}^y\frac{d}{d\hat y}h(\hat y)d\hat y$$

$$=h(y)-\underset{n\to -\infty}{\lim}h(n)+C$$

$$=h(y)-0+C$$

$$=h(y)+C$$

where the bounds make the left-hand side be a CDF on $$y$$ and $$\underset{n\to -\infty}{\lim}h(y)=0$$, given that $$h(y)$$ is monotonically increasing and $$h(y)=z$$ and $$z\in (0,1)$$.

In the case of a decreasingly monotonic function we have

$$-\int\limits_{-\infty}^y\frac{d}{d\hat y}h(\hat y)d\hat y$$

$$=-(h(y)-\underset{n\to -\infty}{\lim}h(n))+C$$

$$=\underset{n\to -\infty}{\lim}h(n)-h(y)+C$$

$$=1-h(y)+C$$

We can set the constant values $$C:=0$$, and thus we have that $$\begin{equation} h(y)=\begin{cases} \int\limits_{-\infty}^yp(\hat y)d\hat y, & h(y) \text{ is monotonically increasing} \\ 1-\int\limits_{-\infty}^yp(\hat y)d\hat y, & h(y) \text{ is monotonically decreasing} \end{cases} \end{equation}$$

Thus, our choice of $$h(y)$$ can be any of these options.

Hope this helps.