I remember being shown the following example in class some time ago, but haven't been able to find any information about it on the internet.
The paradox
Let $(x, y)$ be a uniform random variable on the semi-circle $\{(x, y): x^2 + y^2 < 1, x > 0\}$.
By considering $p(x \vert y \in (-\delta, \delta))$ for small $\delta$ we see that $$p(x \vert y = 0) = 1$$ is a Uniform distribution on $(0, 1)$.
Nevertheless, if we let $\theta = \tan \left(\frac{y}{x} \right)$ be the angle from the $x$-axis, we see by considering $p(x \vert \theta \in (-\delta, \delta))$ that $$p(x \vert \theta=0) = 2x$$ has a linear pdf.
This result is suprising, as naively one would expect for the conditions $y = 0$ and $\theta = 0$ to encode the same information.
Questions
My question is twofold:
- Is there a name / search keyword for this "paradox"
- Under what conditions is it true that the conditional distributions are equivalent, ie $p(x \vert t) = p(x \vert y)$ for some change of variables to $(x, y) \mapsto (x, t)$? what about when $t$ and $x$ are multi-dimensional?
For example if $t = f(y)$ is some differentiable function independent of $x$, we see that $p(x, y) = p(x, t) \lvert \frac{df}{dy} \rvert$ and $p(y) = p(t) \lvert \frac{df}{dy} \rvert$, so the ratios are equal:
$$p(x \vert y) = \frac{p(x, y)}{p(y)} = \frac{p(x, t)}{p(t)} = p(x \vert t),$$
but for (2) I'm interested in if is there a more general condition (eg continuity, or even some dependence on $x$ provided volumes are preserved)?