What does this sentence mean? I only know undergrad-level calculus, probability & linear algebra - and this sentence doesn't make sense to me (Appendix B):
Here is another example usage of this:
Questions:
- What does "low dimensional manifold" mean in this context? (I don't understand wikipedia explanations here)
- Why is it only low dimensional... does a higher dimensional manifold not have this property? I would think that higher-dimesional something has all the same attributes of a lower-dimensional something.
- What do I need to learn to understand this?
I hope I phrased the question clearly - if you need more details please let me know.
In this context, "low-dimensional" just means "lower than the dimension of the ambient space". So for instance, if $X$ is a random variable that takes values in $\mathbb{R}^3$, it might actually be concentrated on a subset of $\mathbb{R}^3$ whose dimension is smaller than $3$ (like a line, or a plane, or a surface), and then it could be continuous but not absolutely continuous. For example, if $X$ is uniformly distributed on the unit sphere in $\mathbb{R}^3$, then $X$ is continuous, since it has probability $0$ of being at any single point. However, it is not absolutely continuous, since the unit sphere has Lebesgue measure $0$ in $\mathbb{R}^3$, and $X$ does not have any probability density function in $\mathbb{R}^3$ (if such a thing existed, it would morally have to be sort of like $\delta$-function supported on the sphere but $0$ everywhere else).
For the passages you've shown, it's not important to understand any of this in technical detail (such as the precise definition of a "manifold" or how one defines "dimension" rigorously); these are just examples of why we are interested in probability distributions that do not come from density functions.