Timeline for Probability density of a hyperplane for a Gaussian distribution
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 2, 2022 at 18:49 | vote | accept | etal | ||
| May 2, 2022 at 18:49 | comment | added | etal | Ok, I was able to find the answer to my simple question on the stats stack exchange here. It was better to not frame it with hyper-planes and instead see it as the result of taking the dot product of a multivariate gaussian with a fixed vector. | |
| May 2, 2022 at 0:01 | comment | added | etal | @IosifPinelis ' answer is good and valuable, so I'm inclined to just keep the question as edited and accept the answer. Does this sound good? If my question doesn't have a 2-line answer for this comment section (which it very well might) should I just create a new question here or on the non-professional math exchange? Thanks! | |
| May 1, 2022 at 23:56 | comment | added | etal | @MattF Although your parsing of my (poorly written) question makes a lot of sense, my question was different. Using your notation, what I wanted was probably simpler, just $\lim_{\epsilon\rightarrow 0^+} \frac{P[\textbf{x}\in f^{-1}([C-\epsilon, C+\epsilon])}{\epsilon}$, where $f(\textbf{x}) = \textbf{x} \cdot \textbf{g}$. | |
| S May 1, 2022 at 23:33 | history | suggested | CommunityBot | CC BY-SA 4.0 |
proper horizontal spacing between f(x) and dx
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| May 1, 2022 at 22:18 | review | Suggested edits | |||
| S May 1, 2022 at 23:33 | |||||
| May 1, 2022 at 16:15 | answer | added | Iosif Pinelis | timeline score: 3 | |
| May 1, 2022 at 7:35 | comment | added | Fedor Petrov | Make a linear change of variables which makes the Gaussian standard, then an orthogonal change of variables which makes your plane orthogonal to a basic vector. | |
| May 1, 2022 at 7:28 | comment | added | user44143 | The standard answer is to define $$P[\textbf{x}\in S|f(\textbf{x})=C]=\lim_{\epsilon\rightarrow 0^+}\frac {P[\textbf{x}\in S \cap f^{-1}([C-\epsilon,C+\epsilon])} {P[\textbf{x}\in f^{-1}([C-\epsilon,C+\epsilon])}$$ where in this case we use $f(\textbf{x})= \textbf{x}\cdot \textbf{g}$. The question is then how to transform this expression into a similar expression without a limit. | |
| May 1, 2022 at 7:07 | history | edited | user44143 | CC BY-SA 4.0 |
clarified question and removed non-answer
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| May 1, 2022 at 1:27 | comment | added | Iosif Pinelis | I cannot attach any meaning to "the probability density of the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure" and also to "the probability density of a one-dimensional gaussian involving $\textbf{x}^*$". | |
| Apr 30, 2022 at 21:02 | history | asked | etal | CC BY-SA 4.0 |