Timeline for Exponential (or other) families of distributions on manifolds.
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2013 at 10:22 | answer | added | Brendan Foreman | timeline score: 4 | |
| Dec 22, 2011 at 14:41 | comment | added | Simon Lyons | Semi-related question: mathoverflow.net/questions/59748/… | |
| Oct 29, 2011 at 21:15 | answer | added | Will Sawin | timeline score: 2 | |
| Oct 29, 2011 at 20:57 | comment | added | Suresh Venkat | @Suvrit that is correct. You can get one kind of normal distribution equivalent using the heat kernel. | |
| Oct 29, 2011 at 8:51 | comment | added | Suvrit | I think if you can define a heat kernel on a manifold, that should give the analogue for a normal distribution, right? | |
| Oct 29, 2011 at 0:32 | comment | added | Robby McKilliam | I would guess that the answer is `not really'. As far as I know there is not a even a universally accepted definition of the 'normal distribution' on a Remanian Manifold. Probably the closest thing to the normal are those distributions that arise from generalisations of Brownian motion on manifolds. math.northwestern.edu/~ehsu/… | |
| Oct 28, 2011 at 22:03 | comment | added | Suresh Venkat | Hi Joe, I'm specifically looking for random vars whose values are points on a manifold. @Deane, information geometry (which I'm familiar with) deals with how to represent families of distributions as manifolds (or submanifolds), which is different to building a probability distribution ON a manifold. | |
| Oct 28, 2011 at 19:51 | answer | added | Suvrit | timeline score: 7 | |
| Oct 28, 2011 at 19:05 | comment | added | Deane Yang | There has been some work on this for a general Riemannian manifold. I think if you google "information geometry", you will find a lot of it. But I suspect that you get the most interesting results if you use a specific Riemannian manifold and define the family of probability distribution according to your specific situation or need. | |
| Oct 28, 2011 at 18:30 | comment | added | Joseph O'Rourke | @Deane: I should not have commented, because clearly I don't understand the question---Sorry! | |
| Oct 28, 2011 at 18:27 | comment | added | Deane Yang | Joseph, isn't that the same thing? | |
| Oct 28, 2011 at 18:25 | comment | added | Joseph O'Rourke | @Deane: I think(?) Suresh is asking for random variables whose values are manifolds, rather than a manifold on which a probability distribution is defined. | |
| Oct 28, 2011 at 18:24 | comment | added | Suresh Venkat | I guess my question isn't so much "can it be done" as much as "has it been done for any specific settings and what references do I need to look at". In particular, in the context of statistical estimation . | |
| Oct 28, 2011 at 18:09 | comment | added | Deane Yang | You just have to make sure that the "exponential function", however you define it, has finite integral. | |
| Oct 28, 2011 at 18:08 | comment | added | Deane Yang | Why not? If you have a complete Riemannian manifold, it's straightforward to define a probability distribution on it that has a distinguished "center" and the distribution decays as, say, the exponential of the negative of the distance from the center. | |
| Oct 28, 2011 at 17:53 | history | asked | Suresh Venkat | CC BY-SA 3.0 |