Timeline for Inferring prior distribution
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 5, 2014 at 0:06 | comment | added | guy | @Cam.Davidson.Pilon I disagree that a Beta can capture "most interesting distributions." For example, no Beta distribution is bimodal, which is the scenario in the data considered by Liu in the paper I mentioned. | |
| Sep 4, 2014 at 23:57 | comment | added | guy | @Cam.Davidson.Pilon I mean in the sense of Bernstein-Von-Mises, when there is only one $p$ to model the prior is asymptotically irrelevant; in the non-hierarchical case using a Beta prior is fine. But nothing like Bernstein-Von-Mises is applicable in a typical hierarchical example, which makes the Beta more suspect. | |
| Sep 4, 2014 at 23:45 | comment | added | Cam.Davidson.Pilon | Good points @guy, for brevity and simplicity I chose the Beta distribution, though I also believe the Beta is flexible enough to capture most interesting distributions. I'm curious about what you mean by "because the prior will swamp the data eventually", have time to elaborate? | |
| Sep 4, 2014 at 15:02 | comment | added | guy | Also, saying "we are dealing with probabilities, so a Beta is reasonable" is only (IMO) an innocuous thing to do if $p_i = p$ is a single parameter of interest, because the prior will swamp the data eventually. Here the parametric assumption really is a parametric assumption. If the goal were to share information in a hierarchichal setting and not infer about $D$ we might use a Beta anyways and hope it doesn't matter, but if $D$ is the target I wouldn't. | |
| Sep 4, 2014 at 14:55 | comment | added | guy | Assuming that $p_i ~ \stackrel{iid}{\sim}\mbox{Beta}(\alpha, \beta)$ seems like a very strong assumption, and if the ultimate goal is inference about $D$ I wouldn't recommend it as a default inference. Either a Dirichlet mixture of Beta distributions (as in Liu, AOS, 1996) or some nonparametric Empirical Bayes technique seems much better. There isn't really any reason to expect $D$ to be a simple 2 parameter distribution. | |
| Sep 4, 2014 at 3:39 | vote | accept | Uwat | ||
| Sep 4, 2014 at 2:42 | comment | added | Cam.Davidson.Pilon | @Uwat, when you have a even somewhat non-trivial model, the only way to perform Bayesian inference is with MCMC. I would recommend Bayesian Methods for Hackers (I'm the author btw) for an intro to Bayesian methods (chapter 1 & 2) and MCMC (chapter 3). For "reconstruct", it's best to not use an individual reconstruction. There are a few choices you could make: 1) use the average values of alpha, beta. 2) Use the MAP of alpha, beta (better). 3) Use the average over the distributions (also better) | |
| Sep 4, 2014 at 0:14 | comment | added | Uwat | Thank you very much about your answer. Unfortunately I don't really have the math knowledge to really understand it yet. Is MCMC (which I don't know about) typically used for that kind of stuff? What specifically should I read about? You say you can "reconstruct possible distributions of D". Would it make sense to take an average of a bunch of reconstructions or would it not be any more valid than any one reconstruction? | |
| Sep 3, 2014 at 18:28 | history | edited | Cam.Davidson.Pilon | CC BY-SA 3.0 |
added 108 characters in body
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| Sep 3, 2014 at 18:23 | comment | added | Cam.Davidson.Pilon | Here's the ipython %hist: gist.github.com/CamDavidsonPilon/1fed2295083e660b776a | |
| Sep 3, 2014 at 18:19 | history | answered | Cam.Davidson.Pilon | CC BY-SA 3.0 |