Timeline for Relationship between Binomial and Beta distributions
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
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| May 13, 2020 at 8:15 | comment | added | Sextus Empiricus | What I mean is that conjugate prior relates to this relationship between the pdf <-> likelihood (and related to that you refer to Bayes, which is not very clear and a bit handwaving). ---- The issue that is in this question is the relationship between cumulative distribution functions. You mention a relationship between the beta distribution and the binomial distribution, but that seems to me to be a different relationship. It is not clear what this has to do with the relationship between the CDF's. | |
| May 13, 2020 at 8:08 | comment | added | Amit Portnoy | I don't know the actual historical progress, but I've seen it presented as starting from the likelihood and finding a constant that makes it into a PDF. The fact that it happens to be the conjugate prior may have been just a nice coincidence (or intuition on Bayes part). | |
| May 13, 2020 at 8:00 | comment | added | Sextus Empiricus | I believe that the beta distribution is foremost the conjugate prior distribution of the binomial distribution. The connection here is the relation $$cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1$$ which is a different thing then the relationship as conjugate prior. | |
| May 13, 2020 at 7:49 | comment | added | Amit Portnoy | Regarding "beta distribution is actually similar to the likelihood for p". I think this is how beta was viewed by Bayes. Start with the binomial likelihood for p, and find the constant that makes the integration works. | |
| May 13, 2020 at 7:42 | comment | added | Amit Portnoy | I don't know, but I think it may have something to do with the support size of the Binomial. In any case, I'm interested to know if there is any known symmetry principle. $1/(n+1)$ comes up analytically from integrating the likelihood of p. | |
| May 13, 2020 at 7:34 | comment | added | Sextus Empiricus | I am saying that it doesn't work for other cases, but I would actually have to check that out. Maybe there is some symmetry principle. | |
| May 13, 2020 at 7:26 | comment | added | Sextus Empiricus | I see now. The first version made me confused. What you seem to be saying now is that the beta distribution is actually similar as the likelihood for $p$ with the binomial distribution. But why is that intuitive? It is not clear why it is like that. We can not do the same trick with other distributions so why does it work here while it doesn't for other cases? Is there some special property that makes it work, or is it just a coincidence? And why did you choose $n+1$? It is a parameter that normalizes the likelihood function, but is that the reason, or is there some more/other logic behind. | |
| May 13, 2020 at 6:45 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 68 characters in body |
| May 12, 2020 at 23:40 | comment | added | Amit Portnoy | Let us continue this discussion in chat. | |
| May 12, 2020 at 23:40 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 121 characters in body |
| May 12, 2020 at 23:10 | comment | added | Amit Portnoy | You're right. I edited my answer. | |
| May 12, 2020 at 23:05 | history | edited | Amit Portnoy | CC BY-SA 4.0 | deleted 9 characters in body |
| May 12, 2020 at 22:45 | history | edited | Amit Portnoy | CC BY-SA 4.0 | deleted 9 characters in body |
| May 12, 2020 at 22:37 | history | edited | Amit Portnoy | CC BY-SA 4.0 | deleted 225 characters in body |
| May 12, 2020 at 20:27 | comment | added | Sextus Empiricus | So it is a likelihood and can be above 1. But still where does the multiplication with a factor n+1 come from. Why is it not something like $$1-\left(1- {n\choose k} \pmb{p}^k (1-\pmb{p})^{n-k}\right)^{n+1}$$ according to $$p(\text{at least one event in k trials}) = 1 -(1 - p(\text{one event in one trial}))^k$$ | |
| May 12, 2020 at 19:49 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 13 characters in body |
| May 12, 2020 at 19:43 | comment | added | Amit Portnoy | Yes, it's a PDF not a PMF. I'll rephrase. | |
| May 12, 2020 at 19:21 | comment | added | Sextus Empiricus | "The chance of at least one of the experiments having exactly k successes is" I have difficulties to imagine how that equals $n+1$ time the probability of having exactly k success in a single run of the experiment. For instance when p=0.5, n=2 and k=1 then the probability equals 1.5 which makes no sense. | |
| May 12, 2020 at 18:31 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 99 characters in body |
| May 12, 2020 at 18:13 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 44 characters in body |
| May 12, 2020 at 18:11 | comment | added | Amit Portnoy | This is supposed to be an intuitive explanation. Nothing more. (I did replace 1 / BetaFunc with the (N+1) * binomial coefficient, is this the issue?) | |
| May 12, 2020 at 16:32 | comment | added | Amit Portnoy | @SextusEmpiricus I think Thomas Bayes derived it :). See youtu.be/UZjlBQbV1KU?t=1074 (though his story is different) | |
| May 12, 2020 at 16:17 | comment | added | Sextus Empiricus | $$(n+1){n\choose k} \pmb{p}^k (1-\pmb{p})^{n-k}$$how did you derive this? | |
| May 12, 2020 at 15:57 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 312 characters in body |
| May 12, 2020 at 15:23 | history | edited | Amit Portnoy | CC BY-SA 4.0 | edited body |
| May 11, 2020 at 9:46 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 52 characters in body |
| May 11, 2020 at 9:25 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 5 characters in body |
| May 11, 2020 at 9:14 | history | edited | Amit Portnoy | CC BY-SA 4.0 | added 32 characters in body |
| May 11, 2020 at 9:08 | history | answered | Amit Portnoy | CC BY-SA 4.0 |