Timeline for Conjugate prior of the Dirichlet distribution?
Current License: CC BY-SA 2.5
13 events
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| Feb 23, 2022 at 0:21 | comment | added | Charlie Parker | why is this answer so long? Why isn't it as follows. The conjugate prior of the Dirichellet is X. Saying what X says. Everything else is nice but makes it harder to extract the information people usually want to look up. | |
| Apr 8, 2021 at 19:27 | comment | added | Legendre17 | @Xi'an, I think it's actually just a difference in sign conventions. In the version of the paper that I can find (core.ac.uk/download/pdf/188707037.pdf), the product $v_t \alpha_t$ appears with a minus sign in the exponential (see eq. (45)), whereas in the derivation on Cross Validated the minus sign is missing. FWIW, I prefer the convention in George's answer since it leads to positive $v_t$'s. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 20, 2016 at 9:44 | comment | added | Joyel | @George_Papandreou, Thank you for your answer and your derivation in Appendix C, which is neat! My question is how does one sample $\mathbf{\alpha}$ from $p(\mathbf{\alpha}\mid \mathbf{\nu}, \eta) \propto \frac{1}{B(\mathbf{\alpha})^\eta}e^{-\sum_{t=1}^D \nu_t,\alpha_t}$ ? I am not sure if I can use rejection sampling while sampling a vector. Looking forward to your help. Thanks so much! | |
| Nov 25, 2015 at 17:39 | comment | added | Xi'an ні війні | @GeorgePapandreou: there is a typo in the reply. The actualisation should be$$v_N[t] = v_0[t] + \sum_{n=1}^N \ln \theta_n[t]$$not$$v_N[t] = v_0[t] - \sum_{n=1}^N \ln \theta_n[t]$$This confused a reader on Cross validated. | |
| May 24, 2015 at 13:17 | comment | added | drevicko | Though the books on bayesian statistics are no doubt a better source, Wikipedia has a useful article on <conjugate priors for exponential family distributions> (such as the Dirichlet). | |
| May 24, 2015 at 12:25 | comment | added | drevicko | @George_Papandreou Thanks for sharing your work here (: That appendix is quite short - perhaps it'd be good to add it here also. | |
| May 24, 2015 at 12:20 | comment | added | drevicko | For others looking for the conjugate prior and it's derivation, it's in Appendix C at the end of the paper. @isarandi: no, it's quite different to a Dirichlet. | |
| Oct 21, 2014 at 0:48 | comment | added | isarandi | So to actually answer the question, is the Dirichlet distribution its own conjugate prior? | |
| Apr 14, 2010 at 3:17 | comment | added | George Papandreou | Hi, at Level-1 we have $p(\mathbf{\theta}|\mathbf{\alpha}$, where $\theta$ is the Dirichlet observation lying on the $D$-dimensional simplex (components between 0 and 1 whose sum is 1), and $\mathbf{\alpha}$ is the Dirichlet distribution parameter vector. At Level-2 we have the conj. prior $p(\mathbf{\alpha}|\eta,\mathbf{v})$ of our paper. In this setup, one observes $\theta$ vectors and updates the hyperparameters $\eta,\mathbf{v}$ that describe the density of $\mathbf{\alpha}$, which itself is considered hidden (it thus makes no sense to measure it). Hope it is clearer now! George | |
| Apr 10, 2010 at 2:25 | comment | added | Neil | Thank you very much for your detailed answer! Am I right that "theta" represents categorical parameter vector observations (i.e., their components are in [0,1])? If the observations are "alpha", the parameter vectors of Dirichlet observations, then the update of v needs to be: v_N[j] = v_0[j] + sum_n [ digamma(sum_i alpha_n[i]) - digamma(alpha_n[j]) ] because that's the expected log of each component value, right? Thanks again. This was very helpful. | |
| Apr 10, 2010 at 0:32 | vote | accept | Neil | ||
| Apr 10, 2010 at 0:00 | history | answered | George Papandreou | CC BY-SA 2.5 |