Timeline for Grid based piecewise-stationary Poisson process test
Current License: CC BY-SA 3.0
19 events
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| Mar 5, 2019 at 5:59 | comment | added | Julian Carlin | Hi! Just thought I'd let you know that paper has been published: academic.oup.com/mnras/article-abstract/482/3/3736/5142868 or arxiv.org/abs/1810.12458 (for the non-paywalled version). Thanks again for your help! | |
| Mar 29, 2018 at 1:18 | comment | added | jbowman | Sure, thanks! John Bowman / Walmart Labs will do nicely. Thanks again! | |
| Mar 28, 2018 at 23:52 | comment | added | Julian Carlin | Would you be willing to be added to the acknowledgements in a paper I am writing which uses part of your answer? If so, would you prefer your full name/affiliation, or just your profile name? It will be published in a astrophysics journal. | |
| Mar 22, 2018 at 2:46 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 22, 2018 at 2:45 | comment | added | jbowman | Sweet! I'm going to have to upgrade my graphics knowledge, I can see that! | |
| Mar 22, 2018 at 0:29 | vote | accept | Julian Carlin | ||
| Mar 22, 2018 at 0:29 | comment | added | Julian Carlin | 1. Thanks for the clarification. 2. You're right, my code did have a silly error in it -- it's all good now! Thanks so much for your help, here's the gif of the posterior grid without the bug, and the marginalisations (without using MLEs for $\lambda_{1/2}$). | |
| Mar 21, 2018 at 13:45 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 21, 2018 at 2:55 | comment | added | jbowman | 1. I have addressed your first issue with an edit at the bottom of the answer. It was probably worthy of a question in its own right! 2. I've used the profile log likelihood, which is an efficient way of calculating the maximum likelihood estimates. See the wikipedia link above, or stats.stackexchange.com/questions/9859/… for more information. If your code shows the MLE on the boundary of the region, I wonder if it's containing an error? I'll try writing some myself. | |
| Mar 21, 2018 at 2:27 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 21, 2018 at 1:30 | comment | added | Julian Carlin | I know that the distribution allows them to be larger, but they are still implicitly functions of one of the input parameters ($T_1$), which would cause a problem when trying to compare different hypotheses (different $T_1$). When I try to implement your likelihood function to scan the full 3D parameter space (assuming $\lambda_{1/2}$, $T_1$ independent) I again find the maximum likelihood points all lie on the grid boundary (gif). Could you explain why you choose to use the MLE for the rates given a specified $T_1$, instead of treating them as free parameters? | |
| Mar 20, 2018 at 4:04 | comment | added | jbowman | It doesn't actually matter if the $x_i$ are all either 0 or 1, the Poisson distribution allows them to be larger, but it is a clever idea for getting rid of the denominator! I may come back to work on this later, i don't feel I've given a clear answer, myself. | |
| Mar 20, 2018 at 2:30 | comment | added | Julian Carlin | I think I get it now, you just need to choose a $t$ time interval discretization such that all events are at least $t$ apart, then your counting functions $x_i$ will always be either 0 or 1, thus disappearing the denominator. You can pull the $t^N$ terms out the front, and as it's constant for a given dataset it's irrelevant to the shape, as you say. I'm still not completely clear as to why my likelihood function doesn't break the degeneracy in the parameters through $N_{1/2}$ being a function of $T_1$, but I certainly agree with most of the arguments you make above! | |
| Mar 19, 2018 at 13:45 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 19, 2018 at 13:40 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 19, 2018 at 12:58 | comment | added | jbowman | With respect to the factorials, they don't matter as they are functions only of the $x_i$, not of the parameters. When working with the log likelihood, they become an additive constant that doesn't change the shape of the likelihood function at all; when being Bayesian, they get cancelled out by the constant of integration of the posterior. To see this latter point, consider a parameter $a$, a likelihood $g$, and posterior $h$: $g(a) \to h(a)$, and $10g(a) \to h(a)$ also as $h(a)$ has to integrate to 1, so the 10 can be ignored. | |
| Mar 19, 2018 at 4:57 | comment | added | Julian Carlin | Thanks for the thorough response, although I'm not sure what you mean when you say "... implicitly assumes there are two periods, one with rate $λ_1 T_1$ and one with rate $λ_2 (T−T_1)$" -- the rates are $\lambda_i$, while the mean number of events is $\lambda_i T_i$. In your likelihood function your units don't seem to match: $\lambda_i$ is next to $T$ in the exponential, but by itself when raised to $N_i$ (Wikipedia hints towards a likelihood function that looks like mine ). I'm also not sure what happened to the factorials in your R code. | |
| Mar 19, 2018 at 4:48 | history | edited | jbowman | CC BY-SA 3.0 |
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| Mar 19, 2018 at 4:19 | history | answered | jbowman | CC BY-SA 3.0 |