Timeline for Recursive parameter estimation for partially observed Ito SDEs
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2016 at 11:19 | vote | accept | S.Surace | ||
| Dec 13, 2016 at 11:18 | vote | accept | S.Surace | ||
| Dec 13, 2016 at 11:19 | |||||
| Dec 13, 2016 at 11:18 | answer | added | S.Surace | timeline score: 3 | |
| May 2, 2015 at 23:03 | history | edited | S.Surace | CC BY-SA 3.0 |
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| May 2, 2015 at 23:00 | comment | added | S.Surace | I will add a few clarifying remarks to the question. | |
| May 2, 2015 at 22:54 | comment | added | S.Surace | I see what you mean. In offline estimation, my $\tau$ is simply the parameter for the offline gradient ascent. So the integral is meant to be evaluated after all the history of $Y_t$ has been observed (and stored). Then I use the usual (in machine learning at least), albeit heuristic practice of making an online rule by simply getting rid of the integral. The thus defined $\hat\theta_t$ should be adapted to the filtration $\mathcal{F}^Y_t$, if I'm not mistaken. | |
| May 2, 2015 at 21:46 | comment | added | ofer zeitouni | I am not sure what is $\tau$, but in any case, in the equation defining $\hat \theta_{ML,offline}$, doesn't $\hat \theta$ depend on the whole path and therefore the stochastic integral is non-adapted? | |
| May 2, 2015 at 20:15 | comment | added | S.Surace | Could you elaborate on your comment re: non-adapted integration? What is unclear about the notation? I think up to $\partial \log P...$ it should be fine, but the step following that (continuum limits etc.) is not clear to me either, and I don't know how to make it rigorous. The main issue seems to be somehow going from the conditional distribution $P(\mathbf{X}|\mathbf{Y},\theta)$ to something resembling a filter (i.e. running forward in time). | |
| May 2, 2015 at 20:12 | comment | added | S.Surace | Thanks for the hint, it just crossed my mind that the innovation process $n_t$ defined above is a Wiener process under the law of $Y$. I haven't seen it called the Kallianpur-Striebel formula, I thought that was the formula for expressing expectations conditioned on observations of $Y_t$ as quotient of expectations under the law where $Y_t$ is a Wiener process (for which one can then derive the DMZ equation)? | |
| May 2, 2015 at 12:38 | comment | added | ofer zeitouni | As for your second question, I am not sure about your notation, but don't you run into the trouble of non-adapted integration? | |
| May 2, 2015 at 12:38 | comment | added | ofer zeitouni | Concerning your first question, $L_t(\theta)$ is the R-N derivative for the law of $Y$ with respect to the reference measure where $Y$ is simply a Wiener process (independent of course of the parameter $\theta$). (In the old filtering literature, this is referred to as the "Kallianpur-Striebel formula"). Since the reference measure does not depend on $\theta$, this seems to me as a good measure of likelihood. | |
| May 1, 2015 at 14:17 | history | edited | S.Surace | CC BY-SA 3.0 |
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| May 1, 2015 at 14:14 | review | First posts | |||
| May 1, 2015 at 14:25 | |||||
| May 1, 2015 at 14:10 | history | asked | S.Surace | CC BY-SA 3.0 |