I want to use the posterior distribution of the model parameters $\theta$ given data in the time frame $[0,t]$ days, $P(\theta|y_{0:t})$; as a prior for the parameters in the time frame $[t+1, t+n]$ days. The problem is; the parameter vector $\theta$ contains multiple parameter $(\theta_1, \theta_2, \theta_3,\theta_4,\theta_5)$ among which two ($\theta_4, \theta_5$) are correlated with each other. In this case, how can I use the posterior as a future prior?

Someone suggested me to use sequential Monte Carlo/particle filtering technique to find the posterior. I am very new to this technique, but I found that this technique is particularly used when the events occur sequentially and we are interested in inference on line (sequentially). Can anyone suggest me if it is going to be the right technique to apply?

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    $\begingroup$ You are not providing enough information about your problem, and even if you were, the question seems too broad to me. Many SMC algorithms used for inference about static parameters are not truly online. $\endgroup$ – Taylor May 9 '19 at 22:50
  • $\begingroup$ @Taylor, Hi. I think I am providing enough information. My question is= how to use posterior (derived using data in timeframe $[0,t]$) as a future prior (data in timeframe $[t+1, t+n]$) when the parameters are correlated, not independent? $\endgroup$ – gultu May 9 '19 at 23:06
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    $\begingroup$ You definitely haven't. It's relatively easy to show that Bayes' rule allows you to use the old posterior as a new prior because \begin{align*} p(\theta \mid y_{0:t+n}) &\propto p(y_{0:t} \mid \theta) p(y_{t+1:t_n} \mid \theta) p(\theta) \\ &\propto p(y_{t+1:n} \mid \theta) p(\theta \mid y_{0:t}). \end{align*} but you haven't told us anything about your model, so, in particular, we don't know whether or not you have samples from the old posterior, or if you know exactly what distribution it is. $\endgroup$ – Taylor May 9 '19 at 23:18
  • $\begingroup$ @Taylor; my data is following a Hawkes process; which is a self-exciting Point process. I know Bayes rule allows us to use old posterior as new prior. In this case, I draw samples for each of the parameters from the posterior (using STAN). Then I find the distribution from this discrete set of samples and use it as a new prior for future data. But what if the parameters are correlated? $\endgroup$ – gultu May 9 '19 at 23:24
  • $\begingroup$ Correlation in which sense? Between the components of $\theta$? between the successive simulations? $\endgroup$ – Xi'an May 10 '19 at 6:21