Factorizable time evolution in a dynamic stochastic process - Cross Validated most recent 30 from stats.stackexchange.com 2019-07-23T10:23:01Z https://stats.stackexchange.com/feeds/question/143058 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/143058 2 Factorizable time evolution in a dynamic stochastic process Antonio Valerio Miceli-Barone https://stats.stackexchange.com/users/21523 2015-03-23T19:49:16Z 2016-07-14T06:16:52Z <p>I have a stationary dynamic system which at each given time $t$ is in state $x_t \in \mathcal{X}$. The set of states $\mathcal{X}$ is assumed to be finite but too large to be enumerated by a practical algorithm.</p> <p>At some initial time $t$ I don't know the exact state but I know a probability distribution over the states, which I assume to be parametric and factorizable:<br> $\Pr[X_t=x] = q(x, \theta_t)$</p> <p>At each step, the state is evolved by a (deterministic) state transition function: $x_{t+1} = f(x_t)$.<br> The problem is that after the state transition the posterior probability distribution over the states $p_{t+1}(x) = \sum_{x'} \Pr[X_{t+1}=x|X_t=x'] \cdot q(x', \theta_t)$ will be in general no longer representable by my parametric factorizable model $q(x, \theta)$.</p> <p>I would like a method to find a parameter vector $\overset{*}{\theta_{t+1}}$ for my model that approximates the true distribution:<br> $\overset{*}{\theta_{t+1}} = argmin_{\theta} D[q(\cdot, \theta), p_{t+1}]$<br> where $D$ is some kind of dissimilarity measure (e.g. the Kullback–Leibler divergence).</p> <p>I was thinking that some kind of variational method might be appropriate. I would like to avoid Monte Carlo methods if possible.</p> <p>Any references or ideas on how to proceed?</p>