3
$\begingroup$

I have an application where some of my states are constant and therefore have no process noise. Over the course of the estimation process, the uncertainty in these states drops several orders of magnitude, so it doesn't work to just use the initial scatter in these dimensions. I've had some success with various ad-hoc techniques, but they all corrupt the uncertainty to some extent. The underlying PDF is not Gaussian. Can anyone suggest techniques to address this problem?

$\endgroup$
  • 2
    $\begingroup$ but theyre random at time 1? yeah this is a difficult problem. theres no real solution to the degeneracy/impoverishment issue here $\endgroup$ – Taylor Jun 21 '17 at 20:52
  • 1
    $\begingroup$ I'd state it as: the uncertainty is very large prior to obtaining observations and then is relatively very small after all the observations are utilized. No problem with a KF and Gaussian representations, but problematic with particles. KF is not applicable due to extreme non-linearity of measurements (observations). $\endgroup$ – Mastiff Jun 21 '17 at 22:23
  • 2
    $\begingroup$ yeah that's a better way to phrase it. and you're experiencing all of your particles have the same values for the portion of your state that's constant? $\endgroup$ – Taylor Jun 21 '17 at 23:17
  • 1
    $\begingroup$ Yeah, without some sort of process noise, the resampling process will eventually collapse the constant dimensions to single values. The simplest way around it is to add some artificial process noise, but this makes the uncertainty bigger than it should be. If I make the (invalid) assumption that the PDF is Gaussian, I can add process noise and then collapse back to the mean in such a way that the mean/covariance is unaffected, but this can ruin the shape of the uncertainty. I've played with other hacks along these basic lines. $\endgroup$ – Mastiff Jun 22 '17 at 0:39
  • 2
    $\begingroup$ I can give some papers, but I don't think that there's any total fix that exists. $\endgroup$ – Taylor Jun 22 '17 at 16:39
2
$\begingroup$

I think "A self-organizing state space model" by Kitagawa was the first person to describe what you're doing. What I think you're doing is replacing your initial model's parameters (random but not dynamic) with dynamic ones. Really the new "parameters" are states of a different state space model. And this larger model's parameters you're thinking of as tuning parameters.

"Combined Parameter and State Estimation in Simulation-Based Filtering" by Liu and West suggests using more intelligent artificial dynamics for the parameters, improving the idea from above. These ways seem to work fine, but it's a totally different model, so it's really apples to oranges here. You're not really looking at $p(\theta|\text{states},\text{data})$ when you're looking at these nice looking samples.

"Following a moving target-monte carlo inference for dynamic bayesian models" by Gilks and Berzuini and "Markov chain monte carlo, sufficient statistics, and particle filters" by Fearnhead suggest the idea of using MCMC moves within a particle filter to “jitter” the sample elements that corresponded to parameters. This is like an extra final step in your particle filter. The goal is to simulate from your parameter posterior.

If you use Metropolis-Hastings with the idea from above to simulate from your parameter posterior, the likelihood that's proportional to this distribution is still hard to evaluate because it depends on your increasing dataset. The Fearnhead paper, "Particle filters in state space models with the presence of unknown static parameters" by Storvik, and "Particle learning and smoothing" by Carvalho et al. get around this algorithmic problem by using recursive formulas for sufficient statistics.

$\endgroup$
  • 1
    $\begingroup$ Thanks. In my case I think the sufficient statistics assumption is not applicable. In particular, I receive observations that are a function of both the dynamic and static states. A joint estimate, where each particle has a full vector of all of these states, provides an estimation benefit that is lost if I look at the dynamic states in aggregate and regenerate the static states at each time step. This isn't my exact application, but a similar one would be estimating the location of a robot and several base stations when given only range measurements between the various elements. $\endgroup$ – Mastiff Jun 29 '17 at 16:16
  • $\begingroup$ There are a few things I don't understand in your comment. Hopefully my first paragraph edit helps a bit more. $\endgroup$ – Taylor Jun 29 '17 at 17:07
  • $\begingroup$ If I understand the last paper properly, the approach requires that the static state particle values are able to be drawn anew as a function of overall statistics of the non-static states. Again, I may not have "got it" yet, but this would imply that relationships between static and dynamic states at the individual particle level are not retained. Is that right? $\endgroup$ – Mastiff Jun 29 '17 at 17:36
  • $\begingroup$ @user2364295 that sounds right if by "static states" you mean "parameters." However, your last sentence is weird. When you say "relationships," I take that to mean all the conditional probability distributions of a specific state space model. Those never change. However,if by "relationships" you mean the exact pair values for each particle ($x^i$ and $\theta^i$), then yes, they change after the jitter. That's the whole point of the algorithm. This sounds like a separate question, though. If you post a separate question I'll probably see it and be able to answer your questions more fully. $\endgroup$ – Taylor Jul 2 '17 at 18:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.