Particle filters (continued…)

1. Particle filters (continued…)

2. Recall • Particle filters • Track state sequence xi given the measurements (y0, y1, …., yi) • Non-linear dynamics • Non-linear measurements Non-Gaussian Non-Gaussian

3. Recall • Maintain a representation of • Two stages • Prediction • Correction (Bayesian) Dynamic model (Markov) Likelihood Prior Posterior

4. 3 Useful tools • Importance sampling • Tool 1: Representing a distribution • Tool 2: Marginalizing • Tool 3: Transforming prior to posterior

5. Tool 1: Representing a distribution • Have a set of samples ui with weights wi • (ui, wi ): Sampled representation off(u) • Expectation under f(u) • Samples used only as a means to evaluate expectations (Not true samples!)

6. Tool 2: Marginalization • Marginalization • Sampled representation • Just retain the required components and ignore the rest! Drop ni

7. Tool 3: From Prior to Posterior • Modify the weights to transform from one distribution to another • Similarly for going from prior to posterior ? To From To From Scale factor is the same for all the samples

8. Simple Particle filter • Prediction • 2 steps • Sampling from joint distribution • Marginalization Dynamic model (Markov) (Notation: Chapter 2) Drop

9. Simple Particle filter • Correction • Modify weights Likelihood Prior Posterior Let Likelihood

10. Improved Particle filter • Simple Particle filter • Many samples have small weights • Number of samples increases at every step • Lots of samples wasted • Resample (Sampling-Importance -Resampling) • Prior: • Predictions: • Resampling also takes care of increasing number of samples

11. Tracking interacting targets* • Using partilce filters to track multiple interacting targets (ants) *Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005.

12. Independent Particle filters • Targets lose identity • Identical appearance • Multiple peaks in the likelihood • Best peak “hijacks” all the nearby targets

13. Alternate view of Particle filters • Notation* Marginalization Posterior Likelihood Prior State at time t Measurement at time t All measurements upto time t *Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005.

14. Alternate view of Particle filters • Sampled representation of prior • Monte-Carlo approximation

15. Alternate view of Particle filters • Sequential Importance Resampling (SIR) • Particles at time t • Weights (easy to verify!) • Prediction and correction in one step Particles sampled from a mixture distribution formed by previous particle set

16. Independent vs. Joint filters • Multiple targets • Joint state space: Union of individual state spaces • Independent targets • Predictions are made independently from respective spaces • Interacting targets • Predictions are from the joint state space • High dimensionality: MCMC better than Importance sampling?

17. Interacting targets • Targets influence the dynamics of others • Particles cannot be propagated independently • Model interactions between targets using Markov Random Fields (MRF) Individual dynamics Pair wise interactions

18. MRF Edges are formed only when templates overlap • Interaction potential • g(Xit ,Xjt) penalizes overlap between targets • Takes care of “hijacking” Overlap is penalized by the interaction potential

19. Joint MRF Particle filter • Sequential Importance Resampling • Particles at time t • Weights • Interactions affect only the weights Equivalent to independent particle filters

20. Target overlap • Targets overlap on each other and then segregate • Overlapped target state “hijacked” • Probably hard to model this?

21. Why MCMC? • Joint MRF Particle filter • Importance sampling in high dimensional spaces • Weights of most particles go to zero • MCMC is used to sample particles directly from the posterior distribution

22. MCMC Joint MRF Particle filter • True samples (no weights) at each step • Stationary distribution for MCMC • Proposal density for Metropolis Hastings (MH) • Select a target randomly • Sample from the single target state proposal density

23. MCMC Joint MRF Particle filter • MCMC-MH iterations are run every time step to obtain particles • “One target at a time” proposal has advantages: • Acceptance probability is simplified • One likelihood evaluation for every MH iteration • Computationally efficient • Requires fewer samples compared to SIR

24. Variable number of targets • Target identifiers kt is a state variable • Each kt determines a corresponding state space • State space is the union of state spaces indexed by kt • Particle filtering • RJMCMC to jump across state spaces Prediction + Correction

25. Thank you!