1. Particle filters (continued…)

2. Recall Particle filters –Track state sequence x i given the measurements ( y 0, y 1, …., y i ) –Non-linear dynamics –Non-linear measurements 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 u i with weights w i ( u i, w i ): Sampled representation of f(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 n i

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) Drop (Notation: Chapter 2)

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* *Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005. State at time t Measurement at time t All measurements upto time t PosteriorPrior Marginalization Likelihood

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 Interaction potential g ( X it, X jt ) penalizes overlap between targets Takes care of “hijacking” Edges are formed only when templates overlap 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 k t is a state variable Each k t determines a corresponding state space State space is the union of state spaces indexed by k t Particle filtering RJMCMC to jump across state spaces Prediction + Correction

25. Thank you!