AAMAS 2009

1. AAMAS 2009 Compact Approximations of Mixture Distributions for State Estimation in Multiagent Settings Prashant Doshi Dept. of Computer Science University of Georgia

2. State Estimation Single agent setting Physical State (Loc, Orient,...)

3. State Estimation Multiagent setting Interactive state Physical State (Loc, Orient,...) (See AAMAS 05)

4. Estimate the interactive state State Estimation in Multiagent Settings Ascribe intentional models (POMDPs) to other agents Update the other agents' beliefs (See JAIR’05)

5. Previous Approaches • Interactive particle filter (I-PF; see JAIR’09) • Generalizes PF to multiagent settings • Approximate simulation of the state estimation • Limitations of the I-PF • Large no. of particles needed even for small state spaces • Distributes particles over the physical state and model spaces • Poor performance when the physical state space is large or continuous • Rao-Blackwellised I-PF (RB-IPF; see AAAI’07) • Marginalize some dimensions of the interactive state space and update them analytically • Sample particles from the remaining and propagate them

6. Factoring the State Estimation Update the physical state space Update other agent's model

7. Factoring the State Estimation Sample particles from just the physical state space Substitute in state estimation Implement using PF Perform as exactly as possible Rao-Blackwellisation of the I-PF

8. Problem with RB-IPF • Starting with a single prior density, becomes a mixture of conditional densities • At most |Aj||j| components after a step of the belief update • (|Aj||j|)tcomponents after t steps of the belief update Number of mixture components grows exponentially with the update

9. Compact Approximation of Mixture Distribution General Idea (Snelson&Ghahramani, 2005) Fit the posterior with a mixture of a constant number, K, densities of exactly the same form as the component densities in the original belief

10. Compact Approximation • Approach • Seek a compact approximation, , parameterized by w, of the mixture, , parameterized by v • Find the approximation that minimizes the KL-Divergencewith the mixture:

11. Compact Approximation • Step 1 • Obtain X samples from the original mixture, • A point estimate of v from D using ML or MAP will not be accurate • We compute a distribution over v,Pr(v|D),using an accurate parameter estimation approach such as Laplace’s

12. Compact Approximation • Step 2 • Sample K parameter sets, vk , k=1...K from Pr(v|D) to obtain • This is a mixture of K component densities, each of the same form as the original component • K < |j| • Fewer components in memory • K may be flexibly varied

13. Compact Approximation • Step 3 • Approximate the integral in computing KLD • Cannot use D and we do not wish to sample again from the original distribution • Generate ‘fake data’ for the approximation • Sample parameters vm, m=1...M to obtain: • Sample (fake data) from the above distribution and approximate the KLD integral:

14. Result Saves on memory and is more efficient to evaluate Proposition: The approximation keeps K sets of parameters in memory, compared to the original density, which keeps |j| parameter sets in memory

15. Discussion & Acknowledgment • State estimation in multiagent settings is a difficult problem • Estimate the state and models of others • Exponentially growing mixture distributions inevitably arise • Compact approximations are needed • Sampling methods and some ingenuity facilitate efficiency • Acknowledgment • This research is supported in part by grant FA9550-08-1-0429 from AFOSR