Dynamic Bayesian Networks And Particle Filtering 1 Time and - PowerPoint PPT Presentation

Dynamic Bayesian Networks And Particle Filtering 1

Time and uncertainty The world changes; we need to track and predict it Diabetes management vs vehicle diagnosis Basic idea: copy state and evidence variables for each time step X t = set of unobservable state variables at time t e.g., BloodSugar t , StomachContents t , etc. E t = set of observable evidence variables at time t e.g., MeasuredBloodSugar t , PulseRate t , FoodEaten t This assumes discrete time ; step size depends on problem Notation: X a : b = X a , X a +1 , . . . , X b − 1 , X b 2

Dynamic Bayesian networks X t , E t contain arbitrarily many variables in a replicated Bayes net BMeter 1 R 0 P(R ) 1 P(R ) 0 t 0.7 Battery 0 Battery 0.7 f 1 0.3 Rain 0 Rain 1 X 0 X 1 R 1 P(U ) 1 t 0.9 f 0.2 X 0 X 1 X t Umbrella 1 Z 1 3

DBNs vs. HMMs Every HMM is a single-variable DBN; every discrete DBN is an HMM X t+1 X t Y Y t t+1 Z Z t t+1 Sparse dependencies ⇒ exponentially fewer parameters; e.g., 20 state variables, three parents each DBN has 20 × 2 3 = 160 parameters, HMM has 2 20 × 2 20 ≈ 10 12 4

DBNs vs Kalman filters Every Kalman filter model is a DBN, but few DBNs are KFs; real world requires non-Gaussian posteriors E.g., where are bin Laden and my keys? What’s the battery charge? BMBroken BMBroken 0 1 BMeter 1 E(Battery|...5555005555...) 5 Battery 0 Battery 1 4 E(Battery|...5555000000...) 3 E(Battery) X 0 X 1 2 P(BMBroken|...5555000000...) X 0 X 1 X 1 t 0 P(BMBroken|...5555005555...) Z 1 -1 15 20 25 30 Time step 5

Exact inference in DBNs Naive method: unroll the network and run any exact algorithm R 0 P(R ) R 0 P(R ) R 0 P(R ) R 0 P(R ) R 0 P(R ) R 0 P(R ) R 0 P(R ) R 0 P(R ) 1 1 1 1 1 1 1 1 P(R ) P(R ) 0 0 t 0.7 t 0.7 t 0.7 t 0.7 t 0.7 t 0.7 t 0.7 t 0.7 0.7 0.7 f 0.3 f 0.3 f 0.3 f 0.3 f 0.3 f 0.3 f 0.3 f 0.3 Rain 0 Rain 1 Rain 0 Rain 1 Rain 2 Rain 3 Rain 4 Rain 5 Rain 6 Rain 7 R 1 P(U ) R 1 P(U ) R 1 P(U ) R 1 P(U ) R 1 P(U ) R 1 P(U ) R 1 P(U ) R 1 P(U ) 1 1 1 1 1 1 1 1 t 0.9 t 0.9 t 0.9 t 0.9 t 0.9 t 0.9 t 0.9 t 0.9 f f f f f f f f 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Umbrella 1 Umbrella 1 Umbrella 2 Umbrella 3 Umbrella 4 Umbrella 5 Umbrella 6 Umbrella 7 Problem: inference cost for each update grows with t Rollup filtering: add slice t + 1 , “sum out” slice t using variable elimination Largest factor is O ( d n +1 ) , update cost O ( d n +2 ) (cf. HMM update cost O ( d 2 n ) ) 6

Likelihood weighting for DBNs Set of weighted samples approximates the belief state Rain 0 Rain 1 Rain 2 Rain 3 Rain 4 Rain 5 Umbrella 1 Umbrella 2 Umbrella 3 Umbrella 4 Umbrella 5 LW samples pay no attention to the evidence! ⇒ fraction “agreeing” falls exponentially with t ⇒ number of samples required grows exponentially with t 1 LW(10) LW(100) LW(1000) 0.8 LW(10000) 0.6 RMS error 0.4 0.2 0 0 5 10 15 20 25 30 35 40 45 50 Time step 7

Particle filtering Basic idea: ensure that the population of samples (“particles”) tracks the high-likelihood regions of the state-space Replicate particles proportional to likelihood for e t Rain t Rain t +1 Rain t +1 Rain t +1 true false (a) Propagate (b) Weight (c) Resample Widely used for tracking nonlinear systems, esp. in vision Also used for simultaneous localization and mapping in mobile robots 10 5 -dimensional state space 8

Particle filtering contd. Assume consistent at time t : N ( x t | e 1: t ) /N = P ( x t | e 1: t ) Propagate forward: populations of x t +1 are N ( x t +1 | e 1: t ) = Σ x t P ( x t +1 | x t ) N ( x t | e 1: t ) Weight samples by their likelihood for e t +1 : W ( x t +1 | e 1: t +1 ) = P ( e t +1 | x t +1 ) N ( x t +1 | e 1: t ) Resample to obtain populations proportional to W : N ( x t +1 | e 1: t +1 ) /N = αW ( x t +1 | e 1: t +1 ) = αP ( e t +1 | x t +1 ) N ( x t +1 | e 1: t ) = αP ( e t +1 | x t +1 ) Σ x t P ( x t +1 | x t ) N ( x t | e 1: t ) = α ′ P ( e t +1 | x t +1 ) Σ x t P ( x t +1 | x t ) P ( x t | e 1: t ) = P ( x t +1 | e 1: t +1 ) 9

Particle filtering performance Approximation error of particle filtering remains bounded over time, at least empirically—theoretical analysis is difficult 1 LW(25) LW(100) LW(1000) 0.8 LW(10000) ER/SOF(25) Avg absolute error 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 40 45 50 Time step 10