1 CS 188: Artificial Intelligence Fall 2011 Lecture 19: Particle Filtering 11/3/2011 Dan Klein – UC Berkeley Presented by Woody Hoburg Announcements Project 4 out TONIGHT: due 11/16 Pick up midterm: Jono’s OH, SDH 730, 2-3:30pm 2 Outline Particle Filtering (sampling-based inference in HMMs) Dynamic Bayes Nets Viterbi Algorithm 3 Recap: Reasoning Over Time Markov models Hidden Markov models X2 X1 X3 X4 rain sun 0.7 0.7 0.3 0.3 X5 X2 E1 X1 X3 X4 E2 E3 E4 E5 X E P rain umbrella 0.9 rain no umbrella 0.1 sun umbrella 0.2 sun no umbrella 0.8 Recap: Filtering Elapse time: compute P( Xt | e1:t-1 ) Observe: compute P( Xt | e1:t ) X2 E1 X1 E2 <0.5, 0.5> Belief: <P(rain), P(sun)> <0.82, 0.18> <0.63, 0.37> <0.88, 0.12> Prior on X1 Observe Elapse time Observe Particle Filtering Sometimes |X| is too big to use exact inference |X| may be too big to even store B(X) E.g. X is continuous |X|2 may be too big to do updates Solution: approximate inference Track samples of X, not all values Samples are called particles Time per step is linear in the number of samples But: number needed may be large This is how robot localization works in practice 0.0 0.1 0.0 0.0 0.0 0.2 0.0 0.2 0.52 Representation: Particles Our representation of P(X) is now a list of N particles (samples) Generally, N << |X| Storing map from X to counts would defeat the point P(x) approximated by number of particles with value x So, many x will have P(x) = 0 More particles, more accuracy Initially, all particles have a weight of 1 7 Particles: (3,3) (2,3) (3,3) (3,2) (3,3) (3,2) (2,1) (3,3) (3,3) (2,1) Particle Filtering: Elapse Time Each particle is moved by sampling its next position from the transition model This is like prior sampling – samples’ frequencies reflect the transition probs Here, most samples move clockwise, but some move in another direction or stay in place This captures the passage of time If we have enough samples, close to the exact values before and after (consistent) Particle Filtering: Observe Slightly trickier: Don’t do rejection sampling (why not?) We don’t sample the observation, we fix it As in likelihood weighting, downweight samples based on the evidence: Note that, as before, the probabilities don’t sum to one, since most have been downweighted (in fact they sum to an approximation of P(e)) Particle Filtering: Resample Rather than tracking weighted samples, we resample N times, we choose from our weighted sample distribution (i.e. draw with replacement) This is analogous to renormalizing the distribution Now the update is complete for this time step, continue with the next one Old Particles: (3,3) w=0.1 (2,1) w=0.9 (2,1) w=0.9 (3,1) w=0.4 (3,2) w=0.3 (2,2) w=0.4 (1,1) w=0.4 (3,1) w=0.4 (2,1) w=0.9 (3,2) w=0.3 New Particles: (2,1) w=1 (2,1) w=1 (2,1) w=1 (3,2) w=1 (2,2) w=1 (2,1) w=1 (1,1) w=1 (3,1) w=1 (2,1) w=1 (1,1) w=1 Robot Localization In robot localization: We know the map, but not the robot’s position Observations may be vectors of range finder readings State space and readings are typically continuous (works basically like a very fine grid) and so we cannot store B(X) Particle filtering is a main technique [Demo] P4: Ghostbusters 2.0 (beta) Goal: Find and kill ghosts Agent is blind, but can hear the ghosts’ banging and clanging. Transition Model: All ghosts move randomly, but are sometimes biased Emission Model: Sensor gives a “noisy” distance to each ghost Noisy distance prob True distance = 8 [Demo]3 Dynamic Bayes Nets (DBNs) We want to track multiple variables over time, using multiple sources of evidence Idea: Repeat a fixed Bayes net structure at each time Variables from time t can condition on those from t-1 DBNs with evidence at leaves are (in principle) HMMs G1a E1a E1b G1b E2a E2b G2b G2a t =1 t =2 E3a E3b G3b t =3 G3a Exact Inference in DBNs Variable elimination applies to dynamic Bayes nets Procedure: “unroll” the network for T time steps, then eliminate variables until P(XT|e1:T) is computed Online belief updates: Eliminate all variables from the previous time step; store factors for current time only 14 G1a E1a E1b G1b E2a E2b G2b G2a E3a E3b G3b t =1 t =2 t =3 G3a DBN Particle Filters A particle is a complete sample for a time step Initialize: Generate prior samples for the t=1 Bayes net Example particle: G1a = (3,3) G1b = (5,3) Elapse time: Sample a successor for each particle Example successor: G2a = (2,3) G2b = (6,3) Observe: Weight each entire sample by the likelihood of the evidence conditioned on the sample Likelihood: P(E1a |G1a ) * P(E1b |G1b ) Resample: Select new particles (tuples of values) in proportion to their likelihood SLAM SLAM = Simultaneous Localization And Mapping We do not know the map or our location Our belief state is over maps and positions! Main techniques: Kalman filtering (Gaussian HMMs) and particle methods [DEMOS] DP-SLAM, Ron Parr Outline Particle Filtering (sampling-based inference in HMMs) Dynamic Bayes Nets Viterbi Algorithm 17 HMMs: MLE Queries HMMs defined by States X Observations E Initial distr: Transitions: Emissions: Query: most likely explanation: X5 X2 E1 X1 X3 X4 E2 E3 E4 E5 184 State Path Trellis State trellis: graph of states and transitions over time Each arc represents some transition Each arc has weight Each path is a sequence of states The product of weights on a path is the seq’s probability Can think of the Forward (and now Viterbi) algorithms as computing sums of all paths (best paths) in this graph sun rain sun rain sun rain sun rain 19 Viterbi Algorithm sun rain sun rain sun rain sun rain 20 Example
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