Document Sample

CS 188: Artificial Intelligence Fall 2009 Lecture 20: Particle Filtering 11/5/2009 Dan Klein – UC Berkeley Announcements Written 3 out: due 10/12 Project 4 out: due 10/19 Written 4 probably axed, Project 5 moving up Course contest update Daily tournaments are now being run! Instructions for registration on the website Qualifying opens soon: 1% on final exam! 2 Recap: Reasoning Over Time Stationary Markov models 0.3 0.7 X1 X2 X3 X4 rain sun 0.7 0.3 Hidden Markov models X E P X1 X2 X3 X4 X5 rain umbrella 0.9 rain no umbrella 0.1 sun umbrella 0.2 E1 E2 E3 E4 E5 sun no umbrella 0.8 Recap: Filtering Elapse time: compute P( Xt | e1:t-1 ) Observe: compute P( Xt | e1:t ) Belief: <P(rain), P(sun)> X1 X2 <0.5, 0.5> Prior on X1 <0.82, 0.18> Observe E1 E2 <0.63, 0.37> Elapse time <0.88, 0.12> Observe Particle Filtering Filtering: approximate solution 0.0 0.1 0.0 Sometimes |X| is too big to use exact inference 0.0 0.0 0.2 |X| may be too big to even store B(X) E.g. X is continuous 0.0 0.2 0.5 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 In memory: list of particles, not states This is how robot localization works in practice 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: We don’t sample the observation, we fix it This is similar to likelihood weighting, so we downweight our 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 equivalent to renormalizing the distribution Now the update is complete for this time step, continue with the next one Particle Filtering Sometimes |X| is too big to use exact inference 0.0 0.1 0.0 |X| may be too big to even store B(X) E.g. X is continuous 0.0 0.0 0.2 |X|2 may be too big to do updates 0.0 0.2 0.5 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 In memory: list of particles, not states This is how robot localization works in practice 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 Particles: (3,3) So, many x will have P(x) = 0! (2,3) More particles, more accuracy (3,3) (3,2) (3,3) For now, all particles have a (3,2) (2,1) weight of 1 (3,3) (3,3) (2,1) 10 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 This is similar to likelihood weighting, so we downweight our 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 Old Particles: Rather than tracking (3,3) w=0.1 weighted samples, (2,1) w=0.9 we resample (2,1) w=0.9 (3,1) w=0.4 (3,2) w=0.3 N times, we choose (2,2) w=0.4 from our weighted (1,1) w=0.4 sample distribution (3,1) w=0.4 (2,1) w=0.9 (i.e. draw with (3,2) w=0.3 replacement) Old Particles: This is equivalent to (2,1) w=1 renormalizing the (2,1) w=1 distribution (2,1) w=1 (3,2) w=1 (2,2) w=1 Now the update is (2,1) w=1 complete for this time (1,1) w=1 (3,1) w=1 step, continue with (2,1) w=1 the next one (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 [Demos] P4: Ghostbusters 2.0 (beta) Noisy distance prob Plot: Pacman's grandfather, Grandpac, True distance = 8 learned to hunt ghosts for sport. 15 13 He was blinded by his power, but could hear the ghosts’ banging and clanging. 11 9 Transition Model: All ghosts move 7 randomly, but are sometimes biased 5 Emission Model: Pacman knows a 3 “noisy” distance to each ghost 1 [Demo] 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 t =1 t =2 t =3 G1 a G2 a G3 a G1 b G2 b G3 b E1a E1b E2a E2b E3a E3b Discrete valued dynamic Bayes nets are also HMMs 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 t =1 t =2 t =3 G1 a G2 a G3 a G1 b G2 b G3 b E1a E1b E2a E2b E3a E3b Online belief updates: Eliminate all variables from the 17 previous time step; store factors for current time only 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 prior samples (tuples of values) in proportion to their likelihood 18 [Demo] 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 Best Explanation Queries X1 X2 X3 X4 X5 E1 E2 E3 E4 E5 Query: most likely seq: 20 State Path Trellis State trellis: graph of states and transitions over time sun sun sun sun rain rain rain rain 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 21 Viterbi Algorithm sun sun sun sun rain rain rain rain 22 Example 23

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 1/24/2012 |

language: | English |

pages: | 23 |

OTHER DOCS BY huanghengdong

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.