Introduction to Artificial Intelligence (AI)

Transcription

1 Introduction to Artificial Intelligence (AI) Computer Science cpsc502, Lecture 10 Oct, 13, 2011 CPSC 502, Lecture 10 Slide 1

2 Today Oct 13 Inference in HMMs More on Robot Localization CPSC 502, Lecture 10 2

3 R&Rsys we'll cover in this course Problem Static Deterministic Arc Consistency SLS Constraint Vars + Satisfaction Search Constraints Query Sequential Planning Representation Reasoning Technique Logics Search STRIPS Search Environment Stochastic Belief Nets Var. Elimination Approx. Inference Temporal. Inference Decision Nets Var. Elimination Markov Processes Value Iteration CPSC 502, Lecture 10 Slide 3

4 How can we minimally extend Markov Chains? Maintaining the Markov and stationary assumption A useful situation to model is the one in which: the reasoning system does not have access to the states but can make observations that give some information about the current state CPSC 502, Lecture 10 Slide 4

5 Hidden Markov Model A Hidden Markov Model (HMM) starts with a Markov chain, and adds a noisy observation about the state at each time step: domain(s) = k domain(o) = h P (S 0 ) specifies initial conditions P (S t+1 S t ) specifies the dynamics P (O t S t ) specifies the sensor model CPSC 502, Lecture 10 Slide 5

6 xample: Localization for Pushed around Robot Localization (where am I?) is a fundamental problem in robotics Suppose a robot is in a circular corridor with 16 locations There are four doors at positions: 2, 4, 7, 11 The Robot initially doesn t know where it is The Robot is pushed around. After a push it can stay in the same location, move left or right. The Robot has Noisy sensor telling whether it is in front of a door CPSC 502, Lecture 10 Slide 6

7 This scenario can be represented as Example Stochastic Dynamics: when pushed, it stays in the same location p=0.2, moves left or right with equal probability P(Loc t + 1 Loc t ) P(Loc 0 ) CPSC 502, Lecture 10 Slide 7

8 This scenario can be represented as Example of Noisy sensor telling whether it is in front of a door. If it is in front of a door P(O t = T) =.8 If not in front of a door P(O t = T) =.1 P(O t Loc t ) CPSC 502, Lecture 10 Slide 8

9 Useful inference in HMMs Localization: Robot starts at an unknown location and it is pushed around times. It wants to determine where it is In general (Filtering): compute the posterior distribution over the current state given all evidence to date P(X t o 0:t ) or P(X t e 0:t ) CPSC 502, Lecture 10 Slide 9

10 Another very useful HMM Inference Most Likely Sequence (given the evidence seen so far) argmax x 0:t P(X 0:t e 0:t ) CPSC 502, Lecture 10 Slide 10

11 HMM: more agile terminology Formal Specification as five-tuple S { s1,..., sn} K { o1,..., om} {1,..., M} { i}, i S A { aij}, i, j S B { bi( ot)}, i S, o t K N j 1 M t 1 aij ( bi ot 1 ) 1 S, K,, A, Set of States B Output Alphabet Initial State Probabilities State Transition Probabilities Symbol Emission Probabilities CPSC 502, Lecture 10 11

12 1. Initialization The forward procedure t( i) P( o1o2... ot, Xt i) ( i) ibi( o1), 1 1 i N 2. Induction t ( j) N t i 1 1( i) aijbj( ot), 1 i N, 1 j N Complexity CPSC 502, Lecture 10 12

13 1( i) ibi( o1) t( j) N ( i) a ( o t 1 ij j t i 1 CPSC 502, Lecture 10 Slide 13 b )

14 Finding the Best State Sequence : arg max P( X X O) The Viterbi Algorithm: Initialization: v 1 (j) = j b j (o 1 ), 1 j N Induction: v t (j) = max 1 i N v t-1 (i) a ij b j (o t ), 1 j N Store backtrace: v t ( j ) : probability of the most probable path that leads to that node bt t ( j ) = argmax 1 i N v t-1 (i) a ij b j (o t ), 1 j N Termination and path readout: P(X O) = max 1 i N v t (i) To recover X start from bt t ( i ) = argmax 1 i N v t (i) and keep going backward CPSC 502, Lecture 10 14

15 Robot Localization: More complex Example Suppose a robot wants to determine its location based on its actions and its sensor readings Three actions: goright, goleft, Stay This can be represented by an augmented HMM CPSC 502, Lecture 10 Slide 15

16 Robot Localization Sensor and Dynamics Model Sample Sensor Model (assume same as for pushed around) Sample Stochastic Dynamics: P(Loc t + 1 Action t, Loc t ) P(Loc t + 1 = L Action t = goright, Loc t = L) = 0.1 P(Loc t + 1 = L+1 Action t = goright, Loc t = L) = 0.8 P(Loc t + 1 = L + 2 Action t = goright, Loc t = L) = P(Loc t + 1 = L Action t = goright, Loc t = L) = for all other locations L All location arithmetic is modulo 16 The action goleft works the same but to the left CPSC 502, Lecture 10 Slide 16

17 Dynamics Model More Details Sample Stochastic Dynamics: P(Loc t + 1 Action, Loc t ) P(Loc t + 1 = L Action t = goright, Loc t = L) = 0.1 P(Loc t + 1 = L+1 Action t = goright, Loc t = L) = 0.8 P(Loc t + 1 = L + 2 Action t = goright, Loc t = L) = P(Loc t + 1 = L Action t = goright, Loc t = L) = for all other locations L CPSC 502, Lecture 10 Slide 17

18 Robot Localization additional sensor Additional Light Sensor: there is light coming through an opening at location 10 P (L t Loc t ) Info from the two sensors is combined : Sensor Fusion CPSC 502, Lecture 10 Slide 18

19 The Robot starts at an unknown location and must determine where it is The model appears to be too ambiguous Sensors are too noisy Dynamics are too stochastic to infer anything But inference actually works pretty well. check: alization/localization.html It uses a generalized form of filtering (not just a sequence of observations, but a sequence of observation pairs (from the two sensors) + the actions CPSC 502, Lecture 10 Slide 19

20 HMMs have many other applications. Natural Language Processing: e.g., Speech Recognition States: phoneme \ word Observations: acoustic signal \ phoneme Bioinformatics: Gene Finding States: coding / non-coding region Observations: DNA Sequences For these problems the critical inference is: find the most likely sequence of states given a sequence of observations CPSC 502, Lecture 10 Slide 20

21 TODO for next Tue Start Reading Chp 9 of textbook (up tp 9.4 included) Work on assignment 2 CPSC 502, Lecture 10 Slide 21

22 Sample scenario to explore in demo Keep making observations without moving. What happens? Then keep moving without making observations. What happens? Assume you are at a certain position alternate moves and observations. CPSC 502, Lecture 10 Slide 22

23 NEED to explain Filtering Because it will be used in POMDPs CPSC 502, Lecture 10 Slide 23

24 Markov Models Markov Chains Simplest Possible Dynamic Bnet Hidden Markov Model Add noisy Observations about the state at time t Add Actions and Values (Rewards) Markov Decision Processes (MDPs) CPSC 502, Lecture 10 Slide 24

25 Answering Query under Uncertainty Probability Theory Static Belief Network & Variable Elimination Monitoring (e.g credit cards) Dynamic Bayesian Network Hidden Markov Models BioInformatics Markov Chains Student Tracing in tutoring Systems Diagnostic Systems (e.g., medicine) spam filters Natural Language Processing CPSC 502, Lecture 10 Slide 25

26 Lecture Overview Recap Temporal Probabilistic Models Start Markov Models Markov Chain Markov Chains in Natural Language Processing CPSC 502, Lecture 10 Slide 26

27 Sampling a discrete probability distribution CPSC 502, Lecture 10 Slide 27

28 Answering Query under Uncertainty Probability Theory Static Belief Network & Variable Elimination Monitoring (e.g credit cards) Dynamic Bayesian Network Hidden Markov Models BioInformatics Markov Chains Student Tracing in tutoring Systems Diagnostic Systems (e.g., medicine) spam filters Natural Language Processing CPSC 502, Lecture 10 Slide 28

29 Answering Queries under Uncertainty Probability Theory Static Belief Network & Variable Elimination Monitoring (e.g credit cards) Diagnostic Systems (e.g., medicine) spam filters Dynamic Bayesian Network Hidden Markov Models BioInformatics Natural Language Processing Robotics Markov Chains Student Tracing in tutoring Systems CPSC 502, Lecture 10 Slide 29

30 Stationary Markov Chain (SMC) A stationary Markov Chain : for all t >0 P (S t+1 S 0,,S t ) = P (S t +1 We only need to specify Simple Model, easy to specify Often the natural model and The network can extend indefinitely and Variations of SMC are at the core of most Natural Language Processing (NLP) applications! CPSC 502, Lecture 10 Slide 30