A Tractable Approach to Probabilistically Accurate Mode Estimation

1. A Tractable Approach to Probabilistically Accurate Mode Estimation Oliver B. Martin Seung H. Chung Brian C. Williams i-SAIRAS 2005 Munich, Germany

2. Valve Stuck open 0.01 Open 0. 01 Open Close 0. 01 Stuck closed Closed 0.01 inflow = outflow = 0

3. Estimation of PCCA(Probabilistic Concurrent Constraint Automata) • Belief State Evolution visualized with a Trellis Diagram • Complete history knowledge is captured in a single belief state by exploiting the Markov property • Belief states are computed using the HMM belief state update equations Compute the belief state for each estimation cycle stuck-open closed open 0.7 0.2 0.1 cmd=openobs=flow cmd=closedobs=no-flow

4. t t+1 Hidden Markov ModelBelief State Update Equations • Propagation step computes aprior probability • Update step computes posterior probability

5. Approximations to PCCA Estimation • k most likely estimates – The belief state can be accurately approximated by maintain the k most likely estimates • Most likely trajectory– The probability of each state can be accurately approximated by the most likely trajectory to that state • 1 or 0 observation probability– The observation probability can be reduced to 1.0 for observations consistent with the state, and 0.0 for observations inconsistent with the state t+1 1.0 or 0.0

6. Livingstone Williams and Nayak, AAAI-96 Livingstone 2 Kurien and Nayak, AAAI-00 Titan Williams et al., IEEE ’03 The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to 1.0 for observations consistent with the state, 0.0 for observations inconsistent with the state 0.50 0.15 0.35 Previous Estimation Approach:Best-First Trajectory Estimation (BFTE) Approximations to Estimation Deep Space One Earth Observing One 0.4 0.2

7. Diagnosis as Approximate Belief State Enumeration for Probabilistic Concurrent Constraint Automata Martin et al., AAAI-05 Increased estimator accuracy by maintaining a compact belief state encoding Minimal overhead over BFTE The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to 1.0 for observations consistent with the state, 0.0 for observations inconsistent with the state 0.65 0.05 0.30 Previous Estimation Approach: Best-First Belief State Enumeration (BFBSE) Approximations to Estimation 0.4 0.2

8. Increased estimator accuracy by properly computing the observation probability Minimal overhead over BFBSE The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to 1.0 for observations consistent with the state, 0.0 for observations inconsistent with the state 0.65 0.30 0.05 New Approach: Best-First Belief State Update (BFBSU) Approximations to Estimation 0.4 0.2

9. Working State Estimate Probability 1 BFBSE 0.8 BFBSU 0.6 State estimated probability 0.4 0.2 0 0 50 100 150 200 250 300 350 400 450 500 Estimation Timestep 0.990 0.001 0 0.995 0.991 0.990 0.010 0.999 1 0.005 0.009 0.010 BFBSE vs. BFBSU: The Consequence ofIncorrect Observation Probability 0.99 Sensor = High Working 0.01 Broken 1 Best-First Belief State Estimation Best-First Belief State Update Assume: P(Sensor = High | Broken) = 1 P(Sensor = High | Broken) = 0.5 1 1 0 0 Observe (Sensor = High) at every time step

10. BSBSU ComputesHMM Belief State Update Equations • Propagation Step • Update Step • Problem: Computing the observation probability for the “otherwise” case can be expensive. • Must compute totalnumber of consistentobservations for k states • Model counting for kproblems

11. Solution:Pre-Compute Observation Probability Rules • Observation Probability Rule (OPR): • Compute OPR for a set of partial states • Compute OPRs using the dissents • Ignore all OPRs with probability of 0 or 1

12. Solve Mode Estimation as anOptimal Constraint Satisfaction Problem • UseConflict-directed A* with a tight admissible heuristic • Use greedy approximation for the cost to go (BFBSE) • Include observation probability within the heuristic (BFBSU)

13. Single-point Failure state estimate probability Nominal state estimate probability ST7A Model Results:State Estimate Accuracy

14. Space Performance Best case: n·b Worst case: bn Time Performance Best case: n2·b·k + n·b·C Worst case: BFBSE: bn(n·k + C) BFBSU: bn(n·k + bn·R + C) ST7A Model Results:State Estimate Performance Number of initial states, k Number of initial states, k

15. EO1 Model Results:State Estimate Performance • Space Performance • Time Performance

16. Conclusion • Best-First Belief State Update • Compute k most likely estimates • Remove most likely trajectory approximation by computing the most likely belief state estimates • Remove 1 or 0 observation probability approximation by computing the proper observation probabilities • Increased PCCA estimator accuracy by computing the Optimal Constraint Satisfaction Problem (OCSP) utility function directly from the HMM propagation and update equations • Maintaining the computational efficiency.