A Continuous tracking Back to our example: Know 1. Model A, B each each t 2. Observation t=17 t=18 t=19 t=20 Can do kalman filterin as before Se A(ok ( ok) A(ok) A(ok) B(ok) B(ok) B(ok) B(ok) a(failed) ok) q(ok) q(ok) q(ok) BAiled) ∧N Kalman filter: @ C(D P(x, d., a,za, z,)=N(o, C) Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16412/6.834 Lecture,15 March2004 Probability of a mode sequence(1/2) Challenge: Can we update p(d.,az.a, z,) efficiently? p(d1…d-1=0k.0k|a121…a1z1-) ok p(d1…d,=ok… ok failed 1, Prediction p(d1…d, d .d P(d, d.-,a, z independence a121…,a1z1)p(dn|d1) Discrete discrete transition probabilityHybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 19 Continuous tracking Back to our example: ok ok ok ok failed A(ok) B(ok) q(ok) A( failed) B( failed) q( failed) A(ok) B(ok) q(ok) A(ok) B(ok) q(ok) A(ok) B(ok) q(ok) t=17 t=18 t=19 … Know: 1. Model A,B each each t 2. Observations 3. Actions Ÿ Can do Kalman filtering as before ( ) ( ) ˆ , i t i xt C Sequence (i) Kalman filter: ( | ... , ... ) (ˆ , ) ( ) ( ) 1 1 1 i t i p xt d dt a z atzt N xt C ( ) ˆ i t x (i) Ct ( ) 1 ˆ i t x  t=20 Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 20 Probability of a mode sequence (1/2) Challenge: Can we update efficiently? ( ... | ... ) 1 t 1 1 t t p d d a z a z a1 b1 Z1 x1 b2 Z2 x2 ( | , ) t 1 t p x a x  Actions Beliefs Observations Continuous states Observable Hidde n ( | ) t t p z x d1 d2 Discrete states ( | ) t 1 t p d d  ok ok ok ok failed ( ... ... | ... ) 1 t1 1 1 t1 t1 p d d ok ok a z a z ( ... ... | ... ) 1 t 1 1 t t p d d ok ok failed a z a z 1. Prediction: ( | ... , ... ) ( ... | ... ) ( ... | ... ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1         t t t t t t t t t t p d d d a z a z p d d a z a z p d d a z a z ( ... | ... ) ( | ) 1 t1 1 1 t1 t1 p dt dt1 p d d a z a z conditional probability independence discrete transition probability