Modeling Uncertainty over time
• Time series of snapshot of the world “state” we are
interested represented as a set of random variables
(RVs)
– Observable
– Hidden
• Stationary process (not static)
• Markovian Property (current state depends only on
finite history – typically just previous time slice)
• Transition Model P(current state/previous state)
• Sensor/Observation Model P(evidence/current state)
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Inference tasks in temporal models
• Filtering: posterior distribution over current state given
evidence = likelihood of evidence
• Prediction: posterior distribution of future state given
evidence to date
• Smoothing: posterior distribution of past state given all
evidence up to the present
• Most likely explanation: given sequence of
observations, most likely sequence of states that has
generated them
• EM-algorithm
– Estimate what transitions occurred and what states
generated the sensor reading and update models
– Updated models provide new estimates and process
iterated until convergence
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Uncertainty and Time
Hidden Markov Models I
p(
|
)
Emission Probs
t
P(
MODEL
Model
) Transition Probs
|
t
t-1
Observations
Hidden
Hidden State
(single
discrete
Observed
variable)
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Uncertainty and Time
Kalman Filtering
• Streams of noisy input data
• Basic idea t->t+1 :
–
–
–
–
–
Prior knowledge of state
Prediction step (based on some model)
Update step (compare prediction to measurements)
Readjust model
Output estimate of state
• Statistically optimal estimate of system state
• Particle filters are another approach
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Kalman Filter
• Linear Gaussian conditional distributions
represent state and sensor models
• LG: P(x/y)=N(ay y + by, σy)(c)
• Next state is linear function of current state plus
some Gaussian noise i.e constant dx/dt
• Forward step: mean + covariance matrix at t
produces mean + covariance matrix at t+1
• Trade-off between observation reliability and
model reliability
• Variants to relax strong assumptions: switching,
extended
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