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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) ICASSP 2013 tutorial1

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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 ICASSP 2013 tutorial2

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Hidden Markov Models I ICASSP 2013 tutorial3 Uncertainty and Time Hidden p( | ) Observed Model P( | ) tt-1 Transition Probs t Emission Probs MODEL Observations Hidden State (single discrete variable)

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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 ICASSP 2013 tutorial4 Uncertainty and Time

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Kalman Filter Linear Gaussian conditional distributions represent state and sensor models LG: P(x/y)=N(a y y + b y, σ y )(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 ICASSP 2013 tutorial5

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