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4_23_09_KalmanFiltering

# 4_23_09_KalmanFiltering - Optimal Probabilistic Adaptation...

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Optimal Probabilistic Adaptation

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When flying from the outside of a canyon into...
… the corridors of the canyon…

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… the statistics of the environment change.
Optimal Probabilistic Adaptation

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Kalman- filtering Adaptation

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The goal is to update the estimate of the environment Λ k at time k to have the prior probability distribution P( I | Λ k ) of inputs I . For this, we use responses measured in the past Ř k = { R k , R k-1 , … , R 0 }. Responses are with adaptation states Ă k = { A k , A k-1 , … , A 0 }, where P( R | I , A k ) is the likelihood function.
Present Environment History of Response P Λ k R k , A k ( ) = We want to estimate History of Adaptation

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Neural Response ( r ) Sensory Input (s) a b c Bayes Theorem P s | r ( ) = P ( r | s ) P s ( ) P r ( ) b a + b = b b + c b + c a + b + c a + b a + b + c P r ( ) = a + b a + b + c P s ( ) = b + c a + b + c P r | s ( ) = b b + c b a + b P s | r ( ) =
Neural Response 1 (R1) Sensory Input (I) a b c Generalization of Bayes Theorem P I | R 1 , R 2 ( ) = P ( R 1 | I , R 2 ) P I | R 2 ( ) P R 1 | R 2 ( ) Neural Response 2 (R2) d A B C

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Present Environment History of Response P Λ k R k , A k ( ) = We want to estimate History of Adaptation KP R k , A k Λ k , R k -1 , A k 1 ( ) P Λ k R k -1 , A k 1 ( ) Measurement Term Prediction Term
Kalman Adaptation Early Measurements Estimated Environment Predicted Environment New Measurements New Estimation: Clean Measurements New Estimation: Noisy Measurements

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How do we estimate this function?
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4_23_09_KalmanFiltering - Optimal Probabilistic Adaptation...

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