Lecture 10 Let s Make a Deal Incorporating New Information The game there are four lego bricks in a bag the lego bricks are either white or red you get to pull one lego brick out of the bag You get 20 if the brick is red and 0 otherwise How much would you pay to play this game Belief 0 0 2 1 0 2 2 0 2 3 0 2 4 0 2 and E S s 0 00 5 00 10 00 15 00 20 00 so you should pay 10 Assume that a red lego is pulled from the bag and then returned How much money should you now expect to make Posterior belief 0 0 1 0 25 2 0 5 3 0 75 4 1 and E S s 0 00 5 00 10 00 15 00 20 00 so you should pay 15 Make another observation Now a white lego is drawn and returned My previous posterior belief is my new prior belief Posterior belief 0 1 1 0 75 2 0 5 3 0 25 4 0 and E S s 0 00 5 00 10 00 15 00 20 00 so you should pay 10 Bayesian State Estimation Using observations to improve estimates of state probabilities Initial belief 0 0 2 1 0 2 2 0 2 3 0 2 4 0 2 Updated belief after drawing a red brick 0 0 1 0 1 2 0 2 3 0 3 4 0 4 Updated belief after drawing a white brick 0 0 1 0 3 2 0 4 3 0 3 4 0 Markov Model of Transitions Updated state probabilities depend only on prior state probabilities This is an example of a Markov Process the state distribution at time n 1 depends only on the state distribution at time n A Markov process generates a probabilistic sequence of states Check yourself Markov Model Correct answer 4 Let N represent the length of the run that starts at n 0 when the initial state is S What is the expected value of N for p 0 03 Analyzing Markov Models We can calculate iterative starting with Check yourself Markov Models Correct answer 3 The difference equation for is What is Check yourself Representing Markov Models Correct answer The probabilities corresponding to S S S is red blue green Combining Observations and Transitions Discrete time steps 0 1 n Random variables for states at each time Random variables for observations Initial State Distribution Observation Model Transition Model Bayesian Estimation of Robot Location Model the location of a robot as a Markov process Transition model can be deterministic or random Estimate the location of the robot from sonar observations Observations could be perfect or noisy We will model processes that combine observations with transitions as Hidden Markov Models Hidden Markov Models state changes probabilistically but cannot be directly observed However we can make observations that are related tot the state of the system
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