Probability theory provides a framework for modeling and reasoning about uncertainty making precise statements about uncertain situations and drawing reliable inferences from unreliable observations Probability theory provides a framework for designing systems that are robust to uncertainty Lecture 9 Module Three Probabilistic Reasoning Topics Probability Bayes theorem Markov Processes Bayesian Inference Lab exercises Localization and mapping Probability Theory Probability Theory Events Probabilities are assigned to events which are possible outcomes of an experiment Example ip three coins in succession There are eight atomic events HHH HHT HTH HTT THH THT TTH TTT Atomic events are mutually exclusive Set of all atomic events is collectively exhaustive cover all cases Set of all possible atomic events is called the sample space U Probability Theory Axioms of Probability The probabilities that are assigned to events must obey three axioms Non negativity Pr A 0 for all events A Scaling Pr U 1 Additivity if A B is empty Pr A B Pr A Pr B Conditional Probability Composing probabilities sequentially using decision tress Example Assume the probability of rain on a given day is 0 1 However if it rains today the probability of rain tomorrow is 0 15 Similarly if it does not rain today the probability of rain tomorrow is 0 05 Check Yourself Conditional Probability Correct answer Oscar has lost his dog in either forest A with a prior probability 0 4 or in forest B with a prior probability 0 6 If the dog is in forest A and Oscar spends a day searching for it in forest A the conditional probability that he will nd the dog that day is 0 25 Similarly if the dog is in forest B and Oscar spends the day looking for it there he will nd the dog that day with probability 0 15 In which forest should Oscar look to maximize the probability that he will nd the dog on the rst day of the search Oscar should look in forest A Check Yourself Conditional Probability Correct answer In which forest should Oscar look to maximize the probability that he will nd the god on the second day of the search Oscar should look in Bayes Rule Decision trees are sequential but set representation is symmetric We can compute the probability of intersections two ways Anti sequential reasoning what is the probability that it rained yesterday given that it rained today Bayes Rule Oral AIDS Test OraSure OMT This test is performed by collecting a sample from your mouth and sending it to the lab for processing The OraSure test has 98 6 sensitivity and 97 7 speci city Sensitivity Pr posTest AIDS 0 986 Speci city Pr negTest noAIDS 977 Check Yourself Bayes Rule Correct answer Given that the patient tests positive what is the probability that the patient has AIDS Random Variables Joint Probability Distributions Reducing Dimensionality A random variable is the probabilistic analog of a deterministic variable While the value of a deterministic variable is a number the value of a random variable is drawn from a distribution Example Let X represent the result of the toss of a die Now we can write Pr X 3 1 6 Probability laws for multi dimensional sample spaces are given by joint probability distributions Let V represent the toss of the rst die and W represent the toss of the second die Pr V W represents the joint probability distribution Pr v w represents the Pr V v and W w The dimensionality of a joint probability distribution can be reduced in two very different ways Marginalizing refers to collapsing one or more dimensions by summing over all possible outcomes along those dimensions Conditioning refers to collapsing dimensions by accounting for new information that restricts outcomes The expected value of a random variable is the weighted sum of all possible values with each value weighted by it probability Example let X represent the result of tossing one fair six sided die Expectation Let s Make a Deal 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 Incorporating New Information 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 Estimating the State of Dynamic System What if the system changes with time Even worse what if it changes probabilistically The new game four white lego bricks are put into a bag behind your back the following process is repeated three times A random brick is removed from the bag a replacement brick that is equally likely to be red or white is added to the bag You 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 Updated state probabilities depend only on prior state probabilities This process is Markov the state distribution at time t depends only on the state distribution at t 1 Check yourself Markov Model of Transitions Correct answer Which of the following processes will generate the transition model shown above Combining Observations and Transitions New game Four lego bricks of unknown color are put into a bag Behind your back the following process is repeated three times You pull out one brick observe its color and replace it a random brick is removed from the bag a replacement brick that is equally likely to be red or white is added to the bag You 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 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 …
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