I have attached a pdf file with the questions and I want all the questions answered.Microsoft Word -...

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I have attached a pdf file with the questions and I want all the questions answered.

Microsoft Word - exam.pdf 1. We are interested in computing whether the event � is more or less probable than event ¬x given an evidence variable Y, i.e., whether P(X = x|Y) is greater than P(X = ¬x|Y). Given the Bayes rule, what is the minimum number of distinct probabilities that we need to have access to in order to answer this question and which ones? 2. Given access to a robot’s motion model and prior belief distribution, how can we compute the probability P(x|u) where x is the latest robot’s state and u is the control applied at a previous state x’? 3. Consider a robot in a square room that is discretized into a map of 16 cells as shown in the image below (left side). Assume that initially the robot is at cell (2,3) with probability 50% or at cell (3,3) with prob. 50%, where rows are indicated first and columns are indicated second. The robot moves and the encoders and compass indicate that we have moved one column to the right. The motion model for such a motion is provided in the image below (right side). The robot is equipped with a range sensor that always points up and measures how many cells away from the end of the grid the robot is. For instance, from cell (1,3) the correct measurement is 3 but from cell (4,2) the correct measurement is 0. The observation model indicates that 80% of the time we get the correct measurement x, 10% of the time we get x − 1 and 10% of the time we get x + 1. The measurement that we get after the robot moves is 1. a) Provide the predictive belief distribution over the map after the motion model is considered given the above prior belief. b) Provide the updated belief distribution (normalized) over the map after the measurement is also taken into account. For the normalization, you can indicate the corresponding fractions. 5. If the robot’s state space is n-dimensional, how many numbers do we need to keep track of to represent a Gaussian distribution in the robot’s belief space in the general case? 6. What are the two main limitations of the basic Kalman filter? 7. The Extended Kalman Filter (EKF) addresses one of the limitations of the basic Kalman filter. Which one? What is the main idea behind the EKF so as to generalize the Kalman filter?

questions-q4jozn0a.pdf

Pawan · Accepted Answer

Solution 1:
The probability P(x|Y) can be computed using bayes rule given by
 
Similarly the probability for P(|Y) is given by
Therefore we need only probabilities to compare the two probabilities.
Solution 2:
P(x|u) can be computed using the bayes rule given by
Therefore, we require the probability P(u|x) that is the probability of the control u given the state x of the robot, the probability of x, and the probability of the state u.
P(u) can be calculated using the prior and likelihood of the control u for given states .
Solution 3:
Part a) The predictive belief of the map can be computed by taking the inner product of the motion map, which is the probability of the motion in all directions irrespective of the position of the robot
The prior of the map is given by the table where each cell corresponds to the position in the map
	0
	0
	0
	0
	0
	0
	0.5
	0
	0
	0
	0.5
	0
	0
	0
	0
	0
Taking the inner product of the map with the motion model, we get
	0
	0
	0.1
	0
	0
	0
	0.15
	0.25
	0
	0
	0.15
	0.25
	0
	0
	0.1
	0
Part b)
The measurement gives the correct value with probability P(x) = 0.8 and P(x-1) = P(x+1) = 0.1
Now, for a cell for example (3, 3), we calculate the poisterior by
Since we know the value of measurement is 1, the final map is
	0
	0
	0.027
	0
	0
	0
	0.324
	0.54
	0
	0
	0.04
	0.067
	0
	0
	0
	0
Solution 4:
NA
Solution 5:
The Gaussian distribution is defined by the mean and the variance of the samples. Therefore for the n-dimensional state-space model we only need to track two variables i.e. the mean and variance.
Solution 6:
The two main limitations of the basic Kalman filter are
1. The Kalman filter assumes a linear model for the system dynamics and the observation processes.
2. A normal distribution is assumed for both the system dynamics and measurements of the process.
Solution 7:
The Extended Kalman filter (EKM) can be implemented for non-linear models. It linearizes the model about the current estimate of mean and covariance. 
Solution 8:
The mean from the q distribution can be calculated by taking samples from the distribution and using the formula
Similarly the expectation can be evaluated using 
For an unknown distribution,

Microsoft Word - exam.pdf 1. We are interested in computing whether the event � is more or less probable than event ¬x given an evidence variable Y, i.e., whether P(X = x|Y) is greater than P(X =...

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