General comments. Send the solutions to me by email. You need to explain your answers. For example, if the homework asks for a number or for an expression, do not just give a number or an expression without any explanation of how you came up with this number or expression. The purpose of such a question is usually to check your ability to follow the corresponding algorithm, not just to guess the correct answer.
1. (Due August 28) Why cannot we just use machine learning to estimate the degree of confidence in its predictions -- and to estimate how accurate are these predictions?
2. (Due August 28) If we know that the subjective probability of an event is between 0.3 and 0.6, and we want to find this probability with higher accuracy, what shall we do?
3. (Due August 28) How many questions do we need to ask a user to get his/her subjective probability with accuracy 10%?
4. (Due August 28) If 3 experts out of 5 think that it will rain tomorrow, what is the resulting degree of confidence?
5. (Due September 4) List two cases when we do not know the probability distribution of measurement errors, and explain why.
6. (Due September 4) Assuming that utility is proportional to the square root of money amount, would a person prefer $64 without any condition or $144 with probability 0.5?
7. (Due September 4) Why, out of all possible representation of probabilities, the cumulative distribution function (cdf) is the most appropriate for decision making?
8. (Due September 4) If a value is −1 with probability 0.3, 0 with probability 0.5, and 1 with probability 0.2, what is the value F(−0.5) of the corresponding cumulative distribution function? what is the value of F(0.5)?
9. (Due September 4) If a value is −1 with probability between 0.2 and 0.4, 0 with probability between 0.4 and 0.6, and 1 with the remaining probability, what is the interval value F(−0.5) of the corresponding p-box? what is the value of F(0.5)?
10. (Due September 11) Describe what project you would like to do -- if you have not yet done that.
11. (Due September 11) If a value is −1 with probability 0.25, 0 with probability 0.25, and 1 with probability 0.5, then how many binary questions do we have to ask, on average, to find the exact value?
12. (Due September 11) Suppose that a traffic judge fines a driver $200 for not stopping at a stop sign, but if he knows that a person's annual salary is higher than 100K, he raises this fine to $2000. In this situation, for a highly paid person who was caught not stopping -- and whose salary is supposed to be confidential, what is the cost of revealing the first of 6 digits of his salary? What is the cost of revealing the last of these 6 digits?
13. (Due September 11) Suppose that we have 2 models of El Paso weather in August: according to the general desert model, the probability of rain is 0.1, and according to the monsoon season model, it rains with probability 0.6. If it rains, which models will be selected by the Maximum Likelihood method? Which model will be selected if it does not rain?
14. (Due September 11) Suppose that we have three measurements of the same quantity, resulting in values −1, 0, and 1. Use the Least Squares method to come up with the most probable estimate of the actual value of the measured quantity.
15. (Due September 18) If we know that the accuracy of x1 is described by standard deviation σ1 = 0.3, and the accuracy of x2 is described by standard deviation σ2 = 0.2, what is the accuracy of y = x1 + 2x2?
16. (Due October 2) Write and test a program that, based on the given function f(x1, ..., xn), given measurement results x1, ..., xn and given standard deviations σ1, ..., σn, use Monte-Carlo technique to estimate the standard deviation σ of the result of data processing. You can assume that the function f is given. For this (and for other programming assignments), submit the printout of the code and of the result of running this code. Feel free to use any programming language where code is easy to read; for example, Java, C, Python are perfect.
17. (Due October 9) Report what you have done on your project during the work-on-your-project day.
18. (Due October 9) Describe how to use linearization to estimate the uncertainty of the value y = f(x1, x2) = x12 + x22 when the measurement results are x1 = 1 and x2 = 2 and the upper bounds on the uncertainties are Δ1 = 0.1 and Δ2 = 0.2. Use both actual values of the partial derivatives and the values obtained by numerical differentiation.
19. (Due October 16) Write and test a program that, based on the given function f(x1, ..., xn), given measurement results x1, ..., xn and given upper bounds Δ1, ..., Δn on the measurement accuracies, use Cauchy Monte-Carlo technique to estimate the upper bound Δ on the uncertainty of the result of data processing. You can assume that the function f is given. For this (and for other programming assignments), submit the printout of the code and of the result of running this code. Feel free to use any programming language where code is easy to read; for example, Java, C, Python are perfect.
20. (Due October 23) Suppose that y = f(x1, x2) = x12 + x22. Suppose that:
Hint: The above function y = f(x1, x2) is strictly increasing with respect to both inputs xi -- when they are both positive. So, when the inputs xi are located in intervals, this function:
21. (Due November 6) Suppose that our degree of confidence in a statement A is 0.8, in a statement B is 0.7, and in a statement C is 0.6. Suppose that we use min as "and" and max as "or". What is our estimate for the degree of confidence in a composite statement A & (B \/ not C)?
22. (Due November 6) If we have three alternatives, with gains [1, 2], [0, 3], and [−2, −1], which of them are possibly optimal? definitely optimal? Which of the alternatives should we choose if Hurwicz coefficient is αH is 0.2? 0.5? 0.8?
23. (Due November 6) In the classical logic, implication A → B is defined as B \/ not A. Similarly to what we did in class, use linear interpolation to come up with the fuzzy value for f→(0.7, 0.8).
24. (Due November 18) Explain why, in our derivation of the ReLU activation function, we used the possibility to change the measuring unit, but not the possibility to change the starting point.