Problem 1 [25%]
Suppose that I collected data for a group of machine learning students from last year. For each student, I have a feature X1 = hours studied for the class every week, X2 = overall GPA, and Y = whether the student receives an A. We fit a logistic regression model and produce estimated coefficients, 0 = 6,1 = 0.1,2 = 1.0.
- Estimate the probability of getting an A for a student who studies for 40h and has an undergrad GPA of 2.0
- By how much would the student in part 1 need to improve their GPA or adjust time studied to have a 90% chance of getting an A in the class? Is that likely?
Problem 2 [25%]
Consider a classification problem with two classes T (true) and F (false). Then, suppose that you have the following four prediction models:
- T: The classifier predicts T for each instance (always)
- F: The classifier predicts F for each instance (always)
- C: The classifier predicts the correct label always (100% accuracy)
- W: The classifier predicts the wrong label always (0% accuracy)
You also have a test set with 60% instances labeled T and 40% instances labeled F. Now, compute the following statistics for each one of your algorithms:
| Statistic | Cls. T | Cls. F | Cls. C | Cls. W |
| recalltrue positive rate false positive rate true negative rate specificity precision |
Some of the rows above may be the same.
Problem O2 [30%]
This problem can be substituted for Problem 2 above, for up to 5 points extra credit. The better score from problems 2 and O2 will be considered.
Solve Exercise 3.4 in [Bishop, C. M. (2006). Pattern Recognition and Machine Learning].
1
Problem 3 [25%]
In this problem, you will derive the bias-variance decomposition of MSE as described in Eq. (2.7) in ISL. Let f be the true model, f be the estimated model. Consider fixed instance x0 with the label y0 = f(x0). For simplicity, assume that Var[] = 0, in which case the decomposition becomes:
2
f
Eh(y0 (x0))2i = Var[f(x0)]+E[f(x0) f(x0)] .
{z }
| {z } Variance Bias test MSE Prove that this equality holds.
Hints:
- You may find the following decomposition of variance helpful:
- This link could be useful: https://en.wikipedia.org/wiki/Variance#Basic_properties
Problem 4 [25%]
Please help me. I wrote the following code that computes the MSE, bias, and variance for a test point.
| set.seed(1984) population <- data.frame(year=seq(1790,1970,10),pop=c(uspop)) population.train <- population[1:nrow(population) 1,] population.test <- population[nrow(population),]E <- c() # prediction errors of the different models for(i in 1:10){ pop.lm <- lm(pop ~ year, data = dplyr::sample_n(population.train, 8)) e <- predict(pop.lm, population.test) population.test$pop E <- c(E,e)} cat(glue::glue(MSE: {mean(E^2)} , Bias^2: {mean(E)^2} ,Var: {var(E)} ,Bias^2+Var: {mean(E)^2 + var(E)})) |
## MSE: 2869.61343086216
## Bias^2: 2681.61281912074
## Var: 208.889568601581
## Bias^2+Var: 2890.50238772232
I expected that the MSE would be equal to Bias2 + Variance, but that does not seem to be the case. The MSE is 2402.515 and Bias2 + Variance is 2428.706. Was my assumption wrong or is there a bug in my code? Is it a problem that I am computing the expectation only over 10 trials?
Hint: If you are using Python and need help with this problem, please come to see me (Marek).
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