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, β^ˆ₀ = −6,β^ˆ₁ = −0.1,β^ˆ₂ = 1.0.

1. Estimate the probability of getting an A for a student who studies for 40h and has an undergrad GPA of 2.0
2. 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^ˆ 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

E^h(y₀ ^ˆ(x₀))²ⁱ = Var[f^ˆ(x₀)]+_E[f(x₀) − f^ˆ(x₀)]_ .

{z }

| {z } Variance Bias test MSE Prove that this equality holds.

Hints:

1. You may find the following decomposition of variance helpful:
2. 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 Biasˆ2 + Variance, but that does not seem to be the case. The MSE is 2402.515 and Biasˆ2 + 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).