PS#8

Q1.6 5 Points A teacher wants to evaluate high school students based on their Math and English grades. Students are classified as either successful(group1) or not-successful(group2) in their college applications. The teacher has data on 20 students as following: Logistic regression  PCA  LDA  None of above 

Our data has three boolean input variables a,b,c and a single boolean output K as shown below: According to the Naive Bayes classifier, what is 0.1875 Q2.3 10 Points Assume three word types a,b,c. Consider a Naive Bayes model with the following conditional probability table: Giving a new sample Compute the probability of predicting x as positive. (Please write detailed steps rather than just the result) Q2.3.pdf  Download P ( a = 1 ∧ b = 0 ∧ c = 1∣ K = 1) x = (1, 0, 1) 

If we consider all M-order interaction as features giving original X having N attributes: , what is the computational complexity in general to naively compute feature maps? Q3.2 7 Points Giving and are valid kernels, select all valid kernels from : ( x , x , ..., x ) 1 2 n  O ( N ) 2  O ( M ) 2  O ( N ) M  O ( M ) N K 1 K 2  K ( x , z ) := K ( x , z ) + 1 K ( x , z ) 2 K ( x , z ) := K ( x , z ) − 1 K ( x , z ) 2 (a>0)  K ( x , z ) := aK ( x , z ) 1 (b<0) K ( x , z ) := bK ( x , z ) 1 Note that the matrix K is the Hadamard product (or element-by-element product) of K1 and K2.  K ( x , z ) := K ( x , z ) K ( x , z ) 1 2 p(x): a polynomial function with positive coefficients  K ( x , z ) := p ( K ( x , z )) 1 ( a real valued function) K ( x , z ) := f ( x ) f ( z ) f : R ⟶ n R . ( : another valid kernel over  K ( x , z ) := K ( ϕ ( x ), ϕ ( z )) 3 K 3 R ⟶ d R d

128 / 135 pts QUESTION 1 Gaussian Discriminant Analysis 60 / 60 pts 1.1 (no title) 5 / 5 pts 1.2 (no title) 5 / 5 pts 1.3 (no title) 5 / 5 pts 1.4 (no title) 5 / 5 pts 1.5 (no title) 5 / 5 pts 1.6 (no title) 5 / 5 pts 1.7 (no title) 10 / 10 pts 1.8 (no title) 10 / 10 pts 1.9 (no title) 10 / 10 pts QUESTION 2 Naive Bayes Model 40 / 40 pts 2.1 (no title) 5 / 5 pts 2.2 (no title) 10 / 10 pts 2.3 (no title) 10 / 10 pts 2.4 (no title) 5 / 5 pts 2.5 (no title) 5 / 5 pts 2.6 (no title) 5 / 5 pts QUESTION 3 Kernel Methods (from PS6) 28 / 35 pts 3.1 (no title) 4 / 4 pts 3.2 (no title) 0 / 7 pts 3.3 (no title) 10 / 10 pts 3.4 (no title) 7 / 7 pts 3.5 (no title) 7 / 7 pts