[SOLVED] CMPUT466- Assignment 6

Description

Problem 1

Suppose 90% samples are positive (t=1) and 10% are negative (t=0). Compute P, R, and F1 scores of majority guess (always predicting t=1). If in your derivation the denominator is 0, please compute the limit.

Explain the deficiency of the F1-score in this case, and discuss one possible treatment to resurrect P, R, F1 scores for this problem.

Problem 2.

^{Prove that the sigmoid function}other words, σ(𝑧) =_. 1+exp¹{−𝑧} is symmetric about the point (0,0.5), in

Problem 3. σ(− 𝑧) = 1 − σ(𝑧)

Prove that minimizing the loss                                                                              is equivalent to

minimize the Kullback–Leibler (KL) divergence between𝐽 =− 𝑡 log 𝑦 − (1 −t _and𝑡)ylog, denoted by KL((1 − 𝑦) t _{|| y),  where t=(}

1For two discrete distributions− 𝑡, 𝑡) and y=(1 − 𝑦, 𝑦) are two Bernoulli distributions.P         and Q

= (𝑝1, ···, 𝑝𝐾)                 = (𝑞1, ···, 𝑞𝐾), the KL divergence is defined as

KL(P || Q)= ∑^𝐾 𝑝 log 𝑝_𝑘

Note: KL divergence is not symmetric between𝑘=1 𝑘               𝑞_𝑘                  P and Q. To minimize the KL, Q must cover all the support of P. Thus, the learned distribution may be smoother than it should be.