Hypothesis Testing and Bayes Factor

Question 1

What is the null/alternative hypothesis in a one sided test?

Answer 1

H0 : θ ≤ θ0

H1 : θ > θ1

Question 2

What is the null/alternative hypothesis in an at least one simple hypothesis test?

Answer 2

H0 : θ = θ0

H1 : θ ≠ θ0

Question 3

How do you perform a one sided test?

Answer 3

Compute the two quantities:

1. P(H0|x1,…,xn) = integral(-inf to θ0) π(θ|x1,…, xn) dθ
2. P(H1|x1,…xn) = integral (θ1 to inf) π(θ|x1,…, xn) dθ

Accept hypothesis with greater probability

Question 4

In a one sided test of a normal model with known variance, when is H0 accepted?

Answer 4

Φ( (c-a) / b ) ≤ 0.5

Question 5

How do you perform an at least one simple hypothesis?

Answer 5

if c ∈ [a ± Z(α/2)*b] accept H0

Question 6

Give the Bayes factor for model selection?

Answer 6

B = P(x| M1) / P(x|M2)

where P(x|Mi) = integral(over θi) P(x| θi , Mi) * π (θi|Mi) dθi

Question 7

When does the Bayes factor become substantial evidence?

Answer 7

3 < B

Question 8

Give the Bayes factor for hypothesis testing?

Answer 8

B = [P(H1|X) * P(H0)] / [P(H0|X) * P(H1)]

Question 9

Give the formula for BIC

Answer 9

−2 log L(θˆ|x) + p log n

Question 10

Given two models, based on BIC values which model should be taken?

Answer 10

The model with lower BIC