Hypothesis Testing

Question 1

What are the problems with hypothesis testing in frequentist inference?

Answer 1

• does extremeness make sense?
• evidence is against H0, not for H1
• p-value does not provide weight of evidence for H1, i.e the probability H1 is true
• violates likelihood principal, as calculation of p-value takes into account values not observed

Question 2

Give a simple vs simple hypothesis

Answer 2

H0: θ = θ0
H1: θ = θ1

Question 3

How do you find Pr(H0 | y) with simple vs simple hypotheses?

Answer 3

(f(y|θ0) * p0) / (f(y|θ0) * p0) + (f(y|θ1) * p1)

Question 4

How can you compare multiple hypotheses?

Answer 4

Pr(y | Hi) * pi / sum(to k) pr(y|hj) * pj

Question 5

Define the Bayes Factor generally

Answer 5

Posterior odds / prior odds

[Pr(H0|y) / Pr(H1|y)] / [p0/p1]

Question 6

Give the interpretation of values of Bayes Factor

Answer 6

< 3 no evidence for H0 over H1
>3 positive evidence for H0
>20 strong evidence for H0
>150 very strong evidence for H0

Question 7

Bayes factor for simple vs simple

Answer 7

f(y|θ0) / f(y|θ1)

Question 8

Bayes factor for composite vs composite

Answer 8

[ ∫(over θ in θ0) f(y|θ0)π(θ) dθ / ∫(over θ in θ1) f(y|θ1)π(θ) dθ ] / [p0/p1]

Question 9

Bayes factor for simpled vs composite

Answer 9

f(y|θ0) / ∫f(y|θ)*π(θ) dθ

Question 10

How do you find p0

Answer 10

∫ π(θ) dθ