Bayesian analysis

Question 1

Bayes factor

Answer 1

Compares how well the data support two competing hypotheses: the null hypothesis (H₀) and the alternative hypothesis (H₁).

BF₁₀ = likelihood of data under H₁ / likelihood under H₀

BF₁₀ = 3 → the data are 3 times more likely under H₁ than under H₀.

BF₁₀ = 1 → no evidence for either.

BF₁₀ = 1/3 → the data are 3 times more likely under H₀ than under H₁.

Question 2

What does a Bayes factor tell you

Answer 2

How much more likely the data are under one hypothesis compared to the other.

Evidence for either hypothesis (unlike p-values, which only test against the null).

Question 3

P values

Answer 3

A p-value just tells you whether the data are unlikely under the null hypothesis

small p value means that the null hypothesis is not likely to be true - reject

Question 4

Bayes factor below 1

Answer 4

Suggests that the data support the null hypothesis (H₀) more than the alternative hypothesis (H₁).

BF₁₀ < 1: The data are more likely under H₀ than H₁.

Question 5

What does a Bayes factor tell us, and how does this differ from what a p-value tells
us?

Answer 5

A Bayes factor tells us the relative evidence for an effect being present compared to there
being no effect.

In contrast, a p-value provides us with an absolute decision of whether or not to reject the null hypothesis.

This means that the Bayes factor can provide evidence for either an effect or the null, whereas the p-value can only be used to reject the null and
suggests an effect is present

Question 6

Advantages of Bayesian analysis

Answer 6

Directly compares hypotheses - Bayesian methods provide a Bayes Factor that quantifies how much more likely the data are under one hypothesis (e.g., H₁) than another (e.g., H₀).
This allows you to say how much evidence supports one model over another — unlike p-values, which only test against the null.

Can provide evidence for the null hypothesis - Frequentist tests (p-values) can only reject or fail to reject the null.
Bayesian analysis can support H₀, which is useful when you want to show that there’s no effect (e.g., no difference between groups, or a treatment is not effective).

Ability to express subjective beliefs g