Statistics exam 4 Bayes

Question 1

What is the bayes factor?
What does BF10 and BF01 mean?

Answer 1

The ratio of two competing models, represented by their evidence. It’s the predictive updating factor

BF10: x times more likely under H1 than H0
BF01: x times more likely under H0 than H1

Question 2

What does BF = 1 mean?

Answer 2

Both models predicted data equally well

Question 3

What is the beta distribution and when do we use it?

Answer 3

It’s a type of probability distribution for binomial variables

Question 4

What determines the shape of the beta distribution?

Answer 4

A (successes) and B (fails)

If a = b: centered bell shape
If a < b: shape left centered
If a > b: shape right centered
If a and b <1: mass close to 0 and 1
If a = b = infinity: spike
If a = b = 1: uniform (uninformative)

Question 5

What is parsimony?

Answer 5

Specific models are rewarded more when predicting well than non-specific competitors

Question 6

What is transitivity?

Answer 6

BF (BA) = 2
BF (AS) = 2
So: BF (BS) = 2 * 2 = 4

Question 7

What is bayes theorem for posterior and prior beliefs?

Answer 7

Posterior = prior * predictive updating factor

Question 8

What is the difference between likelihood and marginal likelihood?

Answer 8

Likelihood: likelihood of the data for all theta’s. This creates a shaped distribution. The area doesn’t sum to 1

Marginal likelihood: average across all likelihoods, weighted by density at each point

Question 9

What does the following mean:
Likelihood > marginal likelihood
Likelihood < marginal likelihood

Answer 9

L > ML = values of theta that predicted data better than average
L < ML = values of theta that predicted data worse than average

Question 10

What happens to values of theta that were 0 in the prior distribution after updating? Why is a spike not a good prior model?

Answer 10

Values of theta that were 0, can’t be updated, since multiplying by 0 always ends with 0
A spike is blind for updating. The spike has infinitely large value, so multiplying by even the smallest number, will still leave it at infinity.

Question 11

What is truncation?

Answer 11

Some values of theta are assigned 0 density, which makes it a one-sided model

Question 12

From what model do you typically start?

Answer 12

Uninformative model. The less informed the prior, the more the data can speak for itself

Question 13

How do you update the a and b in the beta distribution?

Answer 13

a = a + number of successes
b = b + number of fails

Question 14

How can you estimate a proportion from the posterior? Name two ways

Answer 14

1. Take a median or mean
2. Credible interval: take middle 95%

Question 15

How do you interpret a credible interval?

Answer 15

..% probability the true value of theta is between these borders

Question 16

What happens to the prior beliefs when we think both hypotheses are equally likely?

Answer 16

The prior belief value will be equal to 1

Question 17

What is the Savage Dickey Density ratio and what is it used for? What does it say?

Answer 17

It’s used in bayesian hypothesis testing. It’s the ratio of prior/posterior

If prior > posterior –> evidence for H1

Question 18

What is the test parameter of a bayesian test?

Answer 18

The bayes factor

Question 19

What does =/ , > and < mean for the shape of the beta distribution in testing?

Answer 19

=/ : uniform
< or > : truncated

Question 20

What is the difference in winnings between one-sided and two-sided hypotheses?

Answer 20

Two-sided spreads bets more and therefore receives less winnings –> lower marginal likelihood –> lower BF

One-sided has more winnings if it’s correct. Higher marginal likelihood –> higher BF

Parsimony!

Question 21

What is sequential analysis?

Answer 21

Updating beliefs and seeing evolution of the BF. There is usually more support after n>30

Question 22

What is important for choosing a prior?

Answer 22

• Informed by previous knowledge
• One/two sided: do you want to know a difference or specific difference
• Same domain

Question 23

What are the domains for proportion, correlation and difference in means?

Answer 23

Proportion : 0-1
Correlation: -1 to 1
Means: - infinity to infinity

Question 24

What distribution does bayesian correlation use?

Answer 24

Stretched beta distribution with fitting domain

Question 25

What happens to the credible interval and the Bayes factor when adding more participants?

Answer 25

CI gets more narrow
BF gets higher

Question 26

What test statistic does the bayesian t-test use?

Answer 26

Cohens d (delta sign): standardized difference between groups

Question 27

What distribution does the bayesian t-test use? What determines its shape?

Answer 27

Cauchy distribution. Shape is determined by the width

There is no uniform distribution possible because the range is infinitely large

Question 28

What does the width of the Cauchy distribution indicate?

Answer 28

It indicates the area where 50% of the values are

E.g w = 0,7 –> 50% between -0,7 and 0,7

Question 29

What is a robustness check?

Answer 29

Sensitivity analysis for the T-test. It explores what would have happened with a different prior width

BF is pretty stable for many prior widths, except strong prior settings

Question 30

What are two initiatives that emerged from the reproducibility crisis?

Answer 30

1. Preregistration
2. Open science

Question 31

What are the 6 steps for Bayesian testing?

Answer 31

1. assumptions
2. hypotheses
3. set prior distribution
4. compute likelihood
5. bayes factor
6. conclusion

Question 32

What are the 6 steps for frequentist testing?

Answer 32

0. set alpha
1. assumptions
2. hypotheses
3. test statistic
4. p value
5. conclusion