Statistics exam 4 Bayes Flashcards
What is the bayes factor?
What does BF10 and BF01 mean?
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
What does BF = 1 mean?
Both models predicted data equally well
What is the beta distribution and when do we use it?
It’s a type of probability distribution for binomial variables
What determines the shape of the beta distribution?
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)
What is parsimony?
Specific models are rewarded more when predicting well than non-specific competitors
What is transitivity?
BF (BA) = 2
BF (AS) = 2
So: BF (BS) = 2 * 2 = 4
What is bayes theorem for posterior and prior beliefs?
Posterior = prior * predictive updating factor
What is the difference between likelihood and marginal likelihood?
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
What does the following mean:
Likelihood > marginal likelihood
Likelihood < marginal likelihood
L > ML = values of theta that predicted data better than average
L < ML = values of theta that predicted data worse than average
What happens to values of theta that were 0 in the prior distribution after updating? Why is a spike not a good prior model?
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.
What is truncation?
Some values of theta are assigned 0 density, which makes it a one-sided model
From what model do you typically start?
Uninformative model. The less informed the prior, the more the data can speak for itself
How do you update the a and b in the beta distribution?
a = a + number of successes
b = b + number of fails
How can you estimate a proportion from the posterior? Name two ways
- Take a median or mean
- Credible interval: take middle 95%
How do you interpret a credible interval?
..% probability the true value of theta is between these borders
What happens to the prior beliefs when we think both hypotheses are equally likely?
The prior belief value will be equal to 1
What is the Savage Dickey Density ratio and what is it used for? What does it say?
It’s used in bayesian hypothesis testing. It’s the ratio of prior/posterior
If prior > posterior –> evidence for H1
What is the test parameter of a bayesian test?
The bayes factor
What does =/ , > and < mean for the shape of the beta distribution in testing?
=/ : uniform
< or > : truncated
What is the difference in winnings between one-sided and two-sided hypotheses?
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!
What is sequential analysis?
Updating beliefs and seeing evolution of the BF. There is usually more support after n>30
What is important for choosing a prior?
- Informed by previous knowledge
- One/two sided: do you want to know a difference or specific difference
- Same domain
What are the domains for proportion, correlation and difference in means?
Proportion : 0-1
Correlation: -1 to 1
Means: - infinity to infinity
What distribution does bayesian correlation use?
Stretched beta distribution with fitting domain
What happens to the credible interval and the Bayes factor when adding more participants?
CI gets more narrow
BF gets higher
What test statistic does the bayesian t-test use?
Cohens d (delta sign): standardized difference between groups
What distribution does the bayesian t-test use? What determines its shape?
Cauchy distribution. Shape is determined by the width
There is no uniform distribution possible because the range is infinitely large
What does the width of the Cauchy distribution indicate?
It indicates the area where 50% of the values are
E.g w = 0,7 –> 50% between -0,7 and 0,7
What is a robustness check?
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
What are two initiatives that emerged from the reproducibility crisis?
- Preregistration
- Open science
What are the 6 steps for Bayesian testing?
- assumptions
- hypotheses
- set prior distribution
- compute likelihood
- bayes factor
- conclusion
What are the 6 steps for frequentist testing?
- set alpha
- assumptions
- hypotheses
- test statistic
- p value
- conclusion
