Bayesian Statistics

Question 1

Give two reasons for why statistics is needed

Answer 1

• To make inferences on the population

- Assessing the uncertainty of statements

Question 2

What is conditional probability

Answer 2

Chance of one event occurring given that another event has occurred

Question 3

What does P(0) signify? Give a term for this

Answer 3

The plausibility of a certain value of 0 before seeing the statistics ( prior beliefs)

Question 4

What does P(data|0)/P(data) signify?Give a term for this

Answer 4

How well did this value of 0 predict the data, compared to all other values of theta? (predictive updating factor)

Question 5

What does P(0|data) signify? Give a term for this

Answer 5

The plausibility of 0 being equal to a particular value after seeing the data (Posterior beliefs)

Question 6

Give 3 characteristics regarding beta distributions (range, shape, when values= 1)

Answer 6

• Ranges from 1-0
• Shape is determined by two values a and b
• if a and b equal 1 it is uniform

Question 7

What does a flat prior distribution reflect? (a=1 b=1)

Answer 7

A prior distribution that reflects the
belief that all values of the proportion
are equally plausible, a priori, we call
this an uninformative prior

Question 8

What does a normally distributed prior distribution reflect? (a=5 b=5)

Answer 8

A prior distribution that reflects
the belief that values close to 0.5
are more plausible (i.e., there are
equal number of left/right-handed
people), a priori

Question 9

What does a right skewed prior distribution reflect? (a=2 b=6)

Answer 9

A prior distribution that reflects
the belief that values below 0.5
are more plausible (i.e., there are
more right-handed people), a
priori

Question 10

What statistic is used in this situation?

Answer 10

The observed proportion

Question 11

How do we obtain the x% central credible interval?

Answer 11

We take x% of the most central posterior mass and see which 2 points are the thresholds

Question 12

What three flaws are presented in Bayesian Statistics?

Answer 12

1. p-values measured against a sample (fixed size) statistic with some stopping intention changes with change in intention and sample size. i.e If two persons work on the same data and have different stopping intention, they may get two different p- values for the same data, which is undesirable.
   2- Confidence Interval (C.I) like p-value depends heavily on the sample size. This makes the stopping potential absolutely absurd since no matter how many persons perform the tests on the same data, the results should be consistent.
   3- Confidence Intervals (C.I) are not probability distributions therefore they do not provide the most probable value for a parameter and the most probable values.

Question 13

What are parameters?

Answer 13

factors in the models affecting the observed data

Question 14

What is meant by the term models

Answer 14

Models are the mathematical formulation of the observed events

Question 15

What is signified by P(data)?

Answer 15

P(D) is the evidence. This is the probability of data as determined by summing (or integrating) across all possible values of θ, weighted by how strongly we believe in those particular values of θ

Question 16

What two models contribute to the posterior belief? P(0/D)

Answer 16

The prior belief P(0) and the likelihood function P(D|θ)

Question 17

What mathematical function is used to represent the prior beliefs?

Answer 17

Beta distribution

Question 18

What is denoted by p(h1)/p(Ho)

Answer 18

Prior beliefs about hypothesis

Question 19

show the mathematical notation for predictive updating factor in bayesian hypothesis testing

Answer 19

P(data|H1)/P(data|Ho)

Question 20

show the mathematical notation for posterior beliefs in bayesian hypothesis testing

Answer 20

P(H1|data)/P(Ho|data)

Question 21

What do prior odds specify?

Answer 21

How plausible a hypothesis is, relative to another hypothesis, before seeing the data

Question 22

What function does the predictive updating factor serve in bayesian hypothesis testing?

Answer 22

Displaying how well the alternative or null hypothesis predicted the data

Question 23

how do you calculate the savage dickey ratio

Answer 23

Take the ratio of the prior density and the posterior density at point of testing

Question 24

What is bayes factor BF10

Answer 24

The predictive updating factor of bayesian hypothesis testing

Question 25

What does it mean if BF10=20?

Answer 25

The data is 20 times more likely under H1 than H0

Question 26

State BF10=1/20 in a different way and what does this mean?

Answer 26

BF01=20; data is 20 times more likely under H0 than H1

Question 27

Give a general interpretation of bayes factor scores

Answer 27

1-3; anecdotal
3-10; Moderate
10-30; Strong
30-100; Very strong
>100; Extreme

Question 28

In two sided hypothesis testing the marginal likelihood will be _______

Answer 28

Lower

Question 29

What name is given to The concept of rewarding a more specific model, while it predicted the data
equally well

Answer 29

Parsimony

Question 30

What is the relative nature of the bayes factor?

Answer 30

A high Bayes factor does not mean the hypothesis is true, it just means it predicted the data better (or
less poorly) than the other hypothesis

Question 31

The prior distribution has an a=2 and b=6. New data comes in with 23 correct and 7 incorrect results. What is the new data for the distribution

Answer 31

a=25 b=13

Question 32

What domain is the prior distribution in for a difference in means

Answer 32

-infinity, infinity

Question 33

What is meant y a stretched beta distribution

Answer 33

the same as Beta, but then stretched to the domain [-1, 1]

Question 34

What happens to the likelihood distribution if the sample size increases?

Answer 34

It grows narrower

Question 35

What else does the sample size effect?

Answer 35

The correlation increases with the sample size

Question 36

What are differences between groups categorized by in psychology?

Answer 36

Delta (d); a standardised difference between groups

Question 37

What is the null hypothesis for a bayesian t test?

Answer 37

H: d=0

Question 38

What is the formula for a bayesian t test

Answer 38

P(d|data)= P(d)(P(data|d)/P(data)

Question 39

What is a cauchy distribution and when do we use it?

Answer 39

The Cauchy distribution is governed by a single shape parameter that determines how wide it is, we use it when the parameter is between infinite and -infinite. It is a t distribution with df=1.

Question 40

What type of prior distribution for a bayesian t test is usually used as an uninformative priori?

Answer 40

when 50% of the values of δ are located

between -0.707 and 0.707

Question 41

What priori do we usually use for a two sided alternate hypothesis?

Answer 41

A prior distribution that reflects
the belief that 50% of the values
of δ are located between -1.5 and
1.5 , a priori

Question 42

What priori do we usually use for a one sided alternate hypothesis?

Answer 42

A prior distribution that reflects
the belief that only positive values
of δ are possible and that 50% of
the values of δ are between 0
and 0.707, a priori

Question 43

What is meant by the Bayes factor

robustness check, or sensitivity analysis?

Answer 43

Exploring what would have happened if we had chosen a different value for the prior width.

Question 44

What does pre registration help prevent?

Answer 44

presenting exploratory research as confirmatory

research: exploratory research is used to generate theories and hypotheses; confirmatory research is used to test those

Question 45

What is conjugacy?

Answer 45

Using a beta distribution as a prior and a binomial likelihood to produce a posterior beta distribution.