A; the probability that bob's IQ is below 70 based on prior knowledge B; the probability that bob's IQ is below 70 based on posterior knowledge C; the IQ score that is most probable. D; the height difference tells us the difference in likeliness, so an IQ score of 73 (C) is almost 1.5 times as likely as an IQ score of 70 E; the credibility interval; we're 95% sure that the IQ score is between 64.99 and 81.66.

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Bayesian reasoning

37 important questions on Bayesian reasoning

What is a disadvantage of NHST, that bayesian statistic solves?

NHST doesn't give you a chance of the hypothesis being true, bayesian statistics does.

What is an objective probability?

A long-run relative frequency
how much something occurs determines the chance of it occurring in this situation

How do we call the degree of conviction we have in a hypothesis?

Subjective probability

How would you calculate P(L and C)?

P(L) x P(C|L).
it is the chance of L happening, and the chance of C happening given L.

How can you portray P(H and D)?

P(D) x P(H|D)
P(H) x P(D|H)

How do we call the notion that all the information relevant to inference contained in data is provided by the likelihood?

The likelihood principle

Why do we speak of a probability density distribution when speaking of a continuous variable?

A continuous variable can take any value, to assign probability to each value would be infinite
instead probability is assigned to intervals of the continuous variable
The probability that the variable has a value in one or other of the intervals is the sum of the areas for

each interval.

so a probability density distribution tells you how probable it is that the variable takes a value in any interval.

What is the definition of a p value in NHST?

The probability of encountering a test statistic at least as extreme as the one observed, given that the null hypothesis is true.

When talking about frequentist statistics, how do you know for sure that a statement is false?

When the statements assign probabilities to hypotheses or parameters
so if you say "the probability that the true mean equals..." instantly false
you can only assign probability to data occurring

What is the definition of a 95% confidence interval?

If we were to repeat the experiment over and over, in 95% of the cases the population parameter would fall between the values that the confidence interval that belongs to the case produced.

What does it mean to say that an objective probability is a long run relative frequency?

The proportion of occurrence when a particular experiment is repeated infinitely often (under different circumstances)

What are the problem with the classical definition of probability?

It can't be applied to unique or singular events
reference class problem

Describe the reference class problem.

When you do want to assign probability to an event you can determine the probability of events equal to the event.
the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case.
for different classes you attribute the event to the probability can be different.

What is the bayesian definition of probability?

Probability doesn't exist
we only have quantified uncertainty

Why can a bayesian assign probability to a single hypothesis?

Due to the definition of probability of a bayesian,
you can quantify the uncertainty that the hypothesis is true.
the plausibility of the hypothesis is judged based on the knowledge that you have.

Describe the bayesian learning cycle?

Prior knowledge
predictions
data
prediction error, difference between data and predictions
knowledge update, or posterior
prior knowledge.

What is the difference between frequentists and bayesians when it comes to the use of data?

Frequentists have beliefs about the state of the world and make predictions about what the data would look like, then look at the data to see whether its in line with their predictions and beliefs
bayesians look at the data and based on it adjust their beliefs of the world.

What is the difference between the p values of bayesian statistic and frequentist statistic?

Freq: p value means the chance of finding data given H0 is true, p(D|H)
bay: p value means the chance of H0 being true given data, P(H|D)

What can you see in a density-parameter diagram?

The extend to which some parameters are probable to be the population parameters

What can you say about the area under the curve of a density-parameter distribution?

The total area under the curve is always 100%
a certain area indicates your confidence of the parameter lying in that interval
so the area between two numbers means that you have x% confidence in the parameter being between these two numbers

Describe A till E

A; the probability that bob's IQ is below 70 based on prior knowledge
B; the probability that bob's IQ is below 70 based on posterior knowledge
C; the IQ score that is most probable.
D; the height difference tells us the difference in likeliness, so an IQ score of 73 (C) is almost 1.5 times as likely as an IQ score of 70
E; the credibility interval; we're 95% sure that the IQ score is between 64.99 and 81.66.

What different terms are used in bayesian vs frequentist?

Bayesian uses plausibility (of a hypothesis) instead of probability
bayesian uses credibility interval instead of CI.

What is the definition of statistical evidence?

A change in conviction (concerning a given hypothesis) brought about by the data.

What are the two hypotheses in bayesian statistics?

H0 is a general law for instance: θ = 1 (all swans are white) or θ = 1⁄2 (people cannot look into the future – performance is at chance).
H1 is the hypothesis that relaxes he restriction imposed by H0
vgm gwn normale definities

Why is it the case that your data always gives evidence towards a hypothesis while in freq. Stats it sometimes doesn't?

If H0 and H1 predict the data equally well, the predictive factor becomes 1, so the probabilities of H0 or H1 getting updated with one.
in freq. Stats you never get evidence towards anything, it's either significant or not. When not significant it's no evidence for H0 either because the data could be not informative.
if the data not informative in bayes stats. Then the predictive factor becomes 1 and nothing happens.

What does it mean when p(data|H1)/p(data|H0) is larger than 1, or smaller than 1, and equal to 1?

Larger: H1 predicts the data better than H0, evidence in favor of H1
smaller: H0 predicts the data better than H1, evidence in favor of H0
equal: no evidence or weak evidence: both hypotheses predict data equally well/bad.

Why is the predictive updating factor so important?

For people for and against a theory their prior and posterior will differ.
but the extend, and the direction to which their beliefs will be shifted by the data is equal for both.

What is the predictive updating factor?

p(data|H1)/p(data|H0)
the ratio of which hypothesis predicts the data better.

What can you say about a bayes factor of 3?

The data are three times more likely under H1 than under H0.

The data change the relative plausibility of H1 vs H0 with a factor of three.
H1 predicted the data three times better than H0.

What are the thresholds of the strength of the evidence described by the bayes factor?

1 – 3 Anecdotal

3 – 10 Moderate
10 – 30 Strong
30 – 100 Very strong
>100 Extreme

How do you get from odds to probability and vice versa?

To convert from a probability to odds, divide the probability by one minus that probability. So if the probability is 10% or 0.10 , then the odds are 0.1/0.9 or '1 to 9' or 0.111.

To convert from odds to a probability, divide the odds by one plus the odds.

What are advantages of the bayes factor?

Provides a continuous degree of evidence without requiring an all-or-none decision.

Allows evidence to be monitored during data collection.
Differentiates between “the data support H0” (evidence for absence) and “the data are not informative” (absence of evidence).

Why is it wrong to monitor p value levels?

When H1 is true, the p value will steadily go towards lower numbers.
but if H0 is true p values don't go towards 1, they drift randomly.
if you then wait long enough, at some point it will become smaller than alpha and you can stop collecting data.

Why is it okay to monitor bayes factors?

If H1 is true BF will drift towards extremes of the H1
if H0 is true BF will drift towards extremes of H0.

Why does the bayes factor change if you formulate H1 to be more similar to H0?

They become harder to distinguish from each other
if the hypotheses become more similar, the differences in the extend to which they predict the data become smaller because there is more overlap.
if the differences in predictive power become smaller the bayes factor will become smaller.

What can you tell from the dots in a graph in jasp?

If the dot on the posterior distribution is lower than the dot on the prior distribution, H1 would be supported
If the dot on the posterior distribution is higher than the dot on the prior distribution, H0 would be supported

Describe the law of likelihood?

a particular set of data supports one statistical hypothesis better than another if the likelihood of the first hypothesis, on the data, exceeds the likelihood of the second hypothesis

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