Lecture 6

Question 1

What is a Galton board, and why is it important to frequentist probability?

Answer 1

A Galton board is a physical demonstration of frequentist probability, showing the natural emergence of the normal distribution through repeated, objective, and data-driven outcomes—without using prior assumptions.

Question 2

How does frequentism define probability?

Answer 2

As the relative frequency with which an event occurs over repeated trials.

Question 3

What happens to relative frequency as the number of trials approaches infinity?

Answer 3

it converges to a mathematical limit, which is the frequentist definition of probability.

Question 4

Why does Borsboom call the frequentist definition a “conceptual masterpiece”?

Answer 4

Because it builds an elegant and objective theoretical structure rooted in repeatable observations.

Question 5

How did Ronald Fisher propose using chance in research?

Answer 5

Through random assignment and random sampling to control confounds and establish objective probability.

Question 6

What is the benefit of random sampling in frequentism?

Answer 6

It allows us to know the sampling distribution of a statistic and estimate probabilities like P(D|H).

Question 7

What does P(D|H) mean in frequentism?

Answer 7

It’s the probability (or relative frequency) of data D occurring, given that hypothesis H is true, over repeated sampling

Question 8

Why is P(D|H) considered objective in frequentism?

Answer 8

Because it’s the same for all researchers, independent of beliefs or preferences—it’s based on repeatable outcomes

Question 9

What is the implication of P(D > d | H)?

Answer 9

It helps quantify uncertainty and allows control of Type I and Type II errors.

Question 10

What guarantees does the null hypothesis test provide?

Answer 10

It ensures at most 5% Type I errors, assuming proper test execution (except in cases of p-hacking).

Question 11

What is the p-value in frequentist statistics?

Answer 11

The probability of obtaining data as extreme or more extreme than the observed, assuming the null hypothesis is true.

Question 12

What is a common misconception about the p-value?

Answer 12

That it’s the probability the null hypothesis is true—it’s not.

Question 13

Why can’t we assign a probability to the truth of a hypothesis in frequentism?

Answer 13

Because truth isn’t a chance event—probability applies to repeatable outcomes, not to static truths.

Question 14

What do Bayesians claim about P(H)?

Answer 14

That it can represent the degree of belief one should attach to hypothesis H.

Question 15

What is Borsboom’s critique of Bayesian use of P(H)?

Answer 15

It replaces objective frequency with subjective belief, making results depend on who does the analysis.

Question 16

What does Borsboom think about the idea that we can calculate P(H|D)?

Answer 16

He sees it as naïve to believe that statistical methods can yield the probability that a hypothesis is true.

Question 17

What is Borsboom’s overall stance on frequentist vs Bayesian tools?

Answer 17

Both are limited but useful; neither is a cure-all.

Question 18

What does Borsboom criticize about Bayesian advocacy?

Answer 18

He believes it is silly to present Bayesian methods as a fix for poor use of tools.

Question 19

What does Borsboom suggest we should focus on in statistical education?

Answer 19

Teaching people to reason with probabilities, rather than just switching to another automated procedure.

Question 20

Wat is een praktisch voordeel van de null hypothesis test m.b.t. onderzoeksontwerpen?

Answer 20

NHT’s kunnen worden gebruikt voor vrijwel elk onderzoeksontwerp.

Question 21

Is Bayesian statistics a magic cure for ignorance?

Answer 21

No. Bayes is not a magic potion against ignorance.

Question 22

Does Bayesian inference absolve researchers from all statistical errors?

Answer 22

No. Bayes does not redeem you from all statistical sins.

Question 23

Does Bayesian inference answer all scientific or statistical questions?

Answer 23

No. Bayes does not answer all questions.

Question 24

Can Bayesian methods be misused?

Answer 24

Yes. Bayes can be abused, just like any other statistical method.

Question 25

When is Bayesian inference particularly useful?

Answer 25

When you want to learn efficiently or adjust plausibility assessments for hypotheses or parameters based on predictive performance.

Question 26

Is Bayesian statistics subjective?

Answer 26

Yes — it uses existing knowledge to ask meaningful questions.

Question 27

Is Bayesian statistics arbitrary?

Answer 27

No — Bayesians with the same background knowledge will draw the same conclusions.

Question 28

What does a Bayesian analysis of a coin toss help illustrate?

Answer 28

That different hypotheses yield different answers, depending on what question you're asking.

Question 29

Why did different hypotheses in the coin example give different answers?

Answer 29

Because different questions were being asked; Bayesian inference responds to the specific hypothesis being tested.

Question 30

What is the key takeaway from the coin example about asking questions?

Answer 30

Always consider the question before giving an answer—Bayesian statistics depends heavily on context and framing.

Question 31

What should you do if the question isn't clear in Bayesian analysis?

Answer 31

You shouldn't demand an answer if the question hasn’t been clearly formulated.

Question 32

What does Bayesian analysis force researchers to clarify?

Answer 32

Their hypotheses, prior beliefs, and the specific questions they are asking.

Question 33

Are all hypotheses equally plausible in Bayesian statistics?

Answer 33

No—some hypotheses are more plausible than others based on prior knowledge or theory.

Question 34

What is just as important as getting an answer in Bayesian thinking?

Answer 34

Asking the right question—interpretation depends entirely on how the problem is framed.

Question 35

What was the research question in the Adam Sandler movie example?

Answer 35

Is there a correlation between movie quality and box office success for movies starring Adam Sandler?

Question 36

What did the frequentist analysis reveal in this case?

Answer 36

It gave little information or evidence—not very informative.

Question 37

What did the Bayesian analysis show in contrast?

Answer 37

It produced a Bayes Factor (BF₀₁) of about 4.5, favoring the null, and provided a much richer, more informative analysis.

Question 38

What did adjusting the prior in the Bayesian analysis allow?

Answer 38

Seeing how evidence changes with different assumptions—knowledge can be updated on the fly

Question 39

What are four practical benefits of Bayesian statistics?

Answer 39

1. Learning from prediction errors 2. Quantifying evidence (also in favor of H₀) 3. Adjusting knowledge dynamically 4. Answering meaningful, context-specific questions

Question 40

Why are philosophical discussions about chance sometimes problematic in education?

Answer 40

They are often used to distract from educators' reluctance or laziness in teaching Bayesian statistics.

Question 41

Is Bayesian statistics still niche?

Answer 41

No. The Bayesian paradigm is becoming mainstream and is increasingly recognized for its utility.

Question 42

What does the undervaluing of Bayesian methods in statistical training represent?

Answer 42

A missed opportunity, given the concrete, practical advantages Bayesian statistics offers.

Question 43

Wat is een belangrijk verschil tussen Bayesian statistics en frequentisme in hoe ze hypotheses evalueren?

Answer 43

Bayesian statistics vergelijkt de prestaties van zowel H₀ als H₁ via Bayes factors (relatief bewijs). Frequentisme kijkt alleen naar hoe verrassend de data zijn als H₀ waar is (p-waarde).

Question 44

Bayesian vs Frequentist approach to hypotheses

Answer 44

- Bayesian: "We should learn efficiently using predictive updating factors." (Gradually update beliefs based on evidence) - Frequentist: "We should always accept or reject hypotheses provisionally." (Make binary accept/reject decisions based on data)

Question 45

Bayesian vs Frequentist interpretation of results

Answer 45

Bayesian: Posterior distribution depends on interaction between prior and new data — no single objective interpretation. Frequentist: P-value always has the same objective interpretation regardless of prior knowledge.

Question 46

Bayesian vs Frequentist view of probability

Answer 46

Bayesian: A numerical measure of how strongly you believe a statement is true, given all the information you currently have. Frequentist: Probability = the relative frequency with which an event occurs.

Question 47

Bayesian vs Frequentist on evidence and hypotheses

Answer 47

Bayesian: If you are skeptical, you need more evidence to change your belief. Frequentist: No need to necessarily specify an alternative hypothesis.

Question 48

Frequentist motivation: error control

Answer 48

Error-control is a primary motivator for the frequentist approach — you can quantify the probability of making errors (like Type I and II errors).