Comparing Frequentist and Bayesian Approaches to Probabilities and Statistics, Slides of Statistics

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Introduction to

Probabilities, Bayesian and

Frequentist Statistics

Moutoshi Pal, Lica Iwaki & Carolin Lingl Research Methods in Clinical Psychology October 30, 2019

Outline of the Presentation

**1. Quick summary of the paper

2. The Frequentist approach
3. Misconceptions of the p-value
4. The Bayesian approach
5. Benefits of Bayesian Statistics
6. Comparison of the two approaches**

2. Frequentist Statistics

Null Hypothesis Significance Testing (NHST)

"surely the most bone-headedly misguided procedure

ever institutionalized in the rote training of science

students" (Rozeboom, 1997, p. 335).

Issues with Frequentist Statistics

● p-hacking

● Data dredging

● Significance chasing

● Optional stopping (a.k.a. data peaking )

● QPRs (questionable research practices)

Definition of the p-value

A p -value is the probability of the observed data, or more

extreme data, under the assumption that the null hypothesis

is true.

2. Frequentist Statistics

Definition of the p-value

A p -value is the probability of the observed data, or more

extreme data, under the assumption that the null hypothesis

is true.

2. Frequentist Statistics

Definition of the p-value

A p -value is the probability of the observed data, or more

extreme data, under the assumption that the null hypothesis

is true.

2. Frequentist Statistics

Now it’s time

for… a Kahoot

Quiz!!

3. Misconceptions of the p-value

The difference

Recall, Frequentist’s definition of p-value:

“The p -value is the probability of the observed data, or more extreme

data, under the assumption that the null hypothesis is true.”

****Emphasis: the p-value is not the probability of a theory or hypothesis, but the probability of the observed data given the hypothesis

P(D|Ho)

The difference

Bayesian approach : probability is an expression of a degree of belief of an event, based on prior knowledge (i.e. previous experiments) or personal beliefs

The p- value is the probability of the hypothesis given the data

P(Ho|D)

P(D|Ho) ≠ P(Ho|D)

Frequentist Bayesian

The probability of the observed data/result given that some hypothesis is true is NOT equivalent to the probability that a hypothesis is true given that some data/result has been observed.

Bayesian Statistics

● Uses the idea of updating beliefs with new information when testing a hypothesis

● Start with a belief about how something works (i.e. “eating sushi is dangerous”)

Prior beliefs x Bayes’ Factor = Posterior belief (= updated, new belief)

Bayes’ Factor : represents the amount of information that we’ve learned about our hypotheses form the data

● Criticism: prior belief can be different from person to person ○ Includes subjective “beliefs” in calculation (subjective, not objective)

Bayesian Statistics

● Prior Distribution ● Likelihood gives the function of a parameter given the data ● Data

Comparison of Frequentist Statistics & Bayesian

Frequentist

1. Parameters are fixed but unknown and data are random
2. Probability is a measure of frequency of repeated events

Bayesian

1. Parameters are random and data are fixed
2. Probability is a degree of certainty about values

Comparison of Frequentist Statistics & Bayesian

Frequentist

Confidence Interval : “there is a 95% probability that when I compute a confidence interval from data of this sort, the true value of θ will fall within it”.

Bayesian Credible Interval: “given our observed data (posterior distribution), there is a 95% probability that the true value of θ falls within the credible region”