Lecture 2

Frequentist vs Bayesian statistics

• We can draw stronger conclusions from Bayesian statistics!– Frequentist P-value: • P(observed data or more extreme | Ho )

– Bayesian P-value:• P(Ho | data)

• Concerns about Bayesian inference– We have to choose a prior and it is often very arbitrary.

)()|()|( fxfxf prior

likelihoodposterior

A reindeer example

• The herders gather hundreds of reindeer each autumn

• Save 20% of the heaviest calves• Variation between years• They have to decide directly after weighing

whether to keep the calf or not

• Looking back at previous years they can see that the average weight has been 45 kg

• But quite large variance between years of 4.0

• The variance within years is • Let X be the unkown weight of the first incoming

calf; assumed normal

• Will the first calf belong to the top 20% that day?• We need to make a guess of the mean weight

that day, ie

0.42

)( 2)(

),(~ 2NX

Theory

• Prior• Likelihood

• Posterior mean

• Easy to update

),( 2N

),( 2N

22

2

22

2

)|(

xxE

)()|()|( fxfxf