Bayes I

Question 1

statistical model =

Answer 1

general statistical model + statement about the parameter value that describes a certain phenomenon

Question 2

we can reflect a models’ statement via a …

Answer 2

probability distribution. based on what these models claim about theta, certain outcomes are more or less likely.

Question 3

models can also state a range of values

Answer 3

binominal distribution!

Question 4

waar is de beta distribution op gebaseerd

Answer 4

a = successes
b = failures

Question 5

what if a and b equal 1?

Answer 5

distribution is uniform

Question 6

what if the a and b are the same

Answer 6

mooie normaal verdeling -> values closer to 0,5 are more plausible.

Question 7

what if a is smaller than b

Answer 7

piek ligt links -> values below 0,5 are more plausible

Question 8

what if a is larger than b

Answer 8

piek ligt rechts -> values above 0,5 are more plausible

Question 9

dus waar ligt de piek van de binominal grafiek

Answer 9

aan de kant met het LAAGSTE getal

Question 10

wat is P(O)

Answer 10

prior knowledge, things we think BEFORE seeing the data

Question 11

wat is P(O|data)

Answer 11

posterior beliefs, after seeing the data

Question 12

wat is de predictive updating factor

Answer 12

how well did each value of theta predict the data, compared to all other values of theta?

Question 13

bayesian learning cycle

Answer 13

prior knowledge -> prediction -> data -> prediction error -> knowledge update -> prior knowledge…

Question 14

wanneer is deduction

Answer 14

bij prediction en data

Question 15

wanneer is induction

Answer 15

bij prediction error and knowledge update

Question 16

predictive updating factor bestaat uit 2 delen

Answer 16

likelihood given a certain value of theta

marginal likelihood, across all values of theta

Question 17

dus likelihood vertelt ons….

Answer 17

This tells us something about
how well a specific value of θ
predicted the data (i.e., it is the
quality of the prediction for this
specific value)

Question 18

dus marginal likelihood vertelt ons…

Answer 18

This tells us something how well
θ predicted the data, averaged
over all possible values of θ (i.e., it
is the average quality of the
prediction of the model)

Question 19

dus de predictive updating factor vertelt ons…

Answer 19

Taken together, this ratio tells us how well each
value of θ predicted the data, relative to all other
values!

je maakt een soort grafiek van alle values en hoe goed ze de data predicten.

Question 20

the likelihood is NOT

Answer 20

a probability distribution!!!!!!! -> suface area =/= 1

Question 21

marginal likelihood

Answer 21

how well did the model on average predict the data -> average quality of the prediction model.
the probability of generating the observed sample from a prior.
how well we can explain the data using all parameters?
which values of theta predicted better than average?

Question 22

likelihood

Answer 22

the likelihood of the data, given a certain value of theta

Question 23

wanneer krijg je een BF > 1 (goede predictie)

Answer 23

likelihood > marginal likelihood.

dan zijn er values van theta die de data beter predicten dan average.

Question 24

wanneer krijg je een BF < 1 (slechte predictie)

Answer 24

marginal likelihood > likelihood

dan is het average van de data dus een betere predictie dan jouw model.

Question 25

dus heel korte beschrijving marginal en likelihood

Answer 25

marginal likelihood: soort van average over alle data (soort van net als y-, vergelijkingspunt)

likelihood: the likelihood of the data, given a certain value of theta. dus dit is eigenlijk een soort van of deze value van thta de data beter kan voorspellen dan de average van het model

Question 26

posterior knowledge =

Answer 26

prior beliefs * predictive updating factor

Question 27

wat gebeurt er tijdens de predictive updating factor

Answer 27

values of theta that predict the data petter than average receive a boost (increase in plausability), values of theta that predict the data worse than average receive a penalty (decrease in plausability)

Question 28

what is the posterior distribution

Answer 28

a probability distribution!

Question 29

verschil likelihood en probability distribution

Answer 29

y assen zijn hetzelfde

x assen (horizontaal)
likelihood: fixes succcesses, variates theta
probability: fixes theta, variates successes (bijvoorbeeld aantal keer heads)

Question 30

wat vertelt de PUF ons

Answer 30

the ratio tells us how well each value of theta predicted the data, relative to all other values

Question 31

likelihood in relatie tot marginal likelihood

Answer 31

als de likelihood hoger is dan de marginal likelihood, heeft deze waarde van theta de data beter dan gemiddeld heeft voorspelt!

Question 32

wat is de PUF als de likelihood groter is dan marginal likelihood

Answer 32

> 1

Question 33

wat is de PUF als de marginal likelihood groter is dan de likelihood

Answer 33

<1 -> geen goede predictie van die value van theta.

Question 34

wat gebeurt er bij een very strong prior (hoge a en hoge b waardes)

Answer 34

A very strong prior that assigned more mass to certain values. The prior
conviction is so strong that the data cannot overthrow this conviction: the posterior is still situated at the prior stated values.

This highlights an important feature of learning: strong beliefs need a lot of data to be convinced otherwise! This also makes Sarah and Paul very poor learners: they are so convinced of their value that there is no updatin

Question 35

wat als de prior distribution een beta distribution is

Answer 35

dan is de posterior distribution dit ook!

met:
a = a + #successes (in our case, #tails)
b = b + #failures (in our case, #heads)

= continuous distribution

Question 36

wat kan je met de posterior distribution doen

Answer 36

een estimation maken!
take the point estimate, mean or median

you can also make a central credible interval

Question 37

hoe maak je een central credible interval

Answer 37

To obtain the x% central credible interval, we take
x% of the most central posterior mass, and see
which 2 points are the thresholds

dus bijvoorbeeld 95% central credible interval -> kijken welke 95% in het midden ligt, en hiervan benoem je dan de grens.

Question 38

hoe interpreteer je deze central credible interval

Answer 38

under this model, there is a x% probability that the true proportion is between … and ….

Question 39

predictive quality (wat bedoelen ze hiermee)

Answer 39

how well does each possible value of the parameter predict the observed data, compared to the other values