Week 10

Question 1

Lecture 28

Answer 1

Bayesian inference

Question 2

Difference between frequentist and Bayesian

Answer 2

Comes down to probability

Freq: when number of experiments goes to infinity (assumes you can do large number of trials the same) REPEATABILITY is key

Bayesian: degree of belief about an event

Question 3

Prove Bayes theorem

Answer 3

Here P(A,B) is P(A n B)

Question 4

What to do if marginal probability is not available but needed

Answer 4

Question 5

Def Bayesian parametric model

Answer 5

Question 6

Change in notation from freq to Bayesian

Answer 6

As θ is a RV therefore f(x|θ) is now a distribution

Question 7

Interpretation of prior dist

Answer 7

Represents the uncertainty about the (true value of) the parameter θ

Question 8

Multi variate prior

Answer 8

Question 9

The prior has to be

Answer 9

Fully specified, hyper parameters (params of prior) must be fully specified

Question 10

Actualisation principle

Answer 10

Aka Bayes update

1) use prior dist
2) collect data
3) posterior dist

Question 11

Def posterior dist

Answer 11

Question 12

Posterior distribution analog in frequentist

Answer 12

Likelihood function

Question 13

Prove posterior dist

Answer 13

Question 14

Derive posterior dist using Bayes theorem

Answer 14

Question 15

Lecture 29

Answer 15

Started around 2cards ago

Question 16

Beta binomial model

Answer 16

Question 17

Def conjugate priors

Answer 17

A family F of probability distributions in Θ is said to be conjugate (or closed under sampling) for a likelihood function f(x|θ) if, for every prior π_Θ € F , the post dist π(θ| x) also belongs to F

Question 18

Lecture 30

Answer 18

Jeffrey’s prior

Question 19

(Im)proper prior ?

Answer 19

Question 20

(Im)proper posterior

Answer 20

Question 21

Relate (im)proper priors to posteriors

Answer 21

All proper priors lead to proper posteriors (by Bayes)

Some improper priors lead to proper posteriors, Some lead to improper posteriors

Question 22

Why do we care about improper priors

Answer 22

Nice theoretical properties: invariance under reparameterisation

Question 23

Jeffrys prior

Answer 23