Bayesian Analysis

Question 1

Other than Linear, GLM, Logistic, there is another way to estimate parameters, which is?

Answer 1

Bayesian Analysis

Question 2

Bayesian analysis method is

Answer 2

Treat unknown var as a random var, use prior distribution

Question 3

How does prior dist work?

Answer 3

It combine the distribution and the prior dist of the parameter

Question 4

What is posterior mean?

Answer 4

Weighted average of prior mean and MLE

Question 5

Relationship between weight of MLE vs amount of data

Answer 5

Increase, increase

Question 6

Relationship between weight of prior mean vs δ

Answer 6

increase, increase

Question 7

Posterior mean (E(λ|D)) formula?
hint: λ|D ~ Dist Type ( α, δ)

Answer 7

α/δ

Question 8

λ|D ~ Dist Type (α, δ) is?

Answer 8

Prior dist

Question 9

What happen if var in prior dist decrease?

Answer 9

Become more relevant prior info

Question 10

What is diffuse/non-informative prior?

Answer 10

letting both α and δ go to 0 and mean unchanged and var is very large

Question 11

what does conjugate prior leads to?

Answer 11

posterior having the same type of dist as prior

Question 12

Predictive distribution is

Answer 12

To predict the next one, so n+1

Question 13

PDF of predictive dist

Answer 13

Look at fromula sheet the pareto dist,
misal: first parameter a, second parameter is b
so, [a times b^a]/[b + x]^a + 1

Question 14

Predicitve mean formula?

Answer 14

use the pareto dist formula sheet, b/(a-1)

Question 15

Bayesian estimate loss function

Answer 15

Diff between true val and estimate, choose that minimize the loss function

Question 16

E(θ-θhat)^2 squared error loss function?

Answer 16

use posterior mean

Question 17

E(|θ-θhat|) absolute error loss function?

Answer 17

Posterior median using the loss function

Question 18

all or nothing loss function?

Answer 18

I = 1, θ=θhat

Question 19

Bayesian interval (credible and prediction)

Answer 19

Credible from θ from posterior dist
Prediction from Xn+1 from predictive dist

Question 20

Steps on calculate mean posterior dist

Answer 20

1. pdf of the current dist masukin the value
2. if have more than 1 val, masukin again on the pdf trs di kali
3. masukin pdf posterior dist using the given parameter on the posterior dist
4. group miu and ecp
5. what dist is that?
6. calc the mean/median/mode depends on what loss function needed! first parameter divide d by second parameter

Question 21

how to calc posterior median and mode?

Answer 21

median use qgamma function in R
mode:
derive the log of the dist with the plugged in parameters and set lamda to 0

Question 22

What is credibility theory

Answer 22

Set of techniques for calculating insurance premiums, estimating number of claims

Question 23

αI vs αc?

Answer 23

αI based on past data while αc based on relevant data

Question 24

Insurance premium formula

Answer 24

(1-Z)αc + ZαI

Question 25

what happen towards Z vs amount of αI and αc

Answer 25

if αI bigger, it also bigger, but if αc smaller, it also smaller

Question 26

Combine the credibility with bayesian approach!

Answer 26

sample data = past data
prior information = collateral of relevant data

Question 27

posterior mean how to simplfy ?

Answer 27

look at summary week 10 first blue notes

Question 28

How to compute predicitve?
hint: steps

Answer 28

1. calc the posterior dist first:
   a. MLE of process times prior dist
   b. group the lambda and e
   c. recognize what dist is that
2. masukin the predictive dist:
   integral infitinity of Xn+1 | lamda times posterior dist

Question 29

Posterior mean in credibility formatt?

Answer 29

after find the posterior dist and know what is the dist if follows, then using the mean from that dist. and then simplify using summary week 10 first blue notes, and take the second last as the Z

Question 30

Other ways to calculate credibility theory other than bayesian?

Answer 30

Empirical bayes approach

Question 31

empirical bayes approach vs bayesian approach?

Answer 31

treat unknown var to random var, but not applyting prior dist

Question 32

α0 formula for EBCT 1

Answer 32

look at formula sheet

Question 33

αk formula for EBCT1

Answer 33

look at formula sheet

Question 34

estimator of E(m(teta)|D)) formula

Answer 34

α0 + sigma n i=1 (αj) xj

Question 35

Yj in EBCT 2 represent?

Answer 35

aggregate claim amount

Question 36

Pj in EBCT2 represent?

Answer 36

risk volume in a year

Question 37

α0 and αk formula for EBCT2

Answer 37

look at formula sheet