Bayesian Analysis Flashcards
Other than Linear, GLM, Logistic, there is another way to estimate parameters, which is?
Bayesian Analysis
Bayesian analysis method is
Treat unknown var as a random var, use prior distribution
How does prior dist work?
It combine the distribution and the prior dist of the parameter
What is posterior mean?
Weighted average of prior mean and MLE
Relationship between weight of MLE vs amount of data
Increase, increase
Relationship between weight of prior mean vs δ
increase, increase
Posterior mean (E(λ|D)) formula?
hint: λ|D ~ Dist Type ( α, δ)
α/δ
λ|D ~ Dist Type (α, δ) is?
Prior dist
What happen if var in prior dist decrease?
Become more relevant prior info
What is diffuse/non-informative prior?
letting both α and δ go to 0 and mean unchanged and var is very large
what does conjugate prior leads to?
posterior having the same type of dist as prior
Predictive distribution is
To predict the next one, so n+1
PDF of predictive dist
Look at fromula sheet the pareto dist,
misal: first parameter a, second parameter is b
so, [a times b^a]/[b + x]^a + 1
Predicitve mean formula?
use the pareto dist formula sheet, b/(a-1)
Bayesian estimate loss function
Diff between true val and estimate, choose that minimize the loss function
E(θ-θhat)^2 squared error loss function?
use posterior mean
E(|θ-θhat|) absolute error loss function?
Posterior median using the loss function
all or nothing loss function?
I = 1, θ=θhat
Bayesian interval (credible and prediction)
Credible from θ from posterior dist
Prediction from Xn+1 from predictive dist
Steps on calculate mean posterior dist
- pdf of the current dist masukin the value
- if have more than 1 val, masukin again on the pdf trs di kali
- masukin pdf posterior dist using the given parameter on the posterior dist
- group miu and ecp
- what dist is that?
- calc the mean/median/mode depends on what loss function needed! first parameter divide d by second parameter
how to calc posterior median and mode?
median use qgamma function in R
mode:
derive the log of the dist with the plugged in parameters and set lamda to 0
What is credibility theory
Set of techniques for calculating insurance premiums, estimating number of claims
αI vs αc?
αI based on past data while αc based on relevant data
Insurance premium formula
(1-Z)αc + ZαI
what happen towards Z vs amount of αI and αc
if αI bigger, it also bigger, but if αc smaller, it also smaller
Combine the credibility with bayesian approach!
sample data = past data
prior information = collateral of relevant data
posterior mean how to simplfy ?
look at summary week 10 first blue notes
How to compute predicitve?
hint: steps
- calc the posterior dist first:
a. MLE of process times prior dist
b. group the lambda and e
c. recognize what dist is that - masukin the predictive dist:
integral infitinity of Xn+1 | lamda times posterior dist
Posterior mean in credibility formatt?
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
Other ways to calculate credibility theory other than bayesian?
Empirical bayes approach
empirical bayes approach vs bayesian approach?
treat unknown var to random var, but not applyting prior dist
α0 formula for EBCT 1
look at formula sheet
αk formula for EBCT1
look at formula sheet
estimator of E(m(teta)|D)) formula
α0 + sigma n i=1 (αj) xj
Yj in EBCT 2 represent?
aggregate claim amount
Pj in EBCT2 represent?
risk volume in a year
α0 and αk formula for EBCT2
look at formula sheet
