Chapter 2 Bayesian Inference

Question 1

What are the qualities/assumptions of a binomial experiment

Answer 1

1. One repeats binomial experiment n times
2. Each trial outcome is success or failure - binary
3. Probability of success p and is the same for each trial
4. Each trial results are independent

Question 2

What is the likelihood

Answer 2

The data - probability of an observed value

Question 3

how do we denote prior of theta

Answer 3

p(theta)

Question 4

how do we denote likelihood function for data as a function of theta

Answer 4

p(y|theta)

Question 5

how do we denote posterior of theta

Answer 5

p(theta|y)

Question 6

What si the relationship between beta distribution and uniform distribution

Answer 6

Beta(1,1) is a special case of the uniform distribution on (0,1)

Question 7

How to find the pdf of beta(a,b) in R

Answer 7

dbeta(theta,a,b)

Question 8

How to find the cdf of beta(a.b) in R that is (P(thetaX<=thetax)

Answer 8

pbeta(theta,a,b)

Question 9

How to generate a ransom sample of size 5 from the beta(a,b) distribution

Answer 9

rbeta(5,a,b)

Question 10

How find quantiles of beta function giving the cdf at a certain point in R

Answer 10

qbeta(q,a,b)

Question 11

What does beta.select() do and how to use it

Answer 11

Gets us to specify two quantiles and find the beta curve that matches these quantiles
Ex: beta.select(list(x=0.55,p=0.5),list(x=0.8, p=0.9)) = 3.06 , 2.56 so we choose a = 3.06 and b= 2.56

Question 12

What does beta_interval (x, shape_par=c(a,b)) do

Answer 12

beta_interval(0.5, shape_par=c(a,b)) plots the middle 50% area of the prior distribution quantile function can calculate the 25th/75th percentile

Question 13

What prior and sampling distribution give rise to a beta posterior

Answer 13

A beta prior and a binomial sampling distribution

Question 14

Define conjugate prior

Answer 14

A class of priors is conjugate for a sampling model p(y|theta) is the prior and the posterior are from the same class of distribution

Question 15

When plotted what will be the difference between the prior and the posterior

Answer 15

The posterior will have variance smaller as we have more information forming the distribution. The prior mean and posterior mean can be pretty similar if the prior assumptions were accurate

Question 16

how can we estimate the posterior mean from sample proportion and the prior mean

Answer 16

Posterior mean will lie somewhere in between

Question 17

Answering a hypothesis - is this a reasonable assumption

Answer 17

Find the probability using pbeta function - gives P(theta<=X)

Question 18

Define bayesian coverage of 95% of theta

Answer 18

This coverage is achieved in interval [L(y),U(y)] based on observed data Y=y if P(L(y)

Question 19

Define frequent coverage

Answer 19

This coverage is achieved in interval [L(y),U(y)] if before the data is observed if P(L(y)

Question 20

What is the probability of a 100x(1-alpha)% credible interval

Answer 20

P(theta is in interval [theta_(quantile alpha/2), theta _(1-quantile alpha/2) | Y=y) = 1- alpha

Question 21

What is the difference between HPD region and a credible interval

Answer 21

All points in HPD region have higher posterior density than points outside the region. This is not the case for a credible interval. Also HPD can have more than one interval region if posterior is multimodal

Question 22

What does HPD stand for

Answer 22

Highest posterior density

Question 23

Explain what the predictive distribution is

Answer 23

The conditional distribution of Y tilde (which si a future unobserved value) given the data y1……yn. It is written p(Y_tilde | y1……yn)

Question 24

What is significant about independence when considering Y_tilde

Answer 24

Y_ tilde is not independent from the data. Once we have the data we learn more about theta which in turn tells us more about Y_tilde