Chapter 3 Monte Carlo Approximation

Question 1

Explain the concept of monte carlo approximation

Answer 1

Monte Carlo experimentation is the use of simulated random numbers to estimate some functions of a probability distribution. It allows for approximating quantities of interest that are hard to calculate

Question 2

How does the monte carlo method approximate the posterior

Answer 2

We could sample S independent theta values from the posterior distribution. Then the empirical distribution fo the sample we take of s values will approximate the posterior the larger the sample size the more accurate the approximation.

Question 3

What to be cautious of when dealing with tail probabilities estimated by monte carlo inference

Answer 3

If events are very very rare alot fo data may eb needed to make the probability accurate as often tail probabilities can be small.

Question 4

What theorem/ result allows to us to approximate quantities of interest from monte carlo inference as about equal to the actual posterior calculated values

Answer 4

The law of large numbers

Question 5

What effect does the sample size have on monte carlo inference

Answer 5

Bigger the sample size the more accurate the approximation - gets closer to the true value

Question 6

How do we estimate credible intervals using Monte Carlo simulation

Answer 6

Using rbeta() and the quantile function

Question 7

What are the parameters of the quantile function

Answer 7

quantile(sample, c(q1,q2))

Question 8

How can the exact solution of a beta credible interval be found?

Answer 8

beta_interval(% of interval, c(a,b),color=crcblue)
Or
qbeta(quantile, a, b)

Question 9

What is the interpretation of the posterior odds for beta posterior

Answer 9

For beta distribution posterior: If o>1 then theta> 0.5. If o<1 then theta<0.5

Question 10

Why is the posterior odds much easier to calculate using monte carlo inference

Answer 10

The odds p(o|y) is a very complicated distribution and formula so calculation is tricky.

Question 11

How do we find p(o|y)?

Answer 11

Sub theta in terms of the odds into the posterior distribution p(theta | y). Theta = o/1+o and sub this into the posterior and multiply by dtheta/do

Question 12

How to draw independent samples from p(o|y)? - the posterior odds size 1000

Answer 12

Betasamples

Question 13

If the variable odds - is sample from p(o|y) how do you investigate the posterior probability that the odds is less than 1 in monte carlo inference

Answer 13

mean(odds<1)

Question 14

If the variable odds - is sample from p(o|y) how do you investigate the posterior mean

Answer 14

mean(odds)

Question 15

If the variable odds - is sample from p(o|y) how do you investigate the posterior credible interval 95%

Answer 15

quantile(odds, c(0.025,0.975))

Question 16

What is the actual distribution of the predictive posterior distribution for beta prior, binomial likelihood and beta posterior and why is normally using monte carlo

Answer 16

Ytheta | Y=y is Beta-Binomial(m, a+y,b+n-y)

This is a complex distribution and monte carlo approximation is mostly easier

Question 17

What is the r command to use to find the beta-binomial predictive posterior distribution

Answer 17

prob

Question 18

What is the r command to use to find numbers that give a range of values in the the beta-binomial predictive posterior distribution that cover a certain probability mass

Answer 18

discint(pred_distribution, 0.9)
or
discint(pred_distribution, 0.8)
etc

Question 19

How do we simulate the posterior predictive distribution in R?

Answer 19

Draw theta samples from the beta posterior and draw from the likelihood binomial using the theta posterior sample as the parameters

Question 20

explain model checking

Answer 20

Does the model fit the data well. Key idea : the observed data should eb similar in some sense to the data predicted form the bayesian model.

Question 21

What do we assume in our bayesian prediction by monte carlo method to carry out model checking

Answer 21

Sample theta from Beta(a+y,b+n-y) distribution and then use this theta in sample of y_tilde from binomial(m,theta). We require m=n to see if the observed data is similar to the predicted simulated data of the same size.

Question 22

What conclusion should be drawn from model checking

Answer 22

If observed data y is different to the samples of y_tilde ‘s form the posterior predictive distribution we have evidence the Bayesian model is not a good fit for the data.

Question 23

Discuss comparing a histogram for the purpose of model checking

Answer 23

Comparing a histogram of the samples of y_tilde our observed y value should sit in the middle of the graph. If it is it is consistent with the simulations of the replicate data from the predictive distribution.

Question 24

Discuss the use of tail probabilities to comment on model checking

Answer 24

If P(y>y_tilde | y) or 1-P(y>y_tilde | y) are very small it suggests the model does not describe y very well.

Question 25

What is the basic need to be able to use monte carlo

Answer 25

Need to be able to sample independently from the posterior distribution