Questions #3

Question 1

What is the main shortcoming of grid approximation?

Answer 1

It scales very poorly in high dimensions.

Question 2

What is the main shortcoming of quadratic approximation?

Answer 2

Scales better than grid, but it can struggle in the face of complex, hierarchical models.

It also does not fare well in the presence of posterior distributions that cannot be well approximated by a Gaussian distribution

Question 3

What is the 3 things we must be able to do to perform MCMC sampling with the metropolis algo

Answer 3

1. We must be able to generate a random value from the proposal distribution (Normal (theta current, sigma^2)
2. We must be able to calculate the unnormalized posterior densities
3. We must be able to generate a uniform random value from 0 to 1 to accept or reject the proposed parameter value

Question 4

In the trace plot, if the standard variation of the proposal distribution is too low, what will happen?

Answer 4

It will take longer to get to the right values

Question 5

In the trace plot, if the standard deviation of the proposal distribution is too high, what will happen?

Answer 5

It will generate far-away proposals that will usually get rejected and won’t explore the posterior distribution well. (graphique en robot)

Question 6

What is the difference between Metropolis and Metropolis-Hasting?

Answer 6

Metropolis-Hasting generalizes the metropolis by allowing assymetric proposals

Question 7

True or false : We need a symmetric proposal function in the metropolis algo

Answer 7

True

Question 8

True or false : In the Gibbs Algorithm, we always accept the proposal

Answer 8

True

Question 9

What are the advantages of Gibbs ?

Answer 9

Efficiency in sampling from the posterior and no tuning of the proposal distribution

Question 10

What are the disadvantages of Gibbs?

Answer 10

1. Ability to compute and sample from conditional posterior distributions
2. Sampling efficiency in models with correlated parameters

Question 11

True or false : If the parameters are graphed in n-dimensional space, the metropolis algorithm movements can be in any direction,. The gibbs movements are always parallel to the axes

Answer 11

True

Question 12

Write the formulas useful for HMC

Answer 12

1. Formula for momentum
2. Formula for Theta
3. Formula for prob of accept

Question 13

What is the consequence in HMC if s is too low?

Answer 13

The proposal distribution is too concentrated and does not have sufficient time to move into the region of high posterior mass

Question 14

What is the conserquence in HMC if s is too high?

Answer 14

Lower proposal rates since the proposals are too far away from the mode of the distribution
Can result in U-turn problem

Question 15

In MCMC, a small number of samples is needed if we want the posterior mean

Answer 15

True

Question 16

In MCMC, a small number of samples is needed if we want the posterior variance and the percentiles extreme

Answer 16

False. We need a lot of samples

Question 17

In a trace plot, if the chains are all representative of the posterior, they should … (2)

Answer 17

1. They should overlap each other and be unrelated to their randomly set starting positions
2. They should also be stationary around the same modal value

Question 18

Can you give me a reason why the trace plot are not mixing well?

Answer 18

The prior is too flat

Question 19

What we need to do if a chain in the trace plot is isolated from the others?

Answer 19

1. Try to run more samples

2. Need to check our model definition, implementation method, or input data for potential issues

Question 20

How can we know how correlated the proposed parameters are through the sampling iterations in MCMC?

Answer 20

With an autocorrelation plot

Question 21

True or false : A correlogram don’t necessarily give an indication if the chain is representative of the posterior distribution, but it will give us a sens of the efficiency of our MCMC algorithm

Answer 21

True

Question 22

True or false : HMC as low-autocorrelation if the parameters are well-tuned

Answer 22

true

Question 23

true or false : A greater number of iterations will be needed to explore the posterior distribution if there is a lot of autocorrelation

Answer 23

true

Question 24

What are the 2 forms of binomial regression

Answer 24

1. Logistic regression. Each record in the data set indicated whether an event occurred or didn’t. We are estimating probability of an event
2. Aggregated binomial regression. Each record in the data set states the size of the population and the number of events that occurred

Question 25

True or false : Mixtures are a way to handle overdispersion

Answer 25

True

Question 26

True or false : WAIC and PSIS can be used for beta-binomial and poisson-gamma models

Answer 26

False.

Question 27

True or false : Cross-validation is the only tool that we can use for Beta Binomial and Poisson gamma models

Answer 27

True

Question 28

What are the advantages of multilevel models?

Answer 28

1. Better estimates ahen there are repeated observations from the same cluster 2. Better estimates for unbalanced data 3. We get estimates of variation between clusters 4. We avoid averaging of data.

Question 29

What are the advantages of multilevel regression

Answer 29

1. We need assumptions for how clusters vary 2. The posteriors are nowhere near normally distributed. MCMC, which is slow, is required 3. Models are harder to understand

Question 30

True or false : In a multilevel model, the parameter in the linear expression for the varying effects is indexed

Answer 30

True

Question 31

Which approximation method need to be used for multilevel models?

Answer 31

MCMC

Question 32

Multilevel models produce shrinkage estimates. The properties of these estimates are

Answer 32

1. Estimates are between the no pooling and complete pooling estimates; they shrink from the individual estimate towards the overall estimate 2. Shrinkage is greater when there is less data for a group 3. Shrinkage is greater when the individual estimate is further from the overall estimate

Question 33

True or false : The lower the variance of the varying intercept, the lower the effective number of parameters

Answer 33

True

Question 34

True or false : Multilevel models control overfitting by limiting variation of less important varying effects

Answer 34

True

Question 35

When post-stratification can be used?

Answer 35

When the sample are biaised because the presence of the data may be a function of the cluster

Question 36

When it is useful to do a non-centered parametrization?

Answer 36

When there is a divergent transitions. When the acceptance rate is not achieved (HMC)

Question 37

What are the advantages of non-centered parametrization?

Answer 37

1. Better for clusters with low variation, or clusters with a lot of units but not much data in each unit 2. It improves efficiency, and the resulting model is equivalent to the original model

Question 38

What are the disadvantages of non-centered parametrzation?

Answer 38

1. Model is harder to understand 2. The fitted varying effect parameters are standardized 3. Sometimes non-centered parametrization doesn't help