Solutions Manual For Bayesian Statistical Methods 1st Edition
By Brian J. Reich; Sujit K. Ghosh 9781032093185 ALL Chapters .
Transition Kernel - ANSWER: denoted 'P' - the transition kernel (or density) uniquely describes the
dynamics of the chain
Under what conditions will the distribution over the states of the Markov Chain converge to a
stationary distribution? - ANSWER: When the chain is 'aperiodic' and 'irreducible'
what does aperiodic mean? - ANSWER: A markov chain is aperiodic if for any state, the chain can
return to that state after a number of transitions that are a multiple of 1 and can also be 1
What does irreducible mean? - ANSWER: A markov chain is irreducible if any state can be reached
within finite time irrespective of the present state.
Pros and Cons of Trace Plots - ANSWER: It's a fairly efficient method but it is NOT robust.
Define Burn-In? - ANSWER: It's the initial realizations of the markov chain that we discard as the chain
had not converged to the stationary distribution yet.
What does it mean in terms of the posterior when there is low autocorrelation? - ANSWER: It means
samples are more representative of the posterior distribution
The autocorrelation plot shows the correlation between what types of samples? - ANSWER:
Successive samples
Define thinning - ANSWER: The process involves taking the kth realization of the markov chain and
discarding the rest
Thinning: Pros & Cons - ANSWER: It reduces autocorrelation, but it also discards potentially good
information.
What does BUGS stand for - ANSWER: Bayesian Inference Using Gibbs Sampling
What can transformation can be useful to aid comparability, interpretability, and MCMC
convergence? - ANSWER: Normalizing the data corresponding to explanatory variable(s)
What do we typically conclude when posterior summaries are similar? - ANSWER: The posterior
distribution is data-driven.
Explain the Gibbs Sampler (not its algorithm) - ANSWER: Gibbs sampler uses the set of full
conditionals of 'pi' to sample indirectly from the full posterior distribution.
Explain the Metropolis-Hastings (not its algorithm). What's important about it? - ANSWER: The MH
algorithm sequentially draws obs. from a distribution, conditional only on the last obs., thus inducing
a markov chain.
Important aspect is that the approximating candidate distribution can be IMPROVED at each step of
the simulation .
Define mixing - ANSWER: The movement around the parameter space.
What can cause poor mixing? - ANSWER: 1) a high rejection probability
2) very small step sizes
, Explain the idea behind Data Augmentation - ANSWER: We treat the missing data (or auxiliary
variables) as additional parameters to be estimated & form the joint posterior over both these
auxiliary variables and models parameters 'theta vector'
Explain the idea behind Hierarchical Models - ANSWER: The idea is to LEARN the prior to use for the
data we are analyzing by looking at related data sets
How are 'no pooling' and 'complete pooling' combined in hierarchical modeling? - ANSWER: We use
the other data sets to choose an appropriate prior for our analysis, giving us a good 'initial guess' for
the parameter value.
What is the Bayes Factor a ratio of? - ANSWER: It's a ratio of posterior odds to prior odds
What is the Bayes Factor under the simple hypotheses equal to? - ANSWER: The likelihood ratio!
What is the Bayes Factor under the composite hypotheses equal to? - ANSWER: A ratio of the
"weighted" likelihoods by the densities p(theta)
When calculating the Bayes Factor what type of prior distribution should be used? why? - ANSWER: A
proper prior! Otherwise, the BF becomes arbitrary
Main part of inversion sampling? - ANSWER: Calculating the inverse CDG for the target distribution.
Main part of rejection sampling? - ANSWER: Using an envelope (rectangular box) to generate points at
random over this region
What's one problem with sample importance resembling (SIR)? Explain briefly what it is. - ANSWER:
Particle Depletion, Where only a few simulated theta values contribute to the majority of the weights.
All direct sampling algorithms suffer from what? - ANSWER: The problem of dimensionality - easier to
implement to obtain posterior estimates of summary statistics in 1 dimension... but it becomes
significantly difficult to implement efficiently in higher dimensions.
Fact: Gamma(1) = ? - ANSWER: Gamma(1) = 1
Fact: 0! = ? - ANSWER: 0! = 1
How does one conduct a Prior Sensitivity Analysis? - ANSWER: We conduct this my rerunning the
MCMC iterations in Nimble using different priors on each of the parameters.
Fact: 1 choose 0 - ANSWER: 1
Fact: 1 choose 1 - ANSWER: 1
Fact: n! in terms of gamma - ANSWER: n! = gamma (n+1)
What are the advantages and disadvantages of Jeffrey's prior? - ANSWER: Advantages: invariant to
bijective transformations
Disadvantages: it's an improper prior (i.e. it doesn't integrate to 1)
What are the advantages and disadvantages of a uniform prior? - ANSWER: Advantages: it's a flat
prior with all equal length intervals having the same probability
Disadvantages: if we consider a non-linear transformation on the density, the prior is non-uniform.
By Brian J. Reich; Sujit K. Ghosh 9781032093185 ALL Chapters .
Transition Kernel - ANSWER: denoted 'P' - the transition kernel (or density) uniquely describes the
dynamics of the chain
Under what conditions will the distribution over the states of the Markov Chain converge to a
stationary distribution? - ANSWER: When the chain is 'aperiodic' and 'irreducible'
what does aperiodic mean? - ANSWER: A markov chain is aperiodic if for any state, the chain can
return to that state after a number of transitions that are a multiple of 1 and can also be 1
What does irreducible mean? - ANSWER: A markov chain is irreducible if any state can be reached
within finite time irrespective of the present state.
Pros and Cons of Trace Plots - ANSWER: It's a fairly efficient method but it is NOT robust.
Define Burn-In? - ANSWER: It's the initial realizations of the markov chain that we discard as the chain
had not converged to the stationary distribution yet.
What does it mean in terms of the posterior when there is low autocorrelation? - ANSWER: It means
samples are more representative of the posterior distribution
The autocorrelation plot shows the correlation between what types of samples? - ANSWER:
Successive samples
Define thinning - ANSWER: The process involves taking the kth realization of the markov chain and
discarding the rest
Thinning: Pros & Cons - ANSWER: It reduces autocorrelation, but it also discards potentially good
information.
What does BUGS stand for - ANSWER: Bayesian Inference Using Gibbs Sampling
What can transformation can be useful to aid comparability, interpretability, and MCMC
convergence? - ANSWER: Normalizing the data corresponding to explanatory variable(s)
What do we typically conclude when posterior summaries are similar? - ANSWER: The posterior
distribution is data-driven.
Explain the Gibbs Sampler (not its algorithm) - ANSWER: Gibbs sampler uses the set of full
conditionals of 'pi' to sample indirectly from the full posterior distribution.
Explain the Metropolis-Hastings (not its algorithm). What's important about it? - ANSWER: The MH
algorithm sequentially draws obs. from a distribution, conditional only on the last obs., thus inducing
a markov chain.
Important aspect is that the approximating candidate distribution can be IMPROVED at each step of
the simulation .
Define mixing - ANSWER: The movement around the parameter space.
What can cause poor mixing? - ANSWER: 1) a high rejection probability
2) very small step sizes
, Explain the idea behind Data Augmentation - ANSWER: We treat the missing data (or auxiliary
variables) as additional parameters to be estimated & form the joint posterior over both these
auxiliary variables and models parameters 'theta vector'
Explain the idea behind Hierarchical Models - ANSWER: The idea is to LEARN the prior to use for the
data we are analyzing by looking at related data sets
How are 'no pooling' and 'complete pooling' combined in hierarchical modeling? - ANSWER: We use
the other data sets to choose an appropriate prior for our analysis, giving us a good 'initial guess' for
the parameter value.
What is the Bayes Factor a ratio of? - ANSWER: It's a ratio of posterior odds to prior odds
What is the Bayes Factor under the simple hypotheses equal to? - ANSWER: The likelihood ratio!
What is the Bayes Factor under the composite hypotheses equal to? - ANSWER: A ratio of the
"weighted" likelihoods by the densities p(theta)
When calculating the Bayes Factor what type of prior distribution should be used? why? - ANSWER: A
proper prior! Otherwise, the BF becomes arbitrary
Main part of inversion sampling? - ANSWER: Calculating the inverse CDG for the target distribution.
Main part of rejection sampling? - ANSWER: Using an envelope (rectangular box) to generate points at
random over this region
What's one problem with sample importance resembling (SIR)? Explain briefly what it is. - ANSWER:
Particle Depletion, Where only a few simulated theta values contribute to the majority of the weights.
All direct sampling algorithms suffer from what? - ANSWER: The problem of dimensionality - easier to
implement to obtain posterior estimates of summary statistics in 1 dimension... but it becomes
significantly difficult to implement efficiently in higher dimensions.
Fact: Gamma(1) = ? - ANSWER: Gamma(1) = 1
Fact: 0! = ? - ANSWER: 0! = 1
How does one conduct a Prior Sensitivity Analysis? - ANSWER: We conduct this my rerunning the
MCMC iterations in Nimble using different priors on each of the parameters.
Fact: 1 choose 0 - ANSWER: 1
Fact: 1 choose 1 - ANSWER: 1
Fact: n! in terms of gamma - ANSWER: n! = gamma (n+1)
What are the advantages and disadvantages of Jeffrey's prior? - ANSWER: Advantages: invariant to
bijective transformations
Disadvantages: it's an improper prior (i.e. it doesn't integrate to 1)
What are the advantages and disadvantages of a uniform prior? - ANSWER: Advantages: it's a flat
prior with all equal length intervals having the same probability
Disadvantages: if we consider a non-linear transformation on the density, the prior is non-uniform.
