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Bayesian Statistics: Concepts & Definitions 100% Verified. Transition Kernel - answerdenoted '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 distributi...

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Instelling Bayesian Statistics
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©THEBRIGHT EXAM STUDY SOLUTIONS 8/22/2024 12:54 PM

Bayesian Statistics: Concepts & Definitions
100% Verified.

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.

, ©THEBRIGHT EXAM STUDY SOLUTIONS 8/22/2024 12:54 PM

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

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