Bayesian Statistics and Posterior Inference
Bayes' theorem as an inferential engine
Posterior computation
Bayesian inference uses , combining a prior with the likelihood to obtain a posterior distribution over parameters.
Bayesian workflow
Model specification, prior choice, posterior computation, posterior predictive checking, and model criticism form a modern Bayesian analysis pipeline.
Which expression is proportional to the posterior density?
Bayes' theorem gives posterior proportional to likelihood times prior.
Correct answer:
What is a conjugate prior?
Conjugacy provides closed-form updating for selected likelihood-prior pairs.
Correct answer: A prior that yields a posterior in the same family as the prior.
Posterior predictive distribution
The posterior predictive for a future observation is . It accounts for parameter uncertainty.
What does the posterior predictive distribution integrate over?
It averages future-data likelihoods over the posterior distribution of the parameters.
Correct answer: Parameter uncertainty
State one advantage of Bayesian inference in complex models.
Bayesian methods can encode prior information and propagate uncertainty through all levels of the model.
Correct answer: It naturally quantifies uncertainty and supports hierarchical modeling.
Frequentist vs Bayesian perspective
Frequentist
- Parameters are fixed but unknown
- Probability refers to long-run behavior
- Inference based on sampling distributions
Bayesian
- Parameters are random under the prior
- Probability quantifies degree of belief
- Inference based on posterior distributions