• We will quickly review selected materials and discuss some answering strategies
• Please refer to the lecture notes for complete materials

• Frequentist
  • parameter: \(\theta\) is fixed
  • data generating mechanism: \(f(x \mid \theta)\)
  • procedure: long-run frequency guarantee
• Bayesian
  • parameter: \(\theta\) is random
  • data generating mechanism: \(f(x,\theta)\)
  • procedure: coherence
• Both Frequentist and Bayesian need to make assumptions and specify model

Find the prior predictive/posterior/posterior predictive…

• Basics
  • use the formulas in Section 1.2 of the lecture notes
  • check if the course materials contain a direct example
  • proportionality constant
    • not required or can be numerically done in plot (Ex2.3.4)
    • required in point estimation (M0 Q3.1)
• Tricks
  • integrate gamma density (Ex1.2.3)
  • use representation (Ex1.3.2)
  • apply law of iterative expectation (useful for discrete r.v.; see M0 Q1.2)

Plot the prior predictive/posterior/posterior predictive…

1. Derive the required density
2. Implement the required density as a function in R
3. Design the plot (useful function: matplot,colorRampPalette,legend)

Comment on the plot…

• Describe the graph
• Link the features with previous parts of the question
• Discuss the impact of data (if any) on the graph

Suggest a conjugate/informative/non-informative/weakly-informative prior…

• Basics
  • refer to the summary in Section 2.1 of the lecture notes
  • check if the course materials contain a direct example
• Notes
  • be careful with the support of \(\theta\) (Ex2.1.2)
  • weakly-informative \(\approx\) large variance
  • additional properties like invariance and properness may be asked
• Issues with \(\infty\) in computation
  • \(\exp\{\ln(\cdot)\}\) “transform” (Ex2.3.4; also see Karte 2)
  • potential machine dependence (Ex2.3.5)

• Bayes estimator
  • commonly used estimators are optimal under specific losses (Theorem 3.2)
  • admissible if unique (Theorem 3.5)
  • minimax under certain conditions (Theorem 3.7 to 3.9)
• Minimax estimator
  • frequentist philosophy which concerns the least favorable situation
  • free of the unknown \(\theta\) fundamental problem in Example 3.11
  • not practical (Example 3.21)

Derive the Bayes estimator…

• Basics
  • derive the posterior
  • check if the loss function appears in Theorem 3.2 and related examples
  • apply Theorem 3.1 or Definition 3.6
• Tricks
  • use Lemma 3.4 (Ex3.1.2) or moment generating function
  • browse the web for closed-form of mean, median, mode etc.

Open-ended analysis related to point estimation

• Bayesian basics (Ex3.1.9)
  • propose a prior (sampling distribution is usually given)
  • choose hyperparameter and discuss information on hand
  • conduct point estimation
  • interpret the estimate and relate with real data (if any)
• Other topics
  • prove theoretical properties
  • compare different estimators via Monte Carlo experiment (M0 Q3.3; also see Karte 4)
  • discuss potential improvement of the analysis

• Answer ALL questions
  • Blank answer must score 0 (do not waste the chance!)
  • We give partial credit if you can show understanding of the materials
• The only thing that you CANNOT do is to communicate with others
• In other words, you can:
  • Use WolframAlpha to check algebra
  • Read Stack Overflow to debug R code
    • But you cannot ask questions there
  • Watch YouTube video tutorials…
• Ask on Blackboard for clarification
• Remember to cite the source
• Submit on time