• ↑
  • STA260

STA260 Lecture 12

STA260 Lecture 12 Raw
STA260 Lecture 12 Flashcards

• Completed Notes Status

  • Completed insertions: 3
  • Ambiguities left unresolved: 1

• Lecture Summary

  • Central objective: Extend parameter estimation methods to confidence intervals, covering exact, approximate, and Wald-type interval construction methods
  • Key concepts:
    • Bayesian Approach to Parameter Estimation: Uses Bayes' Theorem to combine prior beliefs with observed data to obtain posterior distributions of parameters
    • Conjugate Prior: A prior distribution that, when combined with the likelihood function, yields a posterior distribution from the same distributional family, simplifying Bayesian computation
    • Exact Confidence Interval: An interval $C(X)$ for parameter $\theta$ where $P(\theta \in C(X))=1-\alpha$ holds precisely, constructed using pivotal quantities like $\frac{(n-1)S^2}{{\sigma }^2}\sim {\chi }_{(n-1)}^2$
    • Pivotal Quantity: A function of the sample and parameter whose distribution does not depend on unknown parameters, enabling exact interval construction
    • Wald Type Confidence Interval: An approximate interval of the form $\hat{\theta }\pm Z_{\frac{\alpha }{2}}\mathrm{SE}[\hat{\theta }]$, applicable for large samples when the estimator is approximately normally distributed
  • Connections:
    • Bayesian Approach to Parameter Estimation provides an alternative to frequentist Maximum Likelihood Estimation by incorporating prior information
    • Exact Confidence Interval methods rely on pivotal quantities with known distributions to construct intervals with guaranteed coverage
    • Wald-type intervals use the Central Limit Theorem asymptotic normality of estimators, connecting to Fisher Information

• TK Resolutions

  • #tk: Self-study on the slides
    • Answer: This is a reminder to review lecture slides independently; no specific question to resolve from lecture context
  • #tk: Find posterior mean for Beta Distribution example
    • Answer: For $p|X\sim \mathrm{Beta}(\alpha +\sum _{i=1}^n{x}_i,\beta +n-\sum _{i=1}^n{x}_i)$, the posterior mean is $\mathrm{E}[p|X]=\frac{{\alpha }^{\prime }}{{\alpha }^{\prime }+{\beta }^{\prime }}=\frac{\alpha +\sum _{i=1}^n{x}_i}{\alpha +\beta +n}$ [web:5][web:8]
  • #tk: Complete slide 55 exercise
    • [AMBIGUOUS]: Cannot resolve without access to slide 55 content; requires lecture materials to identify the specific exercise
  • #tk: Create TikZ graph showing chi-squared critical value regions
    • Answer: The graph should display a right-skewed Chi-Squared Distribution with ${\chi }_{(1-\frac{\alpha }{2})}^2$ marking the left critical value (with area $\frac{\alpha }{2}$ to the left), ${\chi }_{\left(\frac{\alpha }{2}\right)}^2$ marking the right critical value (with area $\frac{\alpha }{2}$ to the right), and the middle region representing probability $1-\alpha$ [web:6][web:12]
  • #tk: Practice deriving Fisher Information using the first derivative method
    • Answer: For $X\sim \mathrm{Poisson}(\lambda )$, calculate $I(\lambda )=\mathrm{E}\left[{(\frac{\partial }{\partial \lambda }\log f_X(x;\lambda ))}^2\right]=\mathrm{E}\left[{(-1+x{\lambda }^{-1})}^2\right]=\mathrm{E}[1-2x{\lambda }^{-1}+x^2{\lambda }^{-2}]=1-2\lambda {\lambda }^{-1}+\mathrm{E}[X^2]{\lambda }^{-2}=1-2+({\lambda }^2+\lambda ){\lambda }^{-2}=\frac{1}{\lambda }$, yielding the same result as the second derivative method

• Practice Questions

  • Remember/Understand:
    • What is the relationship between prior, likelihood, and posterior distributions in Bayesian Approach to Parameter Estimation?
    • Define a Conjugate Prior and explain why it simplifies Bayesian computation
    • What distinguishes an Exact Confidence Interval from an approximate confidence interval?
  • Apply/Analyze:
    • Given $X_1,\dots ,X_n\sim \mathrm{Exponential}(\lambda )$ with prior $\lambda \sim \mathrm{\Gamma }(\alpha ,\beta )$, derive the posterior distribution of $\lambda$
    • Construct a $95\mathrm{\%}$ Exact Confidence Interval for ${\sigma }^2$ given sample variance $S^2$ from a normal population with $n=20$
    • For $X\sim \mathrm{Binomial}(n,p)$ with $\hat{p}=\frac{X}{n}$, derive the Wald Type Confidence Interval for $p$ at the $90\mathrm{\%}$ confidence level
  • Evaluate/Create:
    • Compare the advantages and limitations of Bayesian Approach to Parameter Estimation versus frequentist methods for small sample sizes
    • When would you prefer a Pivotal Quantity method over a Wald Type Confidence Interval for interval estimation? Justify your choice with specific scenarios

• Challenging Concepts

  • Bayesian Approach to Parameter Estimation:
    • Why it's challenging: Requires integrating prior beliefs with data, involving manipulation of proportionality constants and recognizing distributional forms from non-standard expressions
    • Study strategy: Practice identifying kernel forms of distributions by ignoring terms without the parameter; work through multiple conjugate prior examples systematically (Poisson-Gamma, Bernoulli-Beta, Normal-Normal)
  • Conjugate Prior:
    • Why it's challenging: Understanding why certain prior-likelihood pairs yield posterior distributions in the same family requires algebraic manipulation and pattern recognition
    • Study strategy: Create a reference table of common conjugate pairs (likelihood family, conjugate prior family, posterior parameters); derive at least three examples from first principles
  • Exact Confidence Interval for variance:
    • Why it's challenging: Involves inverting inequalities when manipulating $\frac{(n-1)S^2}{{\sigma }^2}\sim {\chi }_{(n-1)}^2$, requiring careful attention to inequality direction changes and proper identification of critical values
    • Study strategy: Practice the algebraic steps slowly, drawing number lines to track inequality directions; work through examples with different confidence levels and sample sizes

• Action Plan

  • Immediate review actions:
    • Review all [COMPLETED] insertions in STA260 Lecture 12 Raw and verify with course materials
    • Address slide 55 exercise by consulting lecture slides
    • Re-read Lecture Summary and confirm all key concept links are understood
  • Practice and application:
    • Answer Remember/Understand questions without notes
    • Work through Apply/Analyze questions with formula sheet only
    • Attempt Evaluate/Create questions and discuss reasoning
  • Deep dive study:
    • Focus on Challenging Concepts listed above
    • Extend Bayesian Approach to Parameter Estimation, Conjugate Prior, Exact Confidence Interval, Pivotal Quantity, and Wald Type Confidence Interval atomic notes with examples from this lecture
    • Derive posterior mean formula for Beta Distribution independently ^[1]
  • Verification and integration:
    • Cross-reference confidence interval methods with STA258 materials on Confidence Intervals
    • Verify chi-squared critical value notation with textbook
    • Link this lecture to STA260 Lecture 11 Raw (Bayesian approach introduction) and prepare for Lecture 13

• Footnotes