STAT 638 (Fall 2026), Lecture 19

Overview

Samiran Sinha reviews STAT 638 project grading and hierarchical-model exercises, then develops a Bayesian model for comparing 30-day heart-attack mortality across five Texas hospitals with unequal sample sizes. He builds a binomial-beta hierarchy with Gamma priors on its hyperparameters, explains Gibbs sampling and Metropolis-Hastings updates, and shows how partial pooling shrinks Hospital A’s raw 20% mortality estimate to 14.5%.

Key takeaways

In the Normal hierarchical model, conditioning on a group effect gives observation variance σ², while integrating over the random group effect gives marginal variance τ² + σ².
For the hospital example, Yᵢ ~ Binomial(nᵢ, pᵢ) and pᵢ ~ Beta(α, β) let hospitals with small admission counts borrow information from hospitals with larger counts.
The full conditional for each hospital probability is Beta(Yᵢ + α, nᵢ − Yᵢ + β), but α and β do not have standard full conditional distributions.
When deriving α and β conditionals, the Beta density’s Gamma-function normalizing constants must be retained because they depend on the unknown hyperparameters.
Random-walk Metropolis-Hastings proposes α and β on the log scale; the example’s acceptance rates were 0.4029 and 0.4172, respectively.
Hierarchical shrinkage moved Hospital A’s estimated 30-day mortality from a raw 20% to a posterior estimate of 14.5%, while its 95% credible interval remained wider than Hospital B’s because A had far fewer patients.

Chapters

0:00 Project Reports: Peer Grading and Instructor Review

Sinha plans to grade 17 projects and add peer assessments to reduce the influence of any one grader on students’ grades.
Each written report is to be randomly assigned to 2–3 students, with safeguards against assigning anyone their own project.
Students assess the written report only; Sinha evaluates the oral presentation and question-and-answer portion.
Written projects are due by November 18 and will be placed in a shared folder.

4:20 Exercise 8.1: Conditional and Marginal Variance in a Normal Hierarchy

The model gives group parameters θ₁,…,θₘ an IID Normal(μ, τ²) distribution and observations within group j a Normal(θⱼ, σ²) distribution.
Conditional on θⱼ, an observation has variance σ²; accounting for uncertainty in θⱼ raises the marginal variance to τ² + σ².
Writing Yᵢⱼ = θⱼ + εᵢⱼ separates group-level variability from observation-level residual variation.
The residual εᵢⱼ is independent of θⱼ, so their covariance is zero.

8:40 Exercise 8.1: Covariances and Bayes’ Rule

Two observations from the same group are independent conditional on its θ parameter, so their conditional covariance is zero.
Without conditioning on θ, observations in a shared group can covary because they share the same random group effect.
The exercise asks students to calculate six variance and covariance quantities and compare them with their earlier answers.
Bayes’ rule shows that μ given the group parameters θ₁,…,θₘ does not directly depend on the observations Y₁,…,Yₘ; Sinha recommends practicing Exercises 8.2 and 8.3 for the quiz and exam.

15:00 From Normal Hierarchies to Hospital Mortality Data

Chapter 8 introduced hierarchical models in a Normal linear-model setting; Sinha now extends the same information-sharing idea to a Bernoulli/binomial model.
The example compares 30-day mortality after heart attacks at five Texas hospitals; the dataset is hypothetical and was generated with ChatGPT or Claude.
Hospital admission counts differ sharply, from 20 patients at one hospital to 500 at another, making raw mortality rates unequally precise.
Observed mortality rates range from 9% to 20%; Hospital A’s 20% rate comes from only 20 patients.

20:00 Hospital Comparisons: Binomial Likelihood and Beta Hierarchy

For hospital i, deaths Yᵢ follow Binomial(nᵢ, pᵢ), where nᵢ is admissions and pᵢ is the 30-day mortality probability.
A Beta(α, β) distribution models the hospital-specific probabilities pᵢ, allowing mortality information to be shared across hospitals.
Sinha places Gamma priors on α and β rather than fixing them; fixed Beta shape parameters would prevent the intended hierarchical learning.
The model has seven unknowns: five hospital probabilities plus α and β; the slide’s Beta distribution parameter label should read α and β.

25:00 Gibbs Updates: Beta Full Conditionals and Hyperparameter Densities

The full conditional for each pᵢ is Beta(Yᵢ + α, nᵢ − Yᵢ + β), obtained by combining the binomial likelihood with the Beta prior.
The conditional densities for α and β include the product of the five Beta densities and their Gamma priors.
Because the Beta normalizing constants depend on α and β through Gamma functions, those constants cannot be dropped when deriving their conditionals.
The α and β full conditionals are not standard distributions, so direct Gibbs draws are unavailable.

30:00 MCMC Design: Updating All Seven Hospital-Model Parameters

Sinha describes the sampler as a Markov chain Monte Carlo procedure that repeatedly updates p₁,…,p₅, α, and β using their latest available values.
The five hospital probabilities can be drawn from Beta full conditionals; α and β require Metropolis-Hastings proposals.
Parameter-update order is flexible, and the log full-conditional densities are used to evaluate proposed hyperparameter values.
The lecture previews random-walk Metropolis-Hastings, with fuller treatment deferred to Chapter 10.

35:00 R MCMC Setup: Hospital Counts, Priors, and Initial Values

The example uses hospital death counts and admission totals; individual patient records are unnecessary because the death count is sufficient for the binomial model.
Gamma-prior hyperparameters are set illustratively to 2 and 0.1, without a substantive justification; changing them across reasonable values is a sensitivity analysis.
The code runs 50,000 MCMC iterations and stores the five p values in a iterations-by-hospitals matrix, alongside α and β.
Initial p values use (Y + 1)/(n + 2), a smoothed estimate close to the raw sample proportion; Sinha also notes that Y/n could be used.

40:00 Metropolis-Hastings for α and β: Log-Scale Proposals

Each hyperparameter proposal is generated on the log scale from a Normal random walk centered at its previous log value, then transformed back.
The acceptance calculation combines the difference in log conditional densities with the proposal-density ratio.
A Uniform(0,1) draw determines whether to accept the proposal; rejection leaves the parameter at its previous value.
The proposal standard deviations are tuned by trial and error; the example uses 0.5 for both α and β.

45:00 Sampler Diagnostics and Partial-Pooling Mortality Estimates

The reported acceptance rates are 0.4029 for α and 0.4172 for β; Sinha uses acceptance rates to assess and tune proposal scales.
Posterior summaries include medians and the 2.5th and 97.5th percentiles as 95% credible intervals; the lecture notes that medians can be useful for skewed posteriors.
Hospital A’s estimated mortality is 0.145 with a 0.07–0.262 credible interval, down from its raw rate of 20% through hierarchical shrinkage.
Hospital B’s estimate is 0.102 with a 0.078–0.129 interval; the estimates reflect both cross-hospital pooling and differing sample sizes.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Samiran Sinha.