STAT 638 (Fall 2026), Lecture 29
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Overview
Samiran Sinha develops Bayesian inference for multivariate normal means and covariance matrices using normal-inverse-Wishart and independent priors, then extends the framework to partially observed data through data augmentation. Examples include 176 complete records from 178 elderly patients and the 153-day, six-variable New York airquality dataset; the lecture derives posterior updates, explains MCAR, MAR, and MNAR assumptions, and gives a Gibbs-sampling strategy for imputing missing values.
Key takeaways
- With a normal-inverse-Wishart prior, multivariate normal inference remains conjugate: the covariance has an inverse-Wishart posterior and the mean has a normal conditional posterior.
- The elderly-patient analysis uses 176 complete cases from 178 records and sets ν₀ = 5, S₀ = I₃, μ₀ = (100, 2, 1), and τ₀ = 0.1.
- Missingness assumptions determine whether the missingness process can be ignored: MNAR requires modeling dependence on the unobserved outcomes, unlike the usual MCAR or MAR analyses.
- For an observed-data likelihood, integrate the complete-data distribution over each subject's missing components; data augmentation avoids repeatedly evaluating those integrals by sampling the missing values.
- In a multivariate normal model, missing values can be imputed directly from their conditional normal distribution given observed components, after which standard posterior updates for θ and Σ apply.
- AI can assist with calculations or presentation production, but Sinha stresses that users must verify derivations and provide their own structure and substantive ideas.
Chapters
- Samiran Sinha introduces inference for the mean vector and covariance matrix of a multivariate normal distribution.
- Chapter 7 is excluded from the upcoming cumulative exam, which covers the other material.
- Sinha encourages students to bring original scientific ideas to their projects and treats project discussions as an opportunity to learn.
- For IID vectors Y_i from a multivariate normal distribution, the conjugate prior pairs an inverse-Wishart prior for covariance Σ with a conditional normal prior for mean θ given Σ.
- The prior uses scale matrix S₀, degrees of freedom ν₀, prior mean μ₀, and mean precision parameter τ₀.
- The posterior retains the normal-inverse-Wishart form; the updated mean μ_N combines the sample mean with μ₀ as a weighted average.
- Posterior sampling can draw Σ from its inverse-Wishart marginal and then θ conditionally, or alternate draws from the full conditional distributions.
- The blood-work example measures calcium, inorganic phosphorus, and alkaline phosphatase for 178 elderly patients, retaining 176 complete observations.
- The goal is Bayesian inference for the three-dimensional mean vector and covariance matrix.
- Sinha specifies ν₀ = 5, S₀ = I₃, μ₀ = (100, 2, 1), and τ₀ = 0.1, then asks students to implement posterior sampling.
- An alternative prior makes θ and Σ independent, with θ normally distributed and Σ inverse-Wishart distributed.
- Although the joint posterior is not available in the same closed form, the full conditional distributions for θ and Σ support Gibbs sampling.
- To derive the conditionals, collect terms involving Σ, express them using traces and determinants, and complete the square in θ.
- Sinha emphasizes checking algebra rather than relying blindly on AI, and recommends giving tools such as Claude specific ideas when using them to design project presentations.
- Jeffreys prior is constructed from the Fisher information matrix, using the square root of its determinant for the parameterization under discussion.
- For multivariate normal models, the mean contributes P parameters and the symmetric covariance matrix contributes P(P + 1)/2 unique elements.
- Sinha distinguishes Jeffreys-prior forms associated with independent and dependent prior setups; the prior forms described do not involve the mean parameter.
- The R airquality data contain 153 days of New York measurements from May through September 1973 on six variables, providing a second application for estimating means and covariances.
- For each P-dimensional observation Y_i, an indicator M_ij equals 1 when component j is observed and 0 when it is missing; the vector is partitioned into observed and missing components.
- Discarding the two incomplete rows in the 178-patient example wastes information from their observed calcium, phosphorus, or alkaline phosphatase values.
- MCAR means missingness is independent of complete data; MAR means it depends on observed data but not missing values after conditioning on observed data; MNAR allows dependence on the missing values.
- Under an ignorable missingness mechanism, each subject's observed-data likelihood is obtained by integrating the complete-data model over that subject's missing components.
- Integrating out missing components separately for every subject can make posterior computation difficult.
- Data augmentation treats the missing entries Y_M as latent quantities and samples them from their full conditional distributions alongside model parameters.
- Each iteration alternates between imputing missing values conditional on observed data and current parameters, then updating the parameters using the completed data.
- The resulting imputations also provide posterior draws of plausible complete datasets rather than a single fixed replacement.
- Partition θ and Σ according to missing and observed coordinates; the conditional distribution of missing components given observed components is multivariate normal.
- The conditional mean adjusts the missing-coordinate mean using the observed residuals and cross-covariance, while the conditional covariance accounts for the observed coordinates.
- For each subject, draw missing values from that conditional normal distribution, assemble the completed vector, then sample Σ and θ from their full conditionals.
- Repeat the imputation and parameter-update steps many times to obtain Gibbs samples for inference.
- Sinha provides code for the missing-data procedure and says implementation practice is necessary to understand it beyond a superficial level.
- Students are asked to run the code, verify that it works, and then examine it line by line.
Transcript excerpt
0:01 Sinha, Samiran: Good morning, everyone. So, today we'll talk about Chapter 7. Sinha, Samiran: Multimediate analysis. Multimediate normal distribution, how do you make statistical inference of the mean and the covariance parameter? Sinha, Samiran: And, so, next week, next Friday, is your exam, first exam of the semester. I'm sure you all are very excited about that. First exam. So, for that exam. Sinha, Samiran: I will not include this, Chapter 7. Chapter 7 will be excluded from the exam. Sinha, Samiran: But everything else will be there. It's a cumulative exam. Sinha, Samiran: And I have not decided about the time, 1 hour. Sinha, Samiran: Maybe a little bit more than one hour? Okay, I have not decided. There will be two exams I have to write, one for your class, another for another class, so…
1:03 Sinha, Samiran: I'll do that. And also, please work hard on your projects. I'm very excited about those projects. Sinha, Samiran: And why? Because I do not know Sinha, Samiran: lot of science behind those projects, so I'm looking forward to learn something new from you guys, okay? Sinha, Samiran: I… so, basically, you may think that I'm the professor, and you are the student, so I am the… what? Sinha, Samiran: ocean of knowledge. No, that is not true. I'm also collecting knowledge. I'm collecting, like, collecting pebbles at the seashore, okay? So I have more experience. Does not mean I have more knowledge than you. And what I can do, I can understand quickly Sinha, Samiran: than you guys, statistics, methodology, models, does not mean I know everything. So that is why I want to learn something from these projects.
2:04 Sinha, Samiran: So think about the innovation that you can bring in, and excite me. Sinha, Samiran: Alright, so… We have been talking about multimedia normal distribution.
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Related notes
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.