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Description: The main goal
of this course is to expose students to modern ideas in
statistical theory. Whereas classical theory is
concerned with the behavior of statistical estimates
when the number of variables is fixed and the sample
size increases, our emphasis is on statistical inference
in high-dimensional settings where there may be as many,
or more, variables than observations. Our focus is
motivated by always newer technologies, which now
produce extremely large datasets, often with huge
numbers of measurements on each of a comparatively small
number of experimental units.
Prerequisite: Stats 300A and
300B. Knoweldge of probability theory at the level of
Stats 310A and 310B.
Syllabus:
- Testing problems in high dimensions: sparse
alternatives (needle in a haystack) and nonsparse
alternatives, Bonferroni's method, Fisher's test,
ANOVA, higher criticism.
- Multiple testing problems: familywise error
rate (FWER), procedures for controlling FWER, false
discovery rate (FDR), procedures for controlling FDR,
empirical Bayes view of FDR, local FDR.
- Combining results from several tests via e-values.
- Conditional testing, controlled variable
selection, knockoffs.
- Topics in selective inference: false coverage rate, post-selection inference, selection after the LASSO.
- Topics in conformal inference.
- James-Stein estimation, Stein's unbiased risk
estimate (tentative).
- Theory of high-dimensonal regression:
approximate message passing algorithms (tentative).
Textbooks:
We will
not follow a textbook but the students might find the following
references useful for background reading.
- Large-Scale Inference: Empirical Bayes Methods for
Estimation, Testing, and Prediction by B. Efron, IMS Monographs.
- Algorithmic Learning in a Random World by Vladimir Vovk, Alexander Gammerman and Glenn Shafer, Springer New York, NY
The books below provide background for a few probabilistic results
that we shall use.
- High-Dimensional Statistics: A Non-Asymptotic Viewpoint, by M. Wainwright, Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 48.
- Empirical Processes With Applications to Statistics by
G Shorack and J Wellner, Classics in Applied Mathematics.
Handouts: We will post lecture notes
on Overleaf (and perhaps on Canvas and online, see the proper
section).
Course assistant and office hours:
Grading (tentative):
- Homework assignments: 45%
- Homework assignments will generally be distributed on
Wednesdays and due the following
Wednesday.
- Late homeworks will NOT be accepted for grading
(medical emergencies excepted with proof).
- There will be 6 or 7 assignments; the lowest score
will be dropped in the final grade.
- It is encouraged to discuss the problem sets with others, but
everyone needs to turn in a unique personal write-up.
- Scribing of lectures: 10%
- Most lectures are already scribed but we shall adjust some here and there.
- Final project: 45%.
- Most likely a take-home exam.
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Course policies:
- Use of sources (people, books, internet and so on)
without citing them in homework sets results in failing
grade for course.
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