Topics on Statistics for Large Language Models (LLMs)

Course overview

This is a course for senior students to guide students to discover the frontier research problems of mathematics, applied mathematics and statistics, to understand the background and needs of theory and applied research, to broaden students’ horizons in the field of scientific research, and to encourage students to devote themselves to scientific research and work in the related fields of high-level applied mathematics in the future.

In this course, students are required to read and present research papers.

Reading list

Jump to a lecture:

Introduction
Conformal Prediction
Fairness
Watermarking
Beyond Fairness – Black-box Inference
Beyond Watermarking – Higher Criticism
Alignment
Beyond Higher Criticism – Multiple Hypothesis Testing
Beyond Alignment – Reinforcement Learning
Generative AI
Synthetic Data (Help or Hype?)
Privacy
Model Collapse

Lecture 1: Introduction

Ji, W., Yuan, W., Getzen, E., Cho, K., Jordan, M. I., Mei, S., Weston, J. E., Su, W. J., Xu, J., & Zhang, L. (2025). An Overview of Large Language Models for Statisticians (No. arXiv:2502.17814). arXiv. https://doi.org/10.48550/arXiv.2502.17814

Lecture 2: Conformal Prediction

Angelopoulos, A. N., & Bates, S. (2022). A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification (No. arXiv:2107.07511). arXiv. https://doi.org/10.48550/arXiv.2107.07511
Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R. J., & Wasserman, L. (2018). Distribution-Free Predictive Inference for Regression. Journal of the American Statistical Association, 113(523), 1094–1111. https://doi.org/10.1080/01621459.2017.1307116
Barber, R. F., Candès, E. J., Ramdas, A., & Tibshirani, R. J. (2021). Predictive inference with the jackknife+. The Annals of Statistics, 49(1), 486–507. https://doi.org/10.1214/20-AOS1965
Cherian, J., Gibbs, I., & Candes, E. (2024, November 6). Large language model validity via enhanced conformal prediction methods. The Thirty-eighth Annual Conference on Neural Information Processing Systems. https://openreview.net/forum?id=JD3NYpeQ3R

Lecture 3: Fairness

Kim, M. P., Ghorbani, A., & Zou, J. (2018). Multiaccuracy: Black-Box Post-Processing for Fairness in Classification (No. arXiv:1805.12317). arXiv. https://doi.org/10.48550/arXiv.1805.12317
Deng, Z., Zhang, J., Zhang, L., Ye, T., Coley, Y., Su, W. J., & Zou, J. (2022). FIFA: Making Fairness More Generalizable in Classifiers Trained on Imbalanced Data (No. arXiv:2206.02792). arXiv. https://doi.org/10.48550/arXiv.2206.02792
Zhang, L., Roth, A., & Zhang, L. (2024). Fair Risk Control: A Generalized Framework for Calibrating Multi-group Fairness Risks. Proceedings of the 41st International Conference on Machine Learning, 59783–59805. https://proceedings.mlr.press/v235/zhang24be.html

Lecture 4: Watermarking

Li, X., Ruan, F., Wang, H., Long, Q., & Su, W. J. (2025). A statistical framework of watermarks for large language models: Pivot, detection efficiency and optimal rules. The Annals of Statistics, 53(1), 322–351. https://doi.org/10.1214/24-AOS2468
Xie, Y., Li, X., Mallick, T., Su, W., & Zhang, R. (2025). Debiasing Watermarks for Large Language Models via Maximal Coupling. Journal of the American Statistical Association, 0(0), 1–11. https://doi.org/10.1080/01621459.2025.2520455
Li, X., Ruan, F., Wang, H., Long, Q., & Su, W. J. (2025). Robust detection of watermarks for large language models under human edits. Journal of the Royal Statistical Society Series B, qkaf056, https://doi.org/10.1093/jrsssb/qkaf056

Lecture 5: Beyond Fairness – Black-box Inference

Angelopoulos, A. N., Bates, S., Fannjiang, C., Jordan, M. I., & Zrnic, T. (2023). Prediction-powered inference. Science, 382(6671), 669–674. https://doi.org/10.1126/science.adi6000
Zrnic, T., & Candès, E. J. (2024). Cross-prediction-powered inference. Proceedings of the National Academy of Sciences, 121(15), e2322083121. https://doi.org/10.1073/pnas.2322083121
Miao, J., Miao, X., Wu, Y., Zhao, J., & Lu, Q. (2025). Assumption-lean and data-adaptive post-prediction inference. Journal of Machine Learning Research, 26(179), 1–31. https://www.jmlr.org/papers/v26/24-0056.html

Lecture 6: Beyond Watermarking – Higher Criticism

Donoho, D., & Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. The Annals of Statistics, 32(3), 962–994. https://doi.org/10.1214/009053604000000265
Donoho, D., & Jin, J. (2015). Higher Criticism for Large-Scale Inference, Especially for Rare and Weak Effects. Statistical Science, 30(1), 1–25. https://doi.org/10.1214/14-STS506
Li, J., & Siegmund, D. (2015). Higher criticism: p-values and criticism. The Annals of Statistics, 43(3). https://doi.org/10.1214/15-AOS1312

Lecture 7: Alignment

Liu, K., Long, Q., Shi, Z., Su, W. J., & Xiao, J. (2025). Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium (No. arXiv:2503.10990). arXiv. https://doi.org/10.48550/arXiv.2503.10990
Li, Z., Xu, T., Zhang, Y., Lin, Z., Yu, Y., Sun, R., & Luo, Z.-Q. (2024). ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models (No. arXiv:2310.10505). arXiv. https://doi.org/10.48550/arXiv.2310.10505
Rafailov, R., Sharma, A., Mitchell, E., Ermon, S., Manning, C. D., & Finn, C. (2024). Direct Preference Optimization: Your Language Model is Secretly a Reward Model (No. arXiv:2305.18290). arXiv. https://doi.org/10.48550/arXiv.2305.18290

Lecture 8: Beyond Higher Criticism – Multiple Hypothesis Testing

Barnett, I., Mukherjee, R., & Lin, X. (2017). The Generalized Higher Criticism for Testing SNP-Set Effects in Genetic Association Studies. Journal of the American Statistical Association, 112(517), 64–76. https://doi.org/10.1080/01621459.2016.1192039
Liu, Y., & Xie, J. (2020). Cauchy Combination Test: A Powerful Test With Analytic p-Value Calculation Under Arbitrary Dependency Structures. Journal of the American Statistical Association, 115(529), 393–402. https://doi.org/10.1080/01621459.2018.1554485
Vesely, A., Finos, L., & Goeman, J. J. (2023). Permutation-based true discovery guarantee by sum tests. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(3), 664–683. https://doi.org/10.1093/jrsssb/qkad019

Lecture 9: Beyond Alignment – Reinforcement Learning

Sutton, R. S., & Barto, A. (2020). Reinforcement learning: An introduction (Second edition). The MIT Press. http://incompleteideas.net/book/RLbook2020.pdf
https://github.com/zhoubolei/introRL
https://github.com/ucla-rlcourse/RLexample

Lecture 10: Generative AI

Kotelnikov, A., Baranchuk, D., Rubachev, I., & Babenko, A. (2023). TabDDPM: Modelling Tabular Data with Diffusion Models. Proceedings of the 40th International Conference on Machine Learning, 17564–17579. https://proceedings.mlr.press/v202/kotelnikov23a.html
Ho, J., Jain, A., & Abbeel, P. (2020). Denoising Diffusion Probabilistic Models (No. arXiv:2006.11239). arXiv. https://doi.org/10.48550/arXiv.2006.11239
Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. (2021). Score-Based Generative Modeling through Stochastic Differential Equations (No. arXiv:2011.13456). arXiv. https://doi.org/10.48550/arXiv.2011.13456
Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., & Lakshminarayanan, B. (2021). Normalizing Flows for Probabilistic Modeling and Inference. Journal of Machine Learning Research, 22(57), 1–64.
CVPR2022 tutorial (https://cvpr2022-tutorial-diffusion-models.github.io/)
MIT course (https://diffusion.csail.mit.edu/)

Lecture 11: Synthetic Data (Help or Hype?)

Shen, X., Liu, Y., & Shen, R. (2024). Boosting Data Analytics With Synthetic Volume Expansion (No. arXiv:2310.17848). arXiv. https://doi.org/10.48550/arXiv.2310.17848
Tian, X., & Shen, X. (2025). Enhancing Accuracy in Generative Models via Knowledge Transfer (No. arXiv:2405.16837). arXiv. https://doi.org/10.48550/arXiv.2405.16837
Jain, A., Montanari, A., & Sasoglu, E. (2024, November 6). Scaling laws for learning with real and surrogate data. The Thirty-eighth Annual Conference on Neural Information Processing Systems. https://openreview.net/forum?id=NAcHv7vtL2&noteId=J92zK59XDi

Lecture 12: Privacy

Su, W. J. (2025). A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation, and Blackwell’s Theorem. Annual Review of Statistics and Its Application, 12(1), 157–175. https://doi.org/10.1146/annurev-statistics-112723-034158
Yu, D., Naik, S., Backurs, A., Gopi, S., Inan, H. A., Kamath, G., Kulkarni, J., Lee, Y. T., Manoel, A., Wutschitz, L., Yekhanin, S., & Zhang, H. (2022). Differentially Private Fine-tuning of Language Models (No. arXiv:2110.06500). arXiv. https://doi.org/10.48550/arXiv.2110.06500
Dong, J., Roth, A., & Su, W. J. (2022). Gaussian Differential Privacy. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1), 3–37. https://doi.org/10.1111/rssb.12454
Abadi, M., Chu, A., Goodfellow, I., McMahan, H. B., Mironov, I., Talwar, K., & Zhang, L. (2016). Deep Learning with Differential Privacy. Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, 308–318. https://doi.org/10.1145/2976749.2978318
Bu, Z., Dong, J., Long, Q., & Su, W. (2020). Deep Learning With Gaussian Differential Privacy. Harvard Data Science Review, 2(3). https://doi.org/10.1162/99608f92.cfc5dd25

Lecture 13: Model Collapse

Shumailov, I., Shumaylov, Z., Zhao, Y., Papernot, N., Anderson, R., & Gal, Y. (2024). AI models collapse when trained on recursively generated data. Nature, 631(8022), 755–759. https://doi.org/10.1038/s41586-024-07566-y
Gerstgrasser, M., Schaeffer, R., Dey, A., Rafailov, R., Sleight, H., Hughes, J., Korbak, T., Agrawal, R., Pai, D., Gromov, A., Roberts, D. A., Yang, D., Donoho, D. L., & Koyejo, S. (2024). Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data (No. arXiv:2404.01413). arXiv. https://doi.org/10.48550/arXiv.2404.01413
Dohmatob, E., Feng, Y., & Kempe, J. (2024). Model Collapse Demystified: The Case of Regression. Advances in Neural Information Processing Systems, 37, 46979–47013.