Mathematics Ph.D. Dissertations
On the Posterior Consistency and Bernstein-von Mises Phenomenon for the Diaconis-Ylvisaker Prior
Date of Award
2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (Ph.D.)
Department
Statistics
First Advisor
Riddhi Ghosh (Committee Chair)
Second Advisor
John Chen (Committee Member)
Third Advisor
Junfeng Shang (Committee Member)
Fourth Advisor
John Boman (Other)
Abstract
We investigate the asymptotic behavior of the posterior distribution of the canonical parameter within the exponential family when the dimension of the parameter space grows with the sample size, specifically focusing on the Diaconis-Ylvisaker prior. This prior is notable as it acts as a conjugate prior for the exponential family. Our analysis establishes that, under mild conditions on both the true parameter value θ0 and the hyperparameters of the prior, the distance between the posterior distribution and a normal distribution, centered at the maximum likelihood estimator with a variance equal to the inverse of the Fisher information matrix, approaches zero in the expected total variation distance norm. Our Bernstein-von Mises theorem requires only that the dimension of the parameter space d grows linearly with the sample size n, with the condition d = o(n). In the process, we derive a concentration inequality for the quadratic form of the maximum likelihood estimator, circumventing the need for specific assumptions such as sub-Gaussianity. To illustrate our findings, we offer a specific application to the Multinomial-Dirichlet model, extending our analysis to deal with density estimation and Normal mean estimation problems.
Recommended Citation
Jin, Xin, "On the Posterior Consistency and Bernstein-von Mises Phenomenon for the Diaconis-Ylvisaker Prior" (2024). Mathematics Ph.D. Dissertations. 96.
https://scholarworks.bgsu.edu/math_diss/96