Measuring sample quality with Stein's method
Abstract/Contents
- Abstract
- As the size of datasets has grown, classical methods like Markov chain Monte Carlo have become increasingly burdensome from a computational perspective. Practitioners have been turning to biased Markov chain Monte Carlo procedures that are able to trade off asymptotic exactness for computational speed. Unfortunately, previously used diagnostics to aid with these methods are insufficient for assessing the asymptotic bias incurred. We will first introduce a new computable quality measure based on Stein's method that quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. Our first main theoretical contribution will be showing that our measure converges to zero only if a sample converges to its target distribution. Empirically we will show this discrepancy avoids the problems faced by previous diagnostics, e.g., effective sample size. Our next step will be to generalize these ideas to cover a larger class of target distributions. By studying Ito diffusions with fast-mixing rates, we will be able to extend the purview of acceptable target distributions from distributions with strongly log-concave densities with bounded third and fourth derivatives to a much larger class that includes multimodal and heavy-tailed distributions. Finally, in the last chapter, we will study a variation of our previous methods that is computationally feasible to obtain for much larger samples. This variation will combine our ideas with those from reproducing kernel Hilbert spaces to define a closed-form expression for our measure. While other authors have also recommended this faster variation, we show that most common choices for the kernel function are insufficient for controlling convergence in the multivariate setting. A special class of kernel functions that do indeed control convergence will be proposed and shown to dominate the other traditional kernel functions empirically.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2017 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Gorham, Jackson |
|---|---|
| Associated with | Stanford University, Department of Statistics. |
| Primary advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Primary advisor | Mackey, Lester |
| Thesis advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Thesis advisor | Mackey, Lester |
| Thesis advisor | Holmes, Susan |
| Advisor | Holmes, Susan |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Jackson Gorham. |
|---|---|
| Note | Submitted to the Department of Statistics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2017. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2017 by Jack Christopher Gorham
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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