Under Construction
Visualized examples from Dupui's and Mao's and Birrell's papers.
- Dirac Masses: Example 3.1, in page 13
- Gaussian: Theorem 6.8, in page 114
- Unifrom: Example 1, section (4.1) in page 42.
Here we consider a simple example involving Dirac masses where the (f, Γ)-divergence can be explicitly computed using Theorem 3.3. This example further illustrates the two-stage mass-redistribution/mass-transport interpretation of the infimal convolution formula and demonstrates how the location and distribution of probability mass impacts the result.
Case 1 | Case 2 |
---|---|
1) P ~ N(1, 0.5), Q ~ N(3, 0.5) | 2) P ~ N(1, 0.5), Q ~ N(3, 1) | 3) P ~ N(1, 1), Q ~ N(3, 0.5) |
---|---|---|
For given c, b solves: exp(1 − b) − (1 − b) = 1 + c
When c = 0, b = 1 solves the equation. When c > 0 is small, 1 − b is also close to 0. Moreover, 1 − b can be written as an analytic function of √c
around 0 as 1-b = √2√c - c/3 + O(c^(3/2)).
To reproduce the achieved results run the file of your choice with the corresponding arguments. The output is a gif visualizing the Mass Redistribution and Transport respectivly to the chosen case (Dirac, Gaussian or Uniform).
Argument | Default Value | Info | Choices |
---|---|---|---|
--h |
0.1 | [float] Position of |
-0.5 < h < 0.5 |
Note
: Dirac case 1 has no arguments.
Argument | Default Value | Info | Choices |
---|---|---|---|
--m1 |
1.0 | [float] Mean of distribution P | - |
--sd1 |
0.5 | [float] Standard deviation of distribution P | - |
--m2 |
3.0 | [float] Mean of distribution Q | - |
--sd2 |
1.0 | [float] Standard deviation of distribution Q | - |
Note
: Default values corresponds to Case 2.
Argument | Default Value | Info | Choices |
---|---|---|---|
--c |
0.5 | [float] Parameter of distribution P | 0 < c < e-2 |
Note
: c value must be positive and less than e-2, thus 1-b ≥ 0.
@inproceedings{Dupuis2020,
author = {"Paul Dupuis, Yixiang Mao"},
title = {"Formulation and properties of a divergence used to compare probability measures without absolute continuity and its application to uncertainty quantification"},
journal = {arXiv:2011.08441},
year = {2020},
publisher = {"-?-"}
}
@inproceedings{Birrell2022,
author = {"Jeremiah Birrell, Paul Dupuis, Markos A. Katsoulakis, Yannis Pantazis, Luc Rey-Bellet"},
title = {"(f, Γ)-DIVERGENCES: INTERPOLATING BETWEEN, f -DIVERGENCES AND INTEGRAL PROBABILITY METRICS"},
journal = {arXiv preprint arXiv:2011.05953},
year = {2022},
publisher = {"Journal of Machine Learning Research"}
}