Posts

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

More stochastic than?

In some current research, I need a way to compare distributions and talk about one probability measure that is more stochastic than another. After a little effort (wheel re-invention and directed literature search), and I happened on the very cool idea of 2nd-order stochastic dominance. But here is a different approach, just for kicks.

The basic idea is that given two measures with the same expected value, measures that exhibit more variation around the center are dominated by measures that exhibit less. So singleton \(\delta\)-functions are dominates over all other measures.

Let's say that for two measures \(f\) and \(g\) centered at \(0\), measure \(f\) is dominated by measure \(g\) if and only if

\[ \forall x, \quad \int_{0}^{x} \int_{0}^{v} f(u) d u d v \geq \int_{0}^{x} \int_{0}^{v} g(u) d u d v \]

Now, just looking at this doesn't make much sense to me -- no intuition. So to get a better handle, it's useful to actually have an example we can study. One of the easiest measures where we can apply this is the logistic measure. If the probability distribution is

\[p(x;a)=\frac{1}{a \left(e^{x/2a} + e^{-x/2a}\right)^2}\]

then the lowered CDF is

\[P(x; a) := \int_0^{x} p(x,a) dx = \frac{e^{x/2a} - e^{-x/2a}}{2 (e^{x/2a} + e^{-x/2a})}\]

and the integrated CDF which we need for comparison is

\[I(x; a) := \int_{0}^{x} P(x; a) d x = a \ln\left(\frac{e^{-x/2a} + e^{x/2a}}{2} \right)\]

We can plot a few examples (below). And what we see is that as \(a\) increases and the measure gains more variation, the CDF integrated from the mean is lower because there is less probability near the mean. The delta-function which integrates to an absolute value dominates everything. This is provoking, in that it relates scalar probability distributions to a subset of convex functions, and convex functions generally have very useful properties when we consider optimization problems.

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