Concentration of measure and bounds on variance

2011-02-08T17:25:32Z

I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao bound for Gaussian i.i.d. arguments.

The function has a singularity at the point $\mathbb{x}_0$ where $f(\mathbb{x}_0)=0$, in the sense that $\|\nabla{}f\|\sim{}1/f$ as $\mathbb{x}\to\mathbb{x}_0$

The Cramer-Rao bound then doesn't make sense, because it diverges at $\mathbb{x}_0$, while the variance of $f$, obviously, remains bounded. What I am looking for, I guess, some type of "concentration of measure"/"deviation inequality"-type sharp bound on the variance of $f$.

The literature on concentration of measure phenomenon is extensive and deals with fairly advanced topics, whereas I am looking for something rather more basic. If you could point towards some starting point, your help will be appreciated.

Answer by Tom LaGatta for Concentration of measure and bounds on variance

2011-02-08T22:30:54Z

Let $X_i$ and $X'_i$ be i.i.d random variables. Write $f = f(X_1, \dots, X_n),$ and define $$f_i = f(X_1, \dots, X'_i, \dots, X_n)$$ as the same function with the $i$th input replaced by the independent copy $X'_i$.

The Efron-Stein inequality states that $$\operatorname{Var}(f) \le \tfrac 1 2\sum_{i=1}^n \ \mathbb E(f - f_i)^2.$$

Since your random variables are Gaussians, the system satisfies the Poincaré inequality too: $$\operatorname{Var}(F) \le C \ \mathbb E |\nabla f|^2.$$

Concentration of measure and bounds on variance - MathOverflow

Concentration of measure and bounds on variance

Answer by Tom LaGatta for Concentration of measure and bounds on variance