Correcting Finite Sampling Issues in Entropy l-diversity

Abstract

In statistical disclosure control (SDC) anonymized versions of a database table are obtained via generalization and suppression to reduce de-anonymization attacks, ideally with minimal utility loss. This amounts to an optimization problem in which a measure of remaining diversity needs to be improved. The feasible solutions are those that fulfill some privacy criteria, e.g., the entropy l-diversity. In the statistics it is known that the naive computation of an entropy via the Shannon formula systematically underestimates the (real) entropy and thus influences the resulting equivalence classes. In this contribution we implement an asymptotically unbiased estimator for the Shannon entropy and apply it to three test databases. Our results show previously performed systematic miscalculations; we show that by an unbiased estimator one can increase the utility of the data without compromising privacy.

A Efficient Computation of the G(n)

The inductive definition (4) of the G(n) can be rewritten as

$$\begin{aligned} G(2n) = -\gamma -\ln 2 + \sum _{k=0}^{n-1} \frac{2}{2k+1} \end{aligned}$$

Remember that $G(2n+1) := G(2n)$. This equation can simply be rewritten in terms of the Harmonic numbers $H_n := \sum _{k=1}^{n} 1/k$ as

$$\begin{aligned} G(2n) = -\gamma -\ln 2 + 2 H_{2n-1} - H_{n-1} \end{aligned}$$

The Harmonic numbers can also be expressed in terms of the Digamma function $\psi$ as $H_{n-1} = \psi (n) + \gamma$. The Digamma function has the asymptotic series

$$\begin{aligned} \psi (x) = \ln x - \frac{1}{2x} - \sum _{k=1}^{\infty } \frac{B_{2k}}{2k\,x^{2k}}, \end{aligned}$$

using the Bernoulli numbers $B_n$. Combining the last two equations yields the asymptotic expansion for G(2n):

$$\begin{aligned} G(2n) = \ln (2n) + \frac{1}{24\,n^2} - \frac{7}{960\,n^4} + \frac{31}{8064\,n^6} + \mathscr {O} \left( \frac{1}{n^8}\right) \end{aligned}$$

This expansion is accurate to double presicion for $n\ge 50$. For smaller values we used a precomputed table. Note that, beside the logarithm, it suffices to calculate a single division $1/n^2$ and then proceed with nested multiplication.