The Quasi-Multinomial Synthesizer for Categorical Data

Abstract

We present a new synthesizer for categorical data based on the Quasi-Multinomial distribution. Characteristics of the Quasi-Multinomial distribution provide a tuning parameter, which allows a Quasi-Multinomial synthesizer to control the balance of the utility and the disclosure risks of synthetic data. We develop a Quasi-Multinomial synthesizer based on a popular categorical data synthesizer, the Dirichlet process mixtures of products of multinomial distributions. The general sampling methods and algorithm of the Quasi-Multinomial synthesizer are developed and presented. We illustrate its balance of the utility and the disclosure risks by synthesizing a sample from the American Community Survey.

Appendix

1.1 Appendix 1

Proof of Theorem 3: By symmetry it suffices to show the case of $a_1<a_2$. To simplify our argument, write $g(y;a)=\varGamma (a+y)/(\varGamma (a+1)(a+y)^{y-1})$. Then ${p_{BB}(y)}/{p_{QB}(y)}=g(y;a_1)g(n-y;a_2)/g(n;a_\cdot )$. Hence ${p_{BB}(y)}/{p_{QB}(y)}$ is minimized when $g(y;a_1)g(n-y;a_2)$ is minimized. Also we note that $g(y;a)r(y;a)=g(y+1;a)$, where $r(y;a)=(a+y)^y/(a+y+1)^{y}$. Since $r(0,a)=1$ and $r(y,a)<1$ for $y\ge 1$, g(y; a) is monotonically decreasing when y increases. Therefore to minimize $g(y;a_1)g(n-y;a_2)$, we increase $y_1$ or $y_2$ one by one so that $g(y_1;a_1)g(y_2;a_2)$ is more reduced. Denote the ith step values by $(y_1,y_2)_i, i=1,\dots ,n$. At each step $y_1$ increases by one if $r(y_1,a)\le r(y_2,a)$ otherwise $y_2$ increases by one. Actually $r(0,a_1)=r(0,a_2)=1$, but if $0<a_1<a_2$ then $r(1,a_1)<r(1,a_2)$. Hence we begin with $(y_1,y_2)_1=(1,0)$. Observing

$$\begin{aligned} \frac{d\log r(y,a)}{dy}= & {} \log \left( 1-\frac{1}{a+y+1}\right) +\frac{y}{(a+y)(a+y+1)}\\= & {} -\frac{1}{a+y+1}-\frac{1}{2}(\frac{1}{a+y+1})^2-\cdots +\frac{y}{(a+y)(a+y+1)}<0, \end{aligned}$$

we note that $r(y,a)>r(y',a)$ when $y<y'$. Therefore $r(1,a_2)>r(1,a_1)>r(i,a_1)$ for $i\ge 2$, which leads to $(y_1,y_2)_n=(n,0)$.    $\square$

1.2 Appendix 2

A summary of average acceptance rates r of our QB sampler proposed in Sect. 2.

Table 2. Average acceptance rates: $r(0.1/\beta ,0.9/\beta ,n)$

1.3 Appendix 3

The table contains detailed results of regression-based utility of the DPMPM synthetic data in Sect. 4.

Table 3. 95% confidence intervals of logistic regression coefficients based on the original data and based on the m = 20 synthetic data generated by the DPMPM synthesizer.

1.4 Appendix 4

The Table 4 contains the minimum, first quartile, median, third quartile, and maximum of the identification disclosure risks of the DPMPM synthesizer. They are all marked as horizontal lines in Figs. 3, 4 and 5.

Table 4. Table of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum of the identification disclosure risks of the DPMPM synthesizer.