Bayesian synthesis, which generates synthetic data by sampling from the posterior predictive distribution, is a popular approach for privatizing sensitive personal data. However, how attribute disclosure risk is affected by feature dimensionality, the number of individuals in the original dataset, and the amount of released synthetic information remains poorly understood. We propose a mathematically tractable Bayesian synthesizer for multivariate normal data based on a conjugate normal-inverse-Wishart prior for the mean vector and covariance matrix. The conjugate structure yields closed-form posteriors and enables direct investigation of an adversary's ability to infer records under different data and release settings. We then demonstrate several intuitive properties of synthetic data generation through several simulations. Specifically, we show that disclosure risk decreases with the size of the original dataset, but increases with the dimensionality of the feature space and the amount of synthetic information released, whether through the release of larger synthetic datasets or multiple generator realizations. Finally, the proposed synthesizer was used to generate synthetic versions of a polygenic risk score dataset, with the synthetic data exhibiting distributional properties comparable to those of the original data.

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