Control Variates for Constrained Variables

Numerical Bayesian inference methods typified by Markov chain Monte Carlo generate a set of samples from a probability distribution. When using real-valued samples to approximate the expectation of a random variable, the variance of the resulting estimator, obtained by averaging over those samples, decreases as the number of samples increases. However, it is often useful to reduce the variance without increasing the number of samples. Using control variates is one method to achieve such variance reduction and is applicable in contexts where the random variable is unconstrained. To make it possible to use control variates with constrained variables, this paper proposes the use of a non-linear mapping from an unconstrained space to the constrained space. Results indicate that significant reductions in Monte-Carlo error was achieved with negligible additional computational cost.