Research News: Control Variates for Constrained Variables
Yifan Zhou, Post Doc researcher at the Signal Processing Group, provides an overview of his novel research 'Control Variates for Constrained Variables' which has recently been published at IEEE.
Summary
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. It is often useful to reduce the variance without increasing the number of samples. One existing approach is called control variates but it only works for unconstrained random variables. To expand the usage of such approach in real applications, our work 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 is achieved with negligible additional computational cost.
Importance of the research
Numerical Bayesian inference involves drawing samples from a target distribution, which is typically the posterior distribution of parameters given data, in order to make inferences from such data: such inferences can include deriving point estimates, identifying credible intervals and hypothesis testing. The broad applicability of this generic approach has given rise to a diverse range of applications, spanning numerous aspects of signal processing (particularly in image processing, deconvolution and interference mitigation) but also pertinent in the context of, for example, health and finance. Furthermore, there are a growing number of probabilistic programming languages (e.g. Stan, PyMC3, Figaro and Turing.jl) being adopted with the express purpose of easing the process of exploiting numerical Bayesian inference to arbitrary applications.
None-the-less, particularly in the contexts where repeatability is desirable, it can be of significant interest to minimise the Monte-Carlo variance. This can be achieved by simply having a larger number of samples. Unfortunately, however, increasing the size is, in general, computationally expensive.
Control variates have been demonstrated to offer significant reduction in variance (we note there can, albeit infrequently, be circumstances where no variance reduction is achieved), the approaches that have been developed are not applicable in contexts involving constrained variables. The key contribution made by this work is to enable control variates to be applied in such constrained contexts by using a mapping from the constrained space to an unconstrained space and then posing expectations in terms of non-linear functions of the unconstrained samples.
Results from experiments on widely used real-world applications indicate that significant reductions in Monte-Carlo error result with negligible additional computational cost.
What comes next?
Future work will consider:-
1. Other types of control variates.
2. Integrate the code more tightly with Stan, particularly seamless interface in Python.
3. Interface the approach into other probabilistic programming languages such as PyMC3.
4. Apply the work to particle filters (extending an existing application to Sequential Monte Carlo samplers) to improve the effective samples and reduce the computational cost.
References
The published paper can be viewed here on IEEE Explore.
The work is funded by EPSRC Big Hypotheses.