Control variates for stochastic gradient MCMC

It is well known that Markov chain Monte Carlo (MCMC) methods scale poorly with dataset size. A popular class of methods for solving this issue is stochastic gradient MCMC (SGMCMC). These methods use a noisy estimate of the gradient of the log-posterior, which reduces the per iteration computational cost of the algorithm. Despite this, there are a number of results suggesting that stochastic gradient Langevin dynamics (SGLD), probably the most popular of these methods, still has computational cost proportional to the dataset size. We suggest an alternative log-posterior gradient estimate for stochastic gradient MCMC which uses control variates to reduce the variance. We analyse SGLD using this gradient estimate, and show that, under log-concavity assumptions on the target distribution, the computational cost required for a given level of accuracy is independent of the dataset size. Next we show that a different control variate technique, known as zero variance control variates, can be applied to SGMCMC algorithms for free. This post-processing step improves the inference of the algorithm by reducing the variance of the MCMC output. Zero variance control variates rely on the gradient of the log-posterior; we explore how the variance reduction is affected by replacing this with the noisy gradient estimate calculated by SGMCMC

Adjuvants for Leishmania vaccines: from models to clinical application

Two million new cases of leishmaniasis occur every year, with the cutaneous leishmaniasis (CL) presentation accounting for approximately two-thirds of all cases. Despite the high incidence rates and geographic expansion of the disease, CL remains a neglected tropical disease without effective intervention strategies. Efforts to address this deficit have given rise to the experimental murine model of CL. By virtue of its simplicity and pliability, the CL model has been used to provide substantial information regarding cellular immunity, as well as in the discovery and evaluation of various vaccine adjuvants. The CL model has facilitated in vivo studies of the mechanism of action of many adjuvants, including the TLR4 agonist monophosphoryl lipid A, the TLR7/8 agonist imiquimod, the TLR9 agonist CpG, adenoviral vectors, and the immunostimulatory complexes. Together, these studies have helped to unveil the requirement for certain types of immune responses at specific stages of CL disease and provide a basis to aid the design of effective second-generation vaccines for human CL. This review focuses on adjuvants that have been tested in experimental CL, outlining how they have helped advance our understanding of the disease and ultimately, how they have performed when applied within clinical trials against human CL

Stochastic Gradient MCMC for Nonlinear State Space Models

State space models (SSMs) provide a flexible framework for modeling complex time series via a latent stochastic process. Inference for nonlinear, non-Gaussian SSMs is often tackled with particle methods that do not scale well to long time series. The challenge is two-fold: not only do computations scale linearly with time, as in the linear case, but particle filters additionally suffer from increasing particle degeneracy with longer series. Stochastic gradient MCMC methods have been developed to scale inference for hidden Markov models (HMMs) and linear SSMs using buffered stochastic gradient estimates to account for temporal dependencies. We extend these stochastic gradient estimators to nonlinear SSMs using particle methods. We present error bounds that account for both buffering error and particle error in the case of nonlinear SSMs that are log-concave in the latent process. We evaluate our proposed particle buffered stochastic gradient using SGMCMC for inference on both long sequential synthetic and minute-resolution financial returns data, demonstrating the importance of this class of methods

Large-Scale Stochastic Sampling from the Probability Simplex

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches

sgmcmc:An R Package for Stochastic Gradient Markov Chain Monte Carlo

This paper introduces the R package sgmcmc; which can be used for Bayesian inference on problems with large datasets using stochastic gradient Markov chain Monte Carlo (SGMCMC). Traditional Markov chain Monte Carlo (MCMC) methods, such as Metropolis-Hastings, are known to run prohibitively slowly as the dataset size increases. SGMCMC solves this issue by only using a subset of data at each iteration. SGMCMC requires calculating gradients of the log likelihood and log priors, which can be time consuming and error prone to perform by hand. The sgmcmc package calculates these gradients itself using automatic differentiation, making the implementation of these methods much easier. To do this, the package uses the software library TensorFlow, which has a variety of statistical distributions and mathematical operations as standard, meaning a wide class of models can be built using this framework. SGMCMC has become widely adopted in the machine learning literature, but less so in the statistics community. We believe this may be partly due to lack of software; this package aims to bridge this gap