CoSInES

There now exists a huge array of sophisticated MCMC, SMC, ABC, EM and more algorithms proposed to tackle high-dimensional likelihood-based statistical problems. Many are themselves approximate, or admit natural computationally less expensive approximations. While impressive advances in our understanding of these methodologies have been made this century, better understanding of the robustness, scalability and statistical and algorithmic efficiency is urgently required to underpin ever more complex applications. Moreover, better understanding of the nature of approximate likelihood methods is required to more fully understand the trade- off between computational efficiency and exactness. Particular challenges exist for high-dimensional, often high-frequency, time-indexed structured data.

The first algorithms for Bayesian inference for big data have emerged during the last 5 years, although many involve approximations which are difficult to quantify outside the proximity of Gaussianity, and exact methods are currently extremely complex and often slow. Applications demand methods which are robust to non-Gaussianity and practically implementable for a wide range of big data contexts.

Some progress in the design of parallel and distributed statistical algorithms has been made in the last few years. However, architectural considerations and high- performance implementations have often been viewed as a subsidiary engineering challenge, and there is a lack of robust and mature algorithmic approaches and performant code. This is in stark contrast to the continued development of much-needed, architecturally-informed packages from other areas in Data Science, some of which have revolutionised the practice of data analysis and driven theoretical and methodological innovation.

The theoretical understanding of the scaling complexity properties of MCMC and SMC methods is of vital importance for guiding practice, for example in algorithm choice, optimal scaling, deciding how to initialise and terminate runs). The area has made great strides in recent years, although still relatively little is known about highly heterogenous statistical problems which will typically be encountered within CoSInES.

Under model misspecification, a major challenge is to develop statistical approaches that retain the benefits of prob- abilistic models while maintaining computational tractability and robustness to misspecification. More- over, it is extremely common to find MCMC methods to be robust and efficient for simulated data, only to suddenly mix badly for real data. However this phenomenon is not well-understood and there is a dearth of theory describing how MCMC convergence for Bayesian posterior distributions depends on the particular data set observed.

Within both commercial and security contexts, it is increasingly common for data to only be partially available to statisticians due to confidentiality and privacy constraints. Carrying out statistical inference from encrypted data has become a realistic aim due to the introduction of homomorphic encryption and subsequent pioneering work. However currently only very simple statistical analysis is feasible, and new methods will require algorithmic developments to tailor closely with rapidly developing advancements in homomorphic encryption.