Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related
Problems
Ryan Tibshirani (Carnegie Mellon University, USA)
Abstract:
This talk is about a class of univariate piecewise polynomial functions known as discrete splines, whichshare properties analogous to the better-known class of spline functions, but where continuity in derivatives
is replaced by (a suitable notion of) continuity in divided differences. As it happens, discrete splines bearconnections to a wide array of developments in applied mathematics and statistics, from divideddifferences and Newton interpolation (dating back to over 300 years ago) to trend filtering (from the last
15 years). We survey these connections, and contribute some new perspectives and new results along
the way.
Find the paper at: https://www.stat.cmu.edu/~ryantibs/papers/dspline.pdf

Tree-based Gaussian Process for Computer Experiments with Many-Category Qualitative Factors
Ray-Bing Chen
(Department of Statistics and Institute of Data Science, National Cheng Kung University)
Abstract:
In computer experiments, Gaussian process models are commonly used for emulation. However, whenboth qualitative and quantitative factors are involved in the experiments, emulation using Gaussian processmodels becomes challenging. In particular, when the qualitative factors contain many categories in theexperiments, existing methods in the literature become cumbersome due to curse of dimensionality.
Motivated by the computer experiments for the design of a cooling system, a new tree-based Gaussian
process for emulating computer experiments with many-category qualitative factors is proposed. The
proposed method incorporates a tree structure to split the categories in the qualitative factors, andGaussian process models are employed for modeling the simulation outputs in the leaf nodes. The splitting
rule takes into account the cross-correlations between the categories of the qualitative factors, which havebeen shown by a recent theoretical study to be a crucial element for improving the prediction accuracy.
The application to the design of a cooling system indicates that the proposed method not only enjoys
marked computational advantages and produces accurate predictions, but also provides valuable insights
into the cooling system by discovering the tree structure.

Scaling limit of Markov chain/process Monte Carlo methods
Kengo Kamatani (Institute of Statistical Mathematics)
Abstract:
The scaling limit analysis for Markov chain Monte Carlo methods has been developed over the last
decades. The analysis identifies the rate of convergence to the limiting process, typically the Langevindiffusion process. Moreover, this analysis provides useful criteria for parameter tuning in Monte Carlomethods, as the limiting process is usually much simpler than the Markov chain Monte Carlo process andthus easier to optimise with respect to the tuning parameter. After the seminal work of Roberts et al. in1997, many researchers have generalised the assumption and extended the results to more advanced
methods. In the first half of this talk, we will review some basic scaling limit results for Markov chain Monte
Carlo methods.
Recently, piecewise deterministic Markov processes have gained interest in the context of scalable Monte
Carlo integration methods. Two examples of fundamental importance are the Bouncy Particle sampler and
the Zig-Zag sampler. In this talk, we will determine the scaling limits for both algorithms. Finally, we discussa criterion for the tuning parameter of the Bouncy Particle sampler.
The latter part is a joint work with J. Bierkens (TU Delft) and G. O. Roberts (Warwick).