alt="r Subscript p"/> of branch
on a phylogeny as the product of a global treewise mean parameter
and a branch‐specific random effect
. They model the random‐effect
s as independent and identically distributed from a lognormal distribution such that
has mean 1 and variance
under a hierarchical model where
is the scale parameter. To accommodate the difference in scales of the variability in the parameter space for the HMC sampler, the authors adopt preconditioning with adaptive mass matrix informed by the diagonal entries of the Hessian matrix. More precisely, the nonzero diagonal elements of the mass matrix truncate the values from the first
HMC iterations of
so that the matrix remains positive‐definite and numerically stable. They estimate the treewise (fixed‐effect) mean rate
with posterior mean 4.75 (
Bayesian credible interval:
)
substitutions per site per year with rate variability characterized by scale parameter with posterior mean
for serotype 3 of Dengue virus with a sample size of 352 [69].
Figure 1 illustrates the estimated maximum clade credible evolutionary tree of the Dengue virus dataset.
The authors report relative speedup in terms of the effective sample size per second (ESS/s) of the HMC samplers compared to a univariate transition kernel. The “vanilla” HMC sampler with an identity mass matrix gains speedup for the minimum ESS/s and speedup for the median ESS/s, whereas the “preconditioned” HMC sampler gains and speedups, respectively. Critically, the authors make these performance gains available to scientists everywhere through the popular, open‐source software package for viral phylogenetic inference Bayesian evolutionary analysis by sampling trees (BEAST) [75]. In Section 4.1, we discuss how software package such as BEAST addresses Core Challenge 4, the creation of fast, flexible, and friendly statistical algo‐ware.
Figure 1 A nontraditional and critically important application in computational statistics is the reconstruction of evolutionary histories in the form of phylogenetic trees. Here is a maximum clade credible tree of the Dengue virus example. The dataset consists of sequences of the serotype of the Dengue virus. Branches are coded by the posterior means of the branch‐specific evolutionary rates according to the gradient bar on the top left. The concentric circles indicate the timescale with the year numbers. The outer ring indicates the geographic locations of the samples by the color code on the bottom left. ‘I’ and ‘II’ indicate the two Brazilian lineages as in the original study.
4 Core Challenges 4 and 5
Section 3 provides examples of how computational statisticians might address Core Challenges 1–3 (big , big , and big ) for individual models. Such advances in computational methods must be accompanied by easy‐to‐use software to make them accessible to end users. As Gentle et al. [76] put it, “While referees and editors of scholarly journals determine what statistical theory and methods are published, the developers of the major statistical software packages determine what statistical methods are used.” We would like statistical software to be widely applicable yet computationally efficient at the same time. Trade‐offs invariably arise between these two desiderata, but one should nonetheless strive to design algorithms that are general enough to solve an important class of problems and as efficiently as possible in doing so.
Section 4.1 presents Core Challenge 4, achieving “algo‐ware” (a neologism suggesting an equal emphasis on the statistical algorithm and its implementation) that is sufficiently efficient, broad, and user‐friendly to empower everyday statisticians and data scientists. Core Challenge 5 (Section 4.2) explores the mapping of these algorithms to computational hardware for optimal performance. Hardware‐optimized implementations often exploit model‐specific structures, but good, general‐purpose software should also optimize common routines.
4.1 Fast, Flexible, and Friendly Statistical Algo‐Ware
To accommodate the greatest range of models while remaining simple enough to encourage easy implementation, inference methods should rely solely on the quantities that can be computed algorithmically for any given model. The log‐likelihood (or log‐density in the Bayesian setting) is one such quantity, and one can employ the computational graph framework [77, 78] to evaluate conditional log‐likelihoods for any subset of model parameters as well as their gradients via backpropagation [79]. Beyond being efficient in terms of the first three Core Challenges, an algorithm should demonstrate robust performance on a reasonably wide range of problems without extensive tuning if it is to lend itself to successful software deployment.
HMC (Section
