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1. Exact convergence analysis for Metropolis–Hastings independence samplers in Wasserstein distances

   https://doi.org/10.1017/jpr.2023.21  

   Brown, Austin; Jones, Galin L (March 2024, Journal of Applied Probability)

   Abstract Under mild assumptions, we show that the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis–Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show that the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show that the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together. 

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2. Exact selective inference with randomization

   https://doi.org/10.1093/biomet/asae019  

   Panigrahi, Snigdha; Fry, Kevin; Taylor, Jonathan (April 2024, Biometrika)

   We introduce a pivot for exact selective inference with randomization. Not only does our pivot lead to exact inference in Gaussian regression models, but it is also available in closed form. We reduce this problem to inference for a bivariate truncated Gaussian variable. By doing so, we give up some power that is achieved with approximate maximum likelihood estimation in Panigrahi & Taylor (2023). Yet our pivot always produces narrower confidence intervals than a closely related data-splitting procedure. We investigate the trade-off between power and exact selective inference on simulated datasets and an HIV drug resistance dataset. 

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3. The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory

   https://doi.org/10.1146/annurev-physchem-062422-013259  

   Kaplan, Aaron D.; Levy, Mel; Perdew, John P. (April 2023, Annual Review of Physical Chemistry)

   Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that ( a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and ( b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development. 

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4. Convergent Interactive Inference with Leaky Joins

   https://doi.org/10.5441/002/edbt.2017.33  

   Yang, Ying; Kennedy, Oliver (March 2017, Proceedings of the 20th International Conference on Extending Database Technology, EDBT 2017, Venice, Italy, March 21-24, 2017)

   One of the primary challenges in graphical models is inference, or re-constructing a marginal probability from the graphical model’s factorized representation. While tractable for some graphs, the cost of inference grows exponentially with the graphical model’s complexity, necessitating approximation for more complex graphs. For interactive applications, latency is the dominant concern, making approximate inference the only feasible option. Unfortunately, approximate inference can be wasteful for interactive applications, as exact inference can still converge faster, even for moderately complex inference problems. In this paper, we propose a new family of convergent inference algorithms (CIAs) that bridge the gap between approximations and exact solutions, providing early, incrementally improving approximations that become exact after a finite period of time. We describe two specific CIAs based on a cryptographic technique called linear congruential generators, including a novel incremental join algorithm for dense relations called Leaky Joins. We conclude with experiments that demonstrate the utility of Leaky Joins for convergent inference: On both synthetic and real-world probabilistic graphical models, Leaky Joins converge to exact marginal probabilities almost as fast as state of the art exact inference algorithms, while simultaneously achieving approximations that are almost as good as state of the art approximation algorithms. 

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5. An exact version of Life Table Response Experiment analysis, and the R package exactLTRE

   https://doi.org/10.1111/2041-210X.14065  

   Hernández, Christina M.; Ellner, Stephen P.; Adler, Peter B.; Hooker, Giles; Snyder, Robin E. (March 2023, Methods in Ecology and Evolution)

   Abstract Matrix population models are frequently built and used by ecologists to analyse demography and elucidate the processes driving population growth or decline. Life Table Response Experiments (LTREs) are comparative analyses that decompose the realized difference or variance in population growth rate () into contributions from the differences or variances in the vital rates (i.e. the matrix elements). Since their introduction, LTREs have been based on approximations and have not included biologically relevant interaction terms.We used the functional analysis of variance framework to derive an exact LTRE method, which calculates the exact response of to the difference or variance in a given vital rate, for all interactions among vital rates—including higher‐order interactions neglected by the classical methods. We used the publicly available COMADRE and COMPADRE databases to perform a meta‐analysis comparing the results of exact and classical LTRE methods. We analysed 186 and 1487 LTREs for animal and plant matrix population models, respectively.We found that the classical methods often had small errors, but that very high errors were possible. Overall error was related to the difference or variance in the matrices being analysed, consistent with the Taylor series basis of the classical method. Neglected interaction terms accounted for most of the errors in fixed design LTRE, highlighting the importance of two‐way interaction terms. For random design LTRE, errors in the contribution terms present in both classical and exact methods were comparable to errors due to neglected interaction terms. In most examples we analysed, evaluating exact contributions up to three‐way interaction terms was sufficient for interpreting 90% or more of the difference or variance in .Relative error, previously used to evaluate the accuracy of classical LTREs, is not a reliable metric of how closely the classical and exact methods agree. Error compensation between estimated contribution terms and neglected contribution terms can lead to low relative error despite faulty biological interpretation. Trade‐offs or negative covariances among matrix elements can lead to high relative error despite accurate biological interpretation. Exact LTRE provides reliable and accurate biological interpretation, and the R packageexactLTREmakes the exact method accessible to ecologists. 

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