arXiv subjects

Bob Carpenter

Publications and source records attributed to Bob Carpenter.

18 recordsLinked to original sources

arXiv2026

Preconditioning Hamiltonian Monte Carlo by minimizing Fisher Divergence

Although Hamiltonian Monte Carlo (HMC) scales as O(d^(1/4)) in dimension, there is a large constant factor determined by the curvature of the target density. This constant factor can be reduced in most cases through preconditioning, the state of the art for which uses diagonal or dense penalized maximum likelihood estimation of (co)variance based on a sample of warmup draws. These estimates converge slowly in the diagonal case and scale poorly when expanded to the dense case. We propose a more effective estimator based on minimizing the sample Fisher divergence from a linearly transformed density to a standard normal distribution. We present this estimator in three forms, (a) diagonal, (b) dense, and (c) low-rank plus diagonal. Using a collection of 114 models from posteriordb, we demonstrate that the diagonal minimizer of Fisher divergence outperforms the industry-standard variance-based diagonal estimators used by Stan and PyMC by a median factor of 1.3. The low-rank plus diagonal minimizer of the Fisher divergence outperforms Stan and PyMC's diagonal estimators by a median factor of 4.

stat.CO↗

arXiv2026

Leveraging Sparsity to Improve No-U-Turn Sampling Efficiency for Hierarchical Bayesian Models

Analysts routinely use Bayesian hierarchical models to understand natural processes. The no-U-turn sampler (NUTS) is the most widely used algorithm to sample high-dimensional, continuously differentiable models. But NUTS is slowed by high correlations, especially in high dimensions, limiting the complexity of applied analyses. Here we introduce Sparse NUTS (SNUTS), which preconditions (decorrelates and descales) posteriors using a sparse precision matrix ($Q$). We use Template Model Builder (TMB) to efficiently compute $Q$ from the mode of the Laplace approximation to the marginal posterior, then pass the preconditioned posterior to NUTS through the Bayesian software Stan for sampling. We apply SNUTS to seventeen diverse case studies to demonstrate that preconditioning with $Q$ converges one to two orders of magnitude faster than Stan's industry standard diagonal or dense preconditioners. SNUTS also outperforms preconditioning with the inverse of the covariance estimated with Pathfinder variational inference. SNUTS does not improve sampling efficiency for models with the highly varying curvature found in funnels, wide tails, or multiple modes. SNUTS is most advantageous, and can be scaled beyond $10^4$ parameters, in the presence of high dimensionality, sparseness, and high correlations, all of which are widespread in applied statistics. An open-source implementation of SNUTS is provided in the R package SparseNUTS.

stat.CO↗

arXiv2025

The Within-Orbit Adaptive Leapfrog No-U-Turn Sampler

Locally adapting parameters within Markov chain Monte Carlo methods while preserving reversibility is notoriously difficult. The success of the No-U-Turn Sampler (NUTS) largely stems from its clever local adaptation of the integration time in Hamiltonian Monte Carlo via a geometric U-turn condition. However, posterior distributions frequently exhibit multi-scale geometries with extreme variations in scale, making it necessary to also adapt the leapfrog integrator's step size locally and dynamically. Despite its practical importance, this problem has remained largely open since the introduction of NUTS by Hoffman and Gelman (2014). To address this issue, we introduce the Within-orbit Adaptive Leapfrog No-U-Turn Sampler (WALNUTS), a generalization of NUTS that adapts the leapfrog step size at fixed intervals of simulated time as the orbit evolves. At each interval, the algorithm selects the largest step size from a dyadic schedule that keeps the energy error below a user-specified threshold. Like NUTS, WALNUTS employs biased progressive state selection to favor states with positions that are further from the initial point along the orbit. Empirical evaluations on multiscale target distributions, including Neal's funnel and the Stock-Watson stochastic volatility time-series model, demonstrate that WALNUTS achieves substantial improvements in sampling efficiency and robustness compared to standard NUTS.

stat.CO↗

arXiv2025

The No-Underrun Sampler: A Locally-Adaptive, Gradient-Free MCMC Method

In this work, we introduce the No-Underrun Sampler (NURS), a locally-adaptive, gradient-free Markov chain Monte Carlo method that blends ideas from Hit-and-Run and the No-U-Turn Sampler. NURS dynamically adapts to the local scale of the target distribution without requiring gradient evaluations, making it especially suitable for applications where gradients are unavailable or costly. We establish key theoretical properties, including reversibility, formal connections to Hit-and-Run and Random Walk Metropolis, Wasserstein contraction comparable to Hit-and-Run in Gaussian targets, and bounds on the total variation distance between the transition kernels of Hit-and-Run and NURS. Empirical experiments, supported by theoretical insights, illustrate the ability of NURS to sample from Neal's funnel, a challenging multi-scale distribution from Bayesian hierarchical inference.

math.ST↗

arXiv2024

Incorporating Local Step-Size Adaptivity into the No-U-Turn Sampler using Gibbs Self Tuning

Adapting the step size locally in the no-U-turn sampler (NUTS) is challenging because the step-size and path-length tuning parameters are interdependent. The determination of an optimal path length requires a predefined step size, while the ideal step size must account for errors along the selected path. Ensuring reversibility further complicates this tuning problem. In this paper, we present a method for locally adapting the step size in NUTS that is an instance of the Gibbs self-tuning (GIST) framework. Our approach guarantees reversibility with an acceptance probability that depends exclusively on the conditional distribution of the step size. We validate our step-size-adaptive NUTS method on Neal's funnel density and a high-dimensional normal distribution, demonstrating its effectiveness in challenging scenarios.

stat.ME↗

arXiv2024

posteriordb: Testing, Benchmarking and Developing Bayesian Inference Algorithms

The generality and robustness of inference algorithms is critical to the success of widely used probabilistic programming languages such as Stan, PyMC, Pyro, and Turing.jl. When designing a new general-purpose inference algorithm, whether it involves Monte Carlo sampling or variational approximation, the fundamental problem arises in evaluating its accuracy and efficiency across a range of representative target models. To solve this problem, we propose posteriordb, a database of models and data sets defining target densities along with reference Monte Carlo draws. We further provide a guide to the best practices in using posteriordb for model evaluation and comparison. To provide a wide range of realistic target densities, posteriordb currently comprises 120 representative models and has been instrumental in developing several general inference algorithms.

stat.CO↗

arXiv2024

Sampling From Multiscale Densities With Delayed Rejection Generalized Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is the mainstay of applied Bayesian inference for differentiable models. However, HMC still struggles to sample from hierarchical models that induce densities with multiscale geometry: a large step size is needed to efficiently explore low curvature regions while a small step size is needed to accurately explore high curvature regions. We introduce the delayed rejection generalized HMC (DR-G-HMC) sampler that overcomes this challenge by employing dynamic step size selection, inspired by differential equation solvers. In generalized HMC, each iteration does a single leapfrog step. DR-G-HMC sequentially makes proposals with geometrically decreasing step sizes upon rejection of earlier proposals. This simulates Hamiltonian dynamics that can adjust its step size along a (stochastic) Hamiltonian trajectory to deal with regions of high curvature. DR-G-HMC makes generalized HMC competitive by decreasing the number of rejections which otherwise cause inefficient backtracking and prevents directed movement. We present experiments to demonstrate that DR-G-HMC (1) correctly samples from multiscale densities, (2) makes generalized HMC methods competitive with the state of the art No-U-Turn sampler, and (3) is robust to tuning parameters.

stat.CO↗

arXiv2024

Hierarchical Bayesian Crowdsourcing with Item Difficulty

In applied statistics and machine learning, the gold standards used for training are often biased and almost always noisy. Dawid and Skene's justifiably popular crowdsourcing model adjusts for rater sensitivity and specificity, but fails to capture distributional properties of rating data gathered for training, which in turn biases training. In this study, we introduce a general purpose measurement-error model with which we can infer consensus categories by adding item-level effects for difficulty, discriminativeness, and guessability. We further show how to constrain the bimodal posterior of these models to avoid adversarial raters. We validate our model's goodness of fit with posterior predictive checks, the Bayesian analogue of $\chi^2$ tests, and assess its predictive accuracy using leave-one-out cross-validation. We illustrate our new model with two well-studied data sets, binary rating data for caries in dental X-rays and implication in natural language.

cs.LG↗

arXiv2024

GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo

We introduce a novel and flexible framework for constructing locally adaptive Hamiltonian Monte Carlo (HMC) samplers by Gibbs sampling the algorithm's tuning parameters conditionally based on the position and momentum at each step. For adaptively sampling path lengths, our Gibbs self-tuning (GIST) approach encompasses randomized HMC, multinomial HMC, the No-U-Turn Sampler (NUTS), and the Apogee-to-Apogee Path Sampler as special cases. We exemplify the GIST framework with a novel alternative to NUTS for locally adapting path lengths, evaluated with an exact Hamiltonian for a high-dimensional, ill-conditioned Gaussian measure and with the leapfrog integrator for a suite of diverse models.

stat.CO↗

arXiv2022

Ensemble reweighting using Cryo-EM particles

Cryo-electron microscopy (cryo-EM) has recently become a premier method for obtaining high-resolution structures of biological macromolecules. However, it is limited to biomolecular samples with low conformational heterogeneity, where all the conformations can be well-sampled at many projection angles. While cryo-EM technically provides single-molecule data for heterogeneous molecules, most existing reconstruction tools cannot extract the full distribution of possible molecular configurations. To overcome these limitations, we build on a prior Bayesian approach and develop an ensemble refinement framework that estimates the ensemble density from a set of cryo-EM particles by reweighting a prior ensemble of conformations, e.g., from molecular dynamics simulations or structure prediction tools. Our work is a general approach to recovering the equilibrium probability density of the biomolecule directly in conformational space from single-molecule data. To validate the framework, we study the extraction of state populations and free energies for a simple toy model and from synthetic cryo-EM images of a simulated protein that explores multiple folded and unfolded conformations.

q-bio.BM↗

arXiv2021

Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions

The efficiency of Hamiltonian Monte Carlo (HMC) can suffer when sampling a distribution with a wide range of length scales, because the small step sizes needed for stability in high-curvature regions are inefficient elsewhere. To address this we present a delayed rejection variant: if an initial HMC trajectory is rejected, we make one or more subsequent proposals each using a step size geometrically smaller than the last. We extend the standard delayed rejection framework by allowing the probability of a retry to depend on the probability of accepting the previous proposal. We test the scheme in several sampling tasks, including multiscale model distributions such as Neal's funnel, and statistical applications. Delayed rejection enables up to five-fold performance gains over optimally-tuned HMC, as measured by effective sample size per gradient evaluation. Even for simpler distributions, delayed rejection provides increased robustness to step size misspecification. Along the way, we provide an accessible but rigorous review of detailed balance for HMC.

stat.ML↗

arXiv2021

Pathfinder: Parallel quasi-Newton variational inference

We propose Pathfinder, a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the inverse Hessian estimates produced by the optimizer. Pathfinder returns draws from the approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior. We evaluate Pathfinder on a wide range of posterior distributions, demonstrating that its approximate draws are better than those from automatic differentiation variational inference (ADVI) and comparable to those produced by short chains of dynamic Hamiltonian Monte Carlo (HMC), as measured by 1-Wasserstein distance. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors. Importance resampling over multiple runs of Pathfinder improves the diversity of approximate draws, reducing 1-Wasserstein distance further and providing a measure of robustness to optimization failures on plateaus, saddle points, or in minor modes. The Monte Carlo KL divergence estimates are embarrassingly parallelizable in the core Pathfinder algorithm, as are multiple runs in the resampling version, further increasing Pathfinder's speed advantage with multiple cores.

stat.ML↗

arXiv2021

A Bayesian approach for extracting free energy profiles from cryo-electron microscopy experiments using a path collective variable

Cryo-electron microscopy (cryo-EM) extracts single-particle density projections of individual biomolecules. Although cryo-EM is widely used for 3D reconstruction, due to its single-particle nature, it has the potential to provide information about the biomolecule's conformational variability and underlying free energy landscape. However, treating cryo-EM as a single-molecule technique is challenging because of the low signal-to-noise ratio (SNR) in the individual particles. In this work, we developed the cryo-BIFE method, cryo-EM Bayesian Inference of Free Energy profiles, that uses a path collective variable to extract free energy profiles and their uncertainties from cryo-EM images. We tested the framework over several synthetic systems, where we controlled the imaging parameters and conditions. We found that for realistic cryo-EM environments and relevant biomolecular systems, it is possible to recover the underlying free energy, with the pose accuracy and SNR as crucial determinants. Then, we used the method to study the conformational transitions of a calcium-activated channel with real cryo-EM particles. Interestingly, we recover the most probable conformation (used to generate a high resolution reconstruction of the calcium-bound state), and we find two additional meta-stable states, one which corresponds to the calcium-unbound conformation. As expected for turnover transitions within the same sample, the activation barriers are of the order of a couple $k_BT$. Extracting free energy profiles from cryo-EM will enable a more complete characterization of the thermodynamic ensemble of biomolecules.

q-bio.BM↗

arXiv2020

Bayesian Workflow

The Bayesian approach to data analysis provides a powerful way to handle uncertainty in all observations, model parameters, and model structure using probability theory. Probabilistic programming languages make it easier to specify and fit Bayesian models, but this still leaves us with many options regarding constructing, evaluating, and using these models, along with many remaining challenges in computation. Using Bayesian inference to solve real-world problems requires not only statistical skills, subject matter knowledge, and programming, but also awareness of the decisions made in the process of data analysis. All of these aspects can be understood as part of a tangled workflow of applied Bayesian statistics. Beyond inference, the workflow also includes iterative model building, model checking, validation and troubleshooting of computational problems, model understanding, and model comparison. We review all these aspects of workflow in the context of several examples, keeping in mind that in practice we will be fitting many models for any given problem, even if only a subset of them will ultimately be relevant for our conclusions.

stat.ME↗

arXiv2019

Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC

Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic $\widehat{R}$ of Gelman and Rubin (1992) has serious flaws. Traditional $\widehat{R}$ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.

stat.CO↗

arXiv2016

Fitting Bayesian item response models in Stata and Stan

Stata users have access to two easy-to-use implementations of Bayesian inference: Stata's native {\tt bayesmh} function and StataStan, which calls the general Bayesian engine Stan. We compare these on two models that are important for education research: the Rasch model and the hierarchical Rasch model. Stan (as called from Stata) fits a more general range of models than can be fit by {\tt bayesmh} and is also more scalable, in that it could easily fit models with at least ten times more parameters than could be fit using Stata's native Bayesian implementation. In addition, Stan runs between two and ten times faster than {\tt bayesmh} as measured in effective sample size per second: that is, compared to Stan, it takes Stata's built-in Bayesian engine twice to ten times as long to get inferences with equivalent precision. We attribute Stan's advantage in flexibility to its general modeling language, and its advantages in scalability and speed to an efficient sampling algorithm: Hamiltonian Monte Carlo using the no-U-turn sampler. In order to further investigate scalability, we also compared to the package Jags, which performed better than Stata's native Bayesian engine but not as well as StataStan. Given its advantages in speed, generality, and scalability, and that Stan is open-source and can be run directly from Stata using StataStan, we recommend that Stata users adopt Stan as their Bayesian inference engine of choice.

stat.CO↗

arXiv2015

The Stan Math Library: Reverse-Mode Automatic Differentiation in C++

As computational challenges in optimization and statistical inference grow ever harder, algorithms that utilize derivatives are becoming increasingly more important. The implementation of the derivatives that make these algorithms so powerful, however, is a substantial user burden and the practicality of these algorithms depends critically on tools like automatic differentiation that remove the implementation burden entirely. The Stan Math Library is a C++, reverse-mode automatic differentiation library designed to be usable, extensive and extensible, efficient, scalable, stable, portable, and redistributable in order to facilitate the construction and utilization of such algorithms. Usability is achieved through a simple direct interface and a cleanly abstracted functional interface. The extensive built-in library includes functions for matrix operations, linear algebra, differential equation solving, and most common probability functions. Extensibility derives from a straightforward object-oriented framework for expressions, allowing users to easily create custom functions. Efficiency is achieved through a combination of custom memory management, subexpression caching, traits-based metaprogramming, and expression templates. Partial derivatives for compound functions are evaluated lazily for improved scalability. Stability is achieved by taking care with arithmetic precision in algebraic expressions and providing stable, compound functions where possible. For portability, the library is standards-compliant C++ (03) and has been tested for all major compilers for Windows, Mac OS X, and Linux.

cs.MS↗

arXiv1997

Probabilistic Parsing Using Left Corner Language Models

We introduce a novel parser based on a probabilistic version of a left-corner parser. The left-corner strategy is attractive because rule probabilities can be conditioned on both top-down goals and bottom-up derivations. We develop the underlying theory and explain how a grammar can be induced from analyzed data. We show that the left-corner approach provides an advantage over simple top-down probabilistic context-free grammars in parsing the Wall Street Journal using a grammar induced from the Penn Treebank. We also conclude that the Penn Treebank provides a fairly weak testbed due to the flatness of its bracketings and to the obvious overgeneration and undergeneration of its induced grammar.

cmp-lg↗