An official website of the United States government
Sequential ranking-and-selection procedures deliver Bayesian guarantees by repeatedly computing a posterior quantity of interest—for example, the posterior probability of good selection or the posterior expected opportunity cost—and terminating when this quantity crosses some threshold. Computing these posterior quantities entails nontrivial numerical computation. Thus, rather than exactly check such posterior-based stopping rules, it is common practice to use cheaply computable bounds on the posterior quantity of interest, for example, those based on Bonferroni’s or Slepian’s inequalities. The result is a conservative procedure that samples more simulation replications than are necessary. We explore how the time spent simulating these additional replications might be better spent computing the posterior quantity of interest via numerical integration, with the potential for terminating the procedure sooner. To this end, we develop several methods for improving the computational efficiency of exactly checking the stopping rules. Simulation experiments demonstrate that the proposed methods can, in some instances, significantly reduce a procedure’s total sample size. We further show these savings can be attained with little added computational effort by making effective use of a Monte Carlo estimate of the posterior quantity of interest. Summary of Contribution: The widespread use of commercial simulation software in industry has made ranking-and-selection (R&S) algorithms an accessible simulation-optimization tool for operations research practitioners. This paper addresses computational aspects of R&S procedures delivering finite-time Bayesian statistical guarantees, primarily the decision of when to terminate sampling. Checking stopping rules entails computing or approximating posterior quantities of interest perceived as being computationally intensive to evaluate. The main results of this paper show that these quantities can be efficiently computed via numerical integration and can yield substantial savings in sampling relative to the prevailing approach of using conservative bounds. In addition to enhancing the performance of Bayesian R&S procedures, the results have the potential to advance other research in this space, including the development of more efficient allocation rules.
more »
« less
Automated Model Selection with Bayesian Quadrature
We present a novel technique for tailoring Bayesian quadrature (BQ) to model selection. The state-of-the-art for comparing the evidence of multiple models relies on Monte Carlo methods, which converge slowly and are unreliable for computationally expensive models. Although previous research has shown that BQ offers sample efficiency superior to Monte Carlo in computing the evidence of an individual model, applying BQ directly to model comparison may waste computation producing an overly-accurate estimate for the evidence of a clearly poor model. We propose an automated and efficient algorithm for computing the most-relevant quantity for model selection: the posterior model probability. Our technique maximizes the mutual information between this quantity and observations of the models’ likelihoods, yielding efficient sample acquisition across disparate model spaces when likelihood observations are limited. Our method produces more-accurate posterior estimates using fewer likelihood evaluations than standard Bayesian quadrature and Monte Carlo estimators, as we demonstrate on synthetic and real-world examples.
more »
« less
- Award ID(s):
- 1845434
- PAR ID:
- 10140927
- Date Published:
- Journal Name:
- Proceedings of the 36th International Conference on Machine Learning
- Page Range / eLocation ID:
- 931 - 940
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In computational mechanics, multiple models are often present to describe a physical system. While Bayesian model selection is a helpful tool to compare these models using measurement data, it requires the computationally expensive estimation of a multidimensional integral — known as the marginal likelihood or as the model evidence (i.e., the probability of observing the measured data given the model) — over the multidimensional parameter domain. This study presents efficient approaches for estimating this marginal likelihood by transforming it into a one-dimensional integral that is subsequently evaluated using a quadrature rule at multiple adaptively-chosen iso-likelihood contour levels. Three different algorithms are proposed to estimate the probability mass at each adapted likelihood level using samples from importance sampling, stratified sampling, and Markov chain Monte Carlo (MCMC) sampling, respectively. The proposed approach is illustrated — with comparisons to Monte Carlo, nested, and MultiNest sampling — through four numerical examples. The first, an elementary example, shows the accuracies of the three proposed algorithms when the exact value of the marginal likelihood is known. The second example uses an 11-story building subjected to an earthquake excitation with an uncertain hysteretic base isolation layer with two models to describe the isolation layer behavior. The third example considers flow past a cylinder when the inlet velocity is uncertain. Based on the these examples, the method with stratified sampling is by far the most accurate and efficient method for complex model behavior in low dimension, particularly considering that this method can be implemented to exploit parallel computation. In the fourth example, the proposed approach is applied to heat conduction in an inhomogeneous plate with uncertain thermal conductivity modeled through a 100 degree-of-freedom Karhunen–Loève expansion. The results indicate that MultiNest cannot efficiently handle the high-dimensional parameter space, whereas the proposed MCMC-based method more accurately and efficiently explores the parameter space. The marginal likelihood results for the last three examples — when compared with the results obtained from standard Monte Carlo sampling, nested sampling, and MultiNest algorithm — show good agreement.more » « less
-
Abstract Modern macroeconometrics often relies on time series models for which it is time-consuming to evaluate the likelihood function. We demonstrate how Bayesian computations for such models can be drastically accelerated by reweighting and mutating posterior draws from an approximating model that allows for fast likelihood evaluations, into posterior draws from the model of interest, using a sequential Monte Carlo (SMC) algorithm. We apply the technique to the estimation of a vector autoregression with stochastic volatility and two nonlinear dynamic stochastic general equilibrium models. The runtime reductions we obtain range from 27 % to 88 %.more » « less
-
We investigate approximate Bayesian inference techniques for nonlinear systems described by ordinary differential equation (ODE) models. In particular, the approximations will be based on set-valued reachability analysis approaches, yielding approximate models for the posterior distribution. Nonlinear ODEs are widely used to mathematically describe physical and biological models. However, these models are often described by parameters that are not directly measurable and have an impact on the system behaviors. Often, noisy measurement data combined with physical/biological intuition serve as the means for finding appropriate values of these parameters.Our approach operates under a Bayesian framework, given prior distribution over the parameter space and noisy observations under a known sampling distribution. We explore subsets of the space of model parameters, computing bounds on the likelihood for each subset. This is performed using nonlinear set-valued reachability analysis that is made faster by means of linearization around a reference trajectory. The tiling of the parameter space can be adaptively refined to make bounds on the likelihood tighter. We evaluate our approach on a variety of nonlinear benchmarks and compare our results with Markov Chain Monte Carlo and Sequential Monte Carlo approaches.more » « less
-
ABSTRACT We introduce Bilby-MCMC, a Markov chain Monte Carlo sampling algorithm tuned for the analysis of gravitational waves from merging compact objects. Bilby-MCMC provides a parallel-tempered ensemble Metropolis-Hastings sampler with access to a block-updating proposal library including problem-specific and machine learning proposals. We demonstrate that learning proposals can produce over a 10-fold improvement in efficiency by reducing the autocorrelation time. Using a variety of standard and problem-specific tests, we validate the ability of the Bilby-MCMC sampler to produce independent posterior samples and estimate the Bayesian evidence. Compared to the widely used Dynesty nested sampling algorithm, Bilby-MCMC is less efficient in producing independent posterior samples and less accurate in its estimation of the evidence. However, we find that posterior samples drawn from the Bilby-MCMC sampler are more robust: never failing to pass our validation tests. Meanwhile, the Dynesty sampler fails the difficult-to-sample Rosenbrock likelihood test, over constraining the posterior. For CBC problems, this highlights the importance of cross-sampler comparisons to ensure results are robust to sampling error. Finally, Bilby-MCMC can be embarrassingly and asynchronously parallelized making it highly suitable for reducing the analysis wall-time using a High Throughput Computing environment. Bilby-MCMC may be a useful tool for the rapid and robust analysis of gravitational-wave signals during the advanced detector era and we expect it to have utility throughout astrophysics.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
