Nonlinear state-space models are ubiquitous in modeling real-world dynamical systems. Sequential Monte Carlo (SMC) techniques, also known as particle methods, are a well-known class of parameter estimation methods for this general class of state-space models. Existing SMC-based techniques rely on excessive sampling of the parameter space, which makes their computation intractable for large systems or tall data sets. Bayesian optimization techniques have been used for fast inference in state-space models with intractable likelihoods. These techniques aim to find the maximum of the likelihood function by sequential sampling of the parameter space through a single SMC approximator. Various SMC approximators with different fidelities and computational costs are often available for sample- based likelihood approximation. In this paper, we propose a multi-fidelity Bayesian optimization algorithm for the inference of general nonlinear state-space models (MFBO-SSM), which enables simultaneous sequential selection of parameters and approximators. The accuracy and speed of the algorithm are demonstrated by numerical experiments using synthetic gene expression data from a gene regulatory network model and real data from the VIX stock price index.
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Efficient sampling of constrained high-dimensional theoretical spaces with machine learning
Models of physics beyond the Standard Model often contain a large number of parameters. These form a high-dimensional space that is computationally intractable to fully explore. Experimental results project onto a subspace of parameters that are consistent with those observations, but mapping these constraints to the underlying parameters is also typically intractable. Instead, physicists often resort to scanning small subsets of the full parameter space and testing for experimental consistency. We propose an alternative approach that uses generative models to significantly improve the computational efficiency of sampling high-dimensional parameter spaces. To demonstrate this, we sample the constrained and phenomenological Minimal Supersymmetric Standard Models subject to the requirement that the sampled points are consistent with the measured Higgs boson mass. Our method achieves orders of magnitude improvements in sampling efficiency compared to a brute force search.
more » « less- Award ID(s):
- 1915005
- NSF-PAR ID:
- 10362460
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- The European Physical Journal C
- Volume:
- 81
- Issue:
- 12
- ISSN:
- 1434-6044
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Background In systems biology, the dynamics of biological networks are often modeled with ordinary differential equations (ODEs) that encode interacting components in the systems, resulting in highly complex models. In contrast, the amount of experimentally available data is almost always limited, and insufficient to constrain the parameters. In this situation, parameter estimation is a very challenging problem. To address this challenge, two intuitive approaches are to perform experimental design to generate more data, and to perform model reduction to simplify the model. Experimental design and model reduction have been traditionally viewed as two distinct areas, and an extensive literature and excellent reviews exist on each of the two areas. Intriguingly, however, the intrinsic connections between the two areas have not been recognized.
Results Experimental design and model reduction are deeply related, and can be considered as one unified framework. There are two recent methods that can tackle both areas, one based on model manifold and the other based on profile likelihood. We use a simple sum‐of‐two‐exponentials example to discuss the concepts and algorithmic details of both methods, and provide Matlab‐based code and implementation which are useful resources for the dissemination and adoption of experimental design and model reduction in the biology community.
Conclusions From a geometric perspective, we consider the experimental data as a point in a high‐dimensional data space and the mathematical model as a manifold living in this space. Parameter estimation can be viewed as a projection of the data point onto the manifold. By examining the singularity around the projected point on the manifold, we can perform both experimental design and model reduction. Experimental design identifies new experiments that expand the manifold and remove the singularity, whereas model reduction identifies the nearest boundary, which is the nearest singularity that suggests an appropriate form of a reduced model. This geometric interpretation represents one step toward the convergence of experimental design and model reduction as a unified framework.
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Experimental data are often costly to obtain, which makes it difficult to calibrate complex models. For many models an experimental design that produces the best calibration given a limited experimental budget is not obvious. This paper introduces a deep reinforcement learning (RL) algorithm for design of experiments that maximizes the information gain measured by Kullback–Leibler divergence obtained via the Kalman filter (KF). This combination enables experimental design for rapid online experiments where manual trial-and-error is not feasible in the high-dimensional parametric design space. We formulate possible configurations of experiments as a decision tree and a Markov decision process, where a finite choice of actions is available at each incremental step. Once an action is taken, a variety of measurements are used to update the state of the experiment. This new data leads to a Bayesian update of the parameters by the KF, which is used to enhance the state representation. In contrast to the Nash–Sutcliffe efficiency index, which requires additional sampling to test hypotheses for forward predictions, the KF can lower the cost of experiments by directly estimating the values of new data acquired through additional actions. In this work our applications focus on mechanical testing of materials. Numerical experiments with complex, history-dependent models are used to verify the implementation and benchmark the performance of the RL-designed experiments.more » « less
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Abstract High-fidelity simulators that connect theoretical models with observations are indispensable tools in many sciences. If the likelihood is known, inference can proceed using standard techniques. However, when the likelihood is intractable or unknown, a simulator makes it possible to infer the parameters of a theoretical model directly from real and simulated observations when coupled with machine learning. We introduce an extension of the recently proposed likelihood-free frequentist inference (
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https://github.com/AliAlkadhim/ALFFI .
