Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Ito process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Ito process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean-Vlasov-Ito processes. The effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.
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Amortized variational inference produces a posterior approximator that can compute a posterior approximation given any new observation. Unfortunately, there are few guarantees about the quality of these approximate posteriors. We propose Conformalized Amortized Neural Variational Inference (CANVI), a procedure that is scalable, easily implemented, and provides guaranteed marginal coverage. Given a collection of candidate amortized posterior approximators, CANVI constructs conformalized predictors based on each candidate, compares the predictors using a metric known as predictive efficiency, and returns the most efficient predictor. CANVI ensures that the resulting predictor constructs regions that contain the truth with high probability (exactly how high is prespecified by the user). CANVI is agnostic to design decisions in formulating the candidate approximators and only requires access to samples from the forward model, permitting its use in likelihood-free settings. We prove lower bounds on the predictive efficiency of the regions produced by CANVI and explore how the quality of a posterior approximation relates to the predictive efficiency of prediction regions based on that approximation. Finally, we demonstrate the accurate calibration and high predictive efficiency of CANVI on a suite of simulation-based inference benchmark tasks and an important scientific task: analyzing galaxy emission spectra.
The vast majority of literature on evaluating the significance of a treatment effect based on observational data has been confined to discrete treatments. These methods are not applicable to drawing inference for a continuous treatment, which arises in many important applications. To adjust for confounders when evaluating a continuous treatment, existing inference methods often rely on discretizing the treatment or using (possibly misspecified) parametric models for the effect curve. Recently, Kennedy et al. (2017) proposed nonparametric doubly robust estimation for a continuous treatment effect in observational studies. However, inference for the continuous treatment effect is a harder problem. To the best of our knowledge, a completely nonparametric doubly robust approach for inference in this setting is not yet available. We develop such a nonparametric doubly robust procedure in this paper for making inference on the continuous treatment effect curve. Using empirical process techniques for local U- and V-processes, we establish the test statistic's asymptotic distribution. Furthermore, we propose a wild bootstrap procedure for implementing the test in practice. We illustrate the new method via simulations and a study of a constructed dataset relating the effect of nurse staffing hours on hospital performance. We implement our doubly robust dose response test in the R package DRDRtest on CRAN.
Additive spatial statistical models with weakly stationary process assumptions have become standard in spatial statistics. However, one disadvantage of such models is the computation time, which rapidly increases with the number of datapoints. The goal of this article is to apply an existing subsampling strategy to standard spatial additive models and to derive the spatial statistical properties. We call this strategy the ``spatial data subset model'' approach, which can be applied to big datasets in a computationally feasible way. Our approach has the advantage that one does not require any additional restrictive model assumptions. That is, computational gains increase as model assumptions are removed when using our model framework. This provides one solution to the computational bottlenecks that occur when applying methods such as Kriging to ``big data''. We provide several properties of this new spatial data subset model approach in terms of moments, sill, nugget, and range under several sampling designs. The biggest advantage of our approach is that it is scalable to a dataset of any size that can be stored. We present the results of the spatial data subset model approach on simulated datasets, and on a large dataset consists of 150,000 observations of daytime land surface temperatures measured by the MODIS instrument onboard the Terra satellite.
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x \in \{0, 1\}^n$ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of $x$. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $\Omega(n/\log^2 n)$ for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.
The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. In this work we analyze these methods and show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the $\delta$-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also shows that these condition numbers are a reliable criterion for detecting finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues.
This paper aims at obtaining, by means of integral transforms, analytical approximations in short times of solutions to boundary value problems for the one-dimensional reaction-diffusion equation with constant coefficients. The general form of the equation is considered on a bounded generic interval and the three classical types of boundary conditions, i.e., Dirichlet as well as Neumann and mixed boundary conditions are considered in a unified way. The Fourier and Laplace integral transforms are successively applied and an exact solution is obtained in the Laplace domain. This operational solution is proven to be the accurate Laplace transform of the infinite series obtained by the Fourier decomposition method and presented in the literature as solutions to this type of problem. On the basis of this unified operational solution, four cases are distinguished where innovative formulas expressing consistent analytical approximations in short time limits are derived with respect to the behavior of the solution at the boundaries. Compared to the infinite series solutions, the analytical approximations may open new perspectives and applications, among which can be noted the improvement of numerical efficiency in simulations of one-dimensional moving boundary problems, such as in Stefan models.
Reachable sets of nonlinear control systems can in general only be approximated numerically, and these approximations are typically very expensive to compute. In this paper, we explore strategies for choosing the temporal and spatial discretizations of Euler's method for reachable set computation in a non-uniform way to improve the performance of the method.
Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The developed techniques in this paper are applicable from low- to high-dimensional stochastic spaces, and the basic idea is to construct a relevant target distribution by weighting the original random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.
When estimating quantities and fields that are difficult to measure directly, such as the fluidity of ice, from point data sources, such as satellite altimetry, it is important to solve a numerical inverse problem that is formulated with Bayesian consistency. Otherwise, the resultant probability density function for the difficult to measure quantity or field will not be appropriately clustered around the truth. In particular, the inverse problem should be formulated by evaluating the numerical solution at the true point locations for direct comparison with the point data source. If the data are first fitted to a gridded or meshed field on the computational grid or mesh, and the inverse problem formulated by comparing the numerical solution to the fitted field, the benefits of additional point data values below the grid density will be lost. We demonstrate, with examples in the fields of groundwater hydrology and glaciology, that a consistent formulation can increase the accuracy of results and aid discourse between modellers and observationalists. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). This new abstraction is well-suited to automation, including automatic differentiation. We demonstrate this through implementation in Firedrake, which generates highly optimised code for solving Partial Differential Equations (PDEs) with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint, allowing PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation.
Latent variable models have become instrumental in computational neuroscience for reasoning about neural computation. This has fostered the development of powerful offline algorithms for extracting latent neural trajectories from neural recordings. However, despite the potential of real time alternatives to give immediate feedback to experimentalists, and enhance experimental design, they have received markedly less attention. In this work, we introduce the exponential family variational Kalman filter (eVKF), an online recursive Bayesian method aimed at inferring latent trajectories while simultaneously learning the dynamical system generating them. eVKF works for arbitrary likelihoods and utilizes the constant base measure exponential family to model the latent state stochasticity. We derive a closed-form variational analogue to the predict step of the Kalman filter which leads to a provably tighter bound on the ELBO compared to another online variational method. We validate our method on synthetic and real-world data, and, notably, show that it achieves competitive performance
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