The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.
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Empirical Bayes shrinkage methods usually maintain a prior independence assumption: The unknown parameters of interest are independent from the known precision of the estimates. This assumption is theoretically questionable and empirically rejected, and imposing it inappropriately may harm the performance of empirical Bayes methods. We instead model the conditional distribution of the parameter given the standard errors as a location-scale family, leading to a family of methods that we call CLOSE. We establish that (i) CLOSE is rate-optimal for squared error Bayes regret, (ii) squared error regret control is sufficient for an important class of economic decision problems, and (iii) CLOSE is worst-case robust. We use our method to select high-mobility Census tracts targeting a variety of economic mobility measures in the Opportunity Atlas (Chetty et al., 2020; Bergman et al., 2023). Census tracts selected by close are more mobile on average than those selected by the standard shrinkage method. For 6 out of 15 mobility measures considered, the gain of close over the standard shrinkage method is larger than the gain of the standard method over selecting Census tracts uniformly at random.
Causal inference studies whether the presence of a variable influences an observed outcome. As measured by quantities such as the "average treatment effect," this paradigm is employed across numerous biological fields, from vaccine and drug development to policy interventions. Unfortunately, the majority of these methods are often limited to univariate outcomes. Our work generalizes causal estimands to outcomes with any number of dimensions or any measurable space, and formulates traditional causal estimands for nominal variables as causal discrepancy tests. We propose a simple technique for adjusting universally consistent conditional independence tests and prove that these tests are universally consistent causal discrepancy tests. Numerical experiments illustrate that our method, Causal CDcorr, leads to improvements in both finite sample validity and power when compared to existing strategies. Our methods are all open source and available at github.com/ebridge2/cdcorr.
Randomized controlled trials (RCTs) are a cornerstone of comparative effectiveness because they remove the confounding bias present in observational studies. However, RCTs are typically much smaller than observational studies because of financial and ethical considerations. Therefore it is of great interest to be able to incorporate plentiful observational data into the analysis of smaller RCTs. Previous estimators developed for this purpose rely on unrealistic additional assumptions without which the added data can bias the effect estimate. Recent work proposed an alternative method (prognostic adjustment) that imposes no additional assumption and increases efficiency in the analysis of RCTs. The idea is to use the observational data to learn a prognostic model: a regression of the outcome onto the covariates. The predictions from this model, generated from the RCT subjects' baseline variables, are used as a covariate in a linear model. In this work, we extend this framework to work when conducting inference with nonparametric efficient estimators in trial analysis. Using simulations, we find that this approach provides greater power (i.e., smaller standard errors) than without prognostic adjustment, especially when the trial is small. We also find that the method is robust to observed or unobserved shifts between the observational and trial populations and does not introduce bias. Lastly, we showcase this estimator leveraging real-world historical data on a randomized blood transfusion study of trauma patients.
Bayesian model averaging is a practical method for dealing with uncertainty due to model specification. Use of this technique requires the estimation of model probability weights. In this work, we revisit the derivation of estimators for these model weights. Use of the Kullback-Leibler divergence as a starting point leads naturally to a number of alternative information criteria suitable for Bayesian model weight estimation. We explore three such criteria, known to the statistics literature before, in detail: a Bayesian analogue of the Akaike information criterion which we call the BAIC, the Bayesian predictive information criterion (BPIC), and the posterior predictive information criterion (PPIC). We compare the use of these information criteria in numerical analysis problems common in lattice field theory calculations. We find that the PPIC has the most appealing theoretical properties and can give the best performance in terms of model-averaging uncertainty, particularly in the presence of noisy data.
We consider the problem of learning an optimal prescriptive tree (i.e., an interpretable treatment assignment policy in the form of a binary tree) of moderate depth, from observational data. This problem arises in numerous socially important domains such as public health and personalized medicine, where interpretable and data-driven interventions are sought based on data gathered in deployment -- through passive collection of data -- rather than from randomized trials. We propose a method for learning optimal prescriptive trees using mixed-integer optimization (MIO) technology. We show that under mild conditions our method is asymptotically exact in the sense that it converges to an optimal out-of-sample treatment assignment policy as the number of historical data samples tends to infinity. Contrary to existing literature, our approach: 1) does not require data to be randomized, 2) does not impose stringent assumptions on the learned trees, and 3) has the ability to model domain specific constraints. Through extensive computational experiments, we demonstrate that our asymptotic guarantees translate to significant performance improvements in finite samples, as well as showcase our uniquely flexible modeling power by incorporating budget and fairness constraints.
We consider covariance estimation of any subgaussian distribution from finitely many i.i.d. samples that are quantized to one bit of information per entry. Recent work has shown that a reliable estimator can be constructed if uniformly distributed dithers on $[-\lambda,\lambda]$ are used in the one-bit quantizer. This estimator enjoys near-minimax optimal, non-asymptotic error estimates in the operator and Frobenius norms if $\lambda$ is chosen proportional to the largest variance of the distribution. However, this quantity is not known a-priori, and in practice $\lambda$ needs to be carefully tuned to achieve good performance. In this work we resolve this problem by introducing a tuning-free variant of this estimator, which replaces $\lambda$ by a data-driven quantity. We prove that this estimator satisfies the same non-asymptotic error estimates - up to small (logarithmic) losses and a slightly worse probability estimate. Our proof relies on a new version of the Burkholder-Rosenthal inequalities for matrix martingales, which is expected to be of independent interest.
This article aims at the lifetime prognosis of one-shot devices subject to competing causes of failure. Based on the failure count data recorded across several inspection times, statistical inference of the lifetime distribution is studied under the assumption of Lindley distribution. In the presence of outliers in the data set, the conventional maximum likelihood method or Bayesian estimation may fail to provide a good estimate. Therefore, robust estimation based on the weighted minimum density power divergence method is applied both in classical and Bayesian frameworks. Thereafter, the robustness behaviour of the estimators is studied through influence function analysis. Further, in density power divergence based estimation, we propose an optimization criterion for finding the tuning parameter which brings a trade-off between robustness and efficiency in estimation. The article also analyses when the cause of failure is missing for some of the devices. The analytical development has been restudied through a simulation study and a real data analysis where the data is extracted from the SEER database.
We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.
Despite temperature rise being a first-order design constraint, traditional thermal estimation techniques have severe limitations in modeling critical aspects affecting the temperature in modern-day chips. Existing thermal modeling techniques often ignore the effects of parameter variation, which can lead to significant errors. Such methods also ignore the dependence of conductivity on temperature and its variation. Leakage power is also incorporated inadequately by state-of-the-art techniques. Thermal modeling is a process that has to be repeated at least thousands of times in the design cycle, and hence speed is of utmost importance. To overcome these limitations, we propose VarSim, an ultrafast thermal simulator based on Green's functions. Green's functions have been shown to be faster than the traditional finite difference and finite element-based approaches but have rarely been employed in thermal modeling. Hence we propose a new Green's function-based method to capture the effects of leakage power as well as process variation analytically. We provide a closed-form solution for the Green's function considering the effects of variation on the process, temperature, and thermal conductivity. In addition, we propose a novel way of dealing with the anisotropicity introduced by process variation by splitting the Green's functions into shift-variant and shift-invariant components. Since our solutions are analytical expressions, we were able to obtain speedups that were several orders of magnitude over and above state-of-the-art proposals with a mean absolute error limited to 4% for a wide range of test cases. Furthermore, our method accurately captures the steady-state as well as the transient variation in temperature.
In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.
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