Posterior predictive p-values (ppps) have become popular tools for Bayesian model criticism, being general-purpose and easy to use. However, their interpretation can be difficult because their distribution is not uniform under the hypothesis that the model did generate the data. To address this issue, procedures to obtain calibrated ppps (cppps) have been proposed although not used in practice, because they require repeated simulation of new data and model estimation via MCMC. Here we give methods to balance the computational trade-off between the number of calibration replicates and the number of MCMC samples per replicate. Our results suggest that investing in a large number of calibration replicates while using short MCMC chains can save significant computation time compared to naive implementations, without significant loss in accuracy. We propose different estimators for the variance of the cppp that can be used to confirm quickly when the model fits the data well. Variance estimation requires the effective sample sizes of many short MCMC chains; we show that these can be well approximated using the single long MCMC chain from the real-data model. The procedure for cppp is implemented in NIMBLE, a flexible framework for hierarchical modeling that supports many models and discrepancy measures.
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Equivalence testing allows one to conclude that two characteristics are practically equivalent. We propose a framework for fast sample size determination with Bayesian equivalence tests facilitated via posterior probabilities. We assume that data are generated using statistical models with fixed parameters for the purposes of sample size determination. Our framework defines a distribution for the sample size that controls the length of posterior highest density intervals, where targets for the interval length are calibrated to yield desired power for the equivalence test. We prove the normality of the limiting distribution for the sample size and introduce a two-stage approach for estimating this distribution in the nonlimiting case. This approach is much faster than traditional power calculations for Bayesian equivalence tests, and it requires users to make fewer choices than traditional simulation-based methods for Bayesian sample size determination.
Rare and Weak models for multiple hypothesis testing assume that only a small proportion of the tested hypotheses concern non-null effects and the individual effects are only moderately large, so they generally do not stand out individually, for example in a Bonferroni analysis. Such models have been studied in quite a few settings, for example in some cases studies focused on an underlying Gaussian means model for the hypotheses being tested; in some others, Poisson and Binomial. Such seemingly different models have asymptotically the following common structure. Summarizing the evidence of individual tests by the negative logarithm of its P-value, the model is asymptotically equivalent to a situation in which most negative log P-values have a standard exponential distribution but a small fraction of the P-values might have an alternative distribution which is approximately noncentral chisquared on one degree of freedom. This log-chisquared approximation is different from the log-normal approximation of Bahadur which is unsuitable for analyzing Rare and Weak multiple testing models. We characterize the asymptotic performance of global tests combining asymptotic log-chisquared P-values in terms of the chisquared mixture parameters: the scaling parameter controlling heteroscedasticity, the non-centrality parameter describing the effect size whenever it exists, and the parameter controlling the rarity of the non-null effects. In a phase space involving the last two parameters, we derive a region where all tests are asymptotically powerless. Outside of this region, the Berk-Jones and the Higher Criticism tests have maximal power. Inference techniques based on the minimal P-value, false-discovery rate controlling, and Fisher's combination test have sub-optimal asymptotic phase diagrams.
Two-sample multiple testing problems of sparse spatial data are frequently arising in a variety of scientific applications. In this article, we develop a novel neighborhood-assisted and posterior-adjusted (NAPA) approach to incorporate both the spatial smoothness and sparsity type side information to improve the power of the test while controlling the false discovery of multiple testing. We translate the side information into a set of weights to adjust the $p$-values, where the spatial pattern is encoded by the ordering of the locations, and the sparsity structure is encoded by a set of auxiliary covariates. We establish the theoretical properties of the proposed test, including the guaranteed power improvement over some state-of-the-art alternative tests, and the asymptotic false discovery control. We demonstrate the efficacy of the test through intensive simulations and two neuroimaging applications.
The practical utility of causality in decision-making is widespread and brought about by the intertwining of causal discovery and causal inference. Nevertheless, a notable gap exists in the evaluation of causal discovery methods, where insufficient emphasis is placed on downstream inference. To address this gap, we evaluate seven established baseline causal discovery methods including a newly proposed method based on GFlowNets, on the downstream task of treatment effect estimation. Through the implementation of a distribution-level evaluation, we offer valuable and unique insights into the efficacy of these causal discovery methods for treatment effect estimation, considering both synthetic and real-world scenarios, as well as low-data scenarios. The results of our study demonstrate that some of the algorithms studied are able to effectively capture a wide range of useful and diverse ATE modes, while some tend to learn many low-probability modes which impacts the (unrelaxed) recall and precision.
Robust iterative methods for solving large sparse systems of linear algebraic equations often suffer from the problem of optimizing the corresponding tuning parameters. To improve the performance of the problem of interest, specific parameter tuning is required, which in practice can be a time-consuming and tedious task. This paper proposes an optimization algorithm for tuning the numerical method parameters. The algorithm combines the evolution strategy with the pre-trained neural network used to filter the individuals when constructing the new generation. The proposed coupling of two optimization approaches allows to integrate the adaptivity properties of the evolution strategy with a priori knowledge realized by the neural network. The use of the neural network as a preliminary filter allows for significant weakening of the prediction accuracy requirements and reusing the pre-trained network with a wide range of linear systems. The detailed algorithm efficiency evaluation is performed for a set of model linear systems, including the ones from the SuiteSparse Matrix Collection and the systems from the turbulent flow simulations. The obtained results show that the pre-trained neural network can be effectively reused to optimize parameters for various linear systems, and a significant speedup in the calculations can be achieved at the cost of about 100 trial solves. The hybrid evolution strategy decreases the calculation time by more than 6 times for the black box matrices from the SuiteSparse Matrix Collection and by a factor of 1.4-2 for the sequence of linear systems when modeling turbulent flows. This results in a speedup of up to 1.8 times for the turbulent flow simulations performed in the paper.
We describe a new concrete approach to giving predictable error locations for sequential (flow-sensitive) effect systems. Prior implementations of sequential effect systems rely on either computing a bottom-up effect and comparing it to a declaration (e.g., method annotation) or leaning on constraint-based type inference. These approaches do not necessarily report program locations that precisely indicate where a program may "go wrong" at runtime. Instead of relying on constraint solving, we draw on the notion of a residual from literature on ordered algebraic structures. Applying these to effect quantales (a large class of sequential effect systems) yields an implementation approach which accepts exactly the same program as an original effect quantale, but for effect-incorrect programs is guaranteed to fail type-checking with predictable error locations tied to evaluation order. We have implemented this idea in a generic effect system implementation framework for Java, and report on experiences applying effect systems from the literature and novel effect systems to Java programs. We find that the reported error locations with our technique are significantly closer to the program points that lead to failed effect checks.
In this paper, we propose a method for estimating model parameters using Small-Angle Scattering (SAS) data based on the Bayesian inference. Conventional SAS data analyses involve processes of manual parameter adjustment by analysts or optimization using gradient methods. These analysis processes tend to involve heuristic approaches and may lead to local solutions.Furthermore, it is difficult to evaluate the reliability of the results obtained by conventional analysis methods. Our method solves these problems by estimating model parameters as probability distributions from SAS data using the framework of the Bayesian inference. We evaluate the performance of our method through numerical experiments using artificial data of representative measurement target models.From the results of the numerical experiments, we show that our method provides not only high accuracy and reliability of estimation, but also perspectives on the transition point of estimability with respect to the measurement time and the lower bound of the angular domain of the measured data.
In an effort to provide regional decision support for the public healthcare, we design a data-driven compartment-based model of COVID-19 in Sweden. From national hospital statistics we derive parameter priors, and we develop linear filtering techniques to drive the simulations given data in the form of daily healthcare demands. We additionally propose a posterior marginal estimator which provides for an improved temporal resolution of the reproduction number estimate as well as supports robustness checks via a parametric bootstrap procedure. From our computational approach we obtain a Bayesian model of predictive value which provides important insight into the progression of the disease, including estimates of the effective reproduction number, the infection fatality rate, and the regional-level immunity. We successfully validate our posterior model against several different sources, including outputs from extensive screening programs. Since our required data in comparison is easy and non-sensitive to collect, we argue that our approach is particularly promising as a tool to support monitoring and decisions within public health.
Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second.
Modelling in biology must adapt to increasingly complex and massive data. The efficiency of the inference algorithms used to estimate model parameters is therefore questioned. Many of these are based on stochastic optimization processes which waste a significant part of the computation time due to their rejection sampling approaches. We introduce the Fixed Landscape Inference MethOd (flimo), a new likelihood-free inference method for continuous state-space stochastic models. It applies deterministic gradient-based optimization algorithms to obtain a point estimate of the parameters, minimizing the difference between the data and some simulations according to some prescribed summary statistics. In this sense, it is analogous to Approximate Bayesian Computation (ABC). Like ABC, it can also provide an approximation of the distribution of the parameters. Three applications are proposed: a usual theoretical example, namely the inference of the parameters of g-and-k distributions; a population genetics problem, not so simple as it seems, namely the inference of a selective value from time series in a Wright-Fisher model; and simulations from a Ricker model, representing chaotic population dynamics. In the two first applications, the results show a drastic reduction of the computational time needed for the inference phase compared to the other methods, despite an equivalent accuracy. Even when likelihood-based methods are applicable, the simplicity and efficiency of flimo make it a compelling alternative. Implementations in Julia and in R are available on //metabarcoding.org/flimo. To run flimo, the user must simply be able to simulate data according to the chosen model.
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