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An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density γ, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are \"nearly black\" sequences where only a proportion η is allowed to be nonzero, and sequences with normalized ℓpnorm bounded by η, for$\\eta > 0$and$0 < p \\leq 2$. Estimation error is measured by mean qth power loss, for$0 < q \\leq 2$. For all the classes considered, and for all q in (0, 2], the method achieves the optimal estimation rate as n → ∞ and η → 0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q > 1. Simulations show excellent performance. For appropriately chosen functions γ, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.

Background Fatal opioid-involved overdose rates increased precipitously from 5.0 per 100,000 population to 33.5 in Massachusetts between 1999 and 2022. Methods We used spatial rate smoothing techniques to identify persistent opioid overdose-involved fatality clusters at the ZIP Code Tabulation Area (ZCTA) level. Rate smoothing techniques were employed to identify locations of high fatal opioid overdose rates where population counts were low. In Massachusetts, this included areas with both sparse data and low population density. We used Local Indicators of Spatial Association (LISA) cluster analyses with the raw incidence rates, and the Empirical Bayes smoothed rates to identify clusters from 2011 to 2021. We also estimated Empirical Bayes LISA cluster estimates to identify clusters during the same period. We constructed measures of the socio-built environment and potentially inappropriate prescribing using principal components analysis. The resulting measures were used as covariates in Conditional Autoregressive Bayesian models that acknowledge spatial autocorrelation to predict both, if a ZCTA was part of an opioid-involved cluster for fatal overdose rates, as well as the number of times that it was part of a cluster of high incidence rates. Results LISA clusters for smoothed data were able to identify whether a ZCTA was part of a opioid involved fatality incidence cluster earlier in the study period, when compared to LISA clusters based on raw rates. PCA helped in identifying unique socio-environmental factors, such as minoritized populations and poverty, potentially inappropriate prescribing, access to amenities, and rurality by combining socioeconomic, built environment and prescription variables that were highly correlated with each other. In all models except for those that used raw rates to estimate whether a ZCTA was part of a high fatality cluster, opioid overdose fatality clusters in Massachusetts had high percentages of Black and Hispanic residents, and households experiencing poverty. The models that were fitted on Empirical Bayes LISA identified this phenomenon earlier in the study period than the raw rate LISA. However, all the models identified minoritized populations and poverty as significant factors in predicting the persistence of a ZCTA being part of a high opioid overdose cluster during this time period. Conclusion Conducting spatially robust analyses may help inform policies to identify community-level risks for opioid-involved overdose deaths sooner than depending on raw incidence rates alone. The results can help inform policy makers and planners about locations of persistent risk.

The reduction in locational traffic accident risks through appropriate traffic safety management is important to support, maintain, and improve children’s safe and independent mobility. This study proposes and verifies a method to evaluate the risk of elementary school students-vehicle accidents (ESSVAs) at individual intersections on residential roads in Toyohashi city, Japan, considering the difference in travel purposes (i.e., school commuting purpose; SCP or non-school commuting purpose: NSCP), based on a statistical regression model and Empirical Bayes (EB) estimation. The results showed that the ESSVA risk of children’s travel in SCP is lower than that in NSCP, and not only ESSVAs in SCP but also most ESSVAs in NSCP occurred on or near the designated school routes. Therefore, it would make sense to implement traffic safety management and measures focusing on school routes. It was also found that the locational ESSVA risk structure is different depending on whether the purpose of the children’s travels is SCP or NSCP in the statistical model. Finally, it was suggested that evaluation of locational ESSVA risks based on the EB estimation is useful for efficiently extracting locations where traffic safety measures should be implemented compared to that only based on the number of accidents in the past.

Due to non-stationary and noise characteristics of river flow time series data, some pre-processing methods are adopted to address the multi-scale and noise complexity. In this paper, we proposed an improved framework comprising Complete Ensemble Empirical Mode Decomposition with Adaptive Noise-Empirical Bayesian Threshold (CEEMDAN-EBT). The CEEMDAN-EBT is employed to decompose non-stationary river flow time series data into Intrinsic Mode Functions (IMFs). The derived IMFs are divided into two parts; noise-dominant IMFs and noise-free IMFs. Firstly, the noise-dominant IMFs are denoised using empirical Bayesian threshold to integrate the noises and sparsities of IMFs. Secondly, the denoised IMF’s and noise free IMF’s are further used as inputs in data-driven and simple stochastic models respectively to predict the river flow time series data. Finally, the predicted IMF’s are aggregated to get the final prediction. The proposed framework is illustrated by using four rivers of the Indus Basin System. The prediction performance is compared with Mean Square Error, Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE). Our proposed method, CEEMDAN-EBT-MM, produced the smallest MAPE for all four case studies as compared with other methods. This suggests that our proposed hybrid model can be used as an efficient tool for providing the reliable prediction of non-stationary and noisy time series data to policymakers such as for planning power generation and water resource management.

To detect positive selection at individual amino acid sites, most methods use an empirical Bayes approach. After parameters of a Markov process of codon evolution are estimated via maximum likelihood, they are passed to Bayes formula to compute the posterior probability that a site evolved under positive selection. A difficulty with this approach is that parameter estimates with large errors can negatively impact Bayesian classification. By assigning priors to some parameters, Bayes Empirical Bayes (BEB) mitigates this problem. However, as implemented, it imposes uniform priors, which causes it to be overly conservative in some cases. When standard regularity conditions are not met and parameter estimates are unstable, inference, even under BEB, can be negatively impacted. We present an alternative to BEB called smoothed bootstrap aggregation (SBA), which bootstraps site patterns from an alignment of protein coding DNA sequences to accommodate the uncertainty in the parameter estimates. We show that deriving the correction for parameter uncertainty from the data in hand, in combination with kernel smoothing techniques, improves site specific inference of positive selection. We compare BEB to SBA by simulation and real data analysis. Simulation results show that SBA balances accuracy and power at least as well as BEB, and when parameter estimates are unstable, the performance gap between BEB and SBA can widen in favor of SBA. SBA is applicable to a wide variety of other inference problems in molecular evolution.

The yeast centrosome or Spindle Pole Body (SPB) is an organelle situated in the nuclear membrane, where it nucleates spindle microtubules and acts as a signaling hub. Various studies have explored the effects of forcing individual proteins to interact with the yeast SPB, however no systematic study has been performed. We used synthetic physical interactions to detect proteins that inhibit growth when forced to associate with the SPB. We found the SPB to be especially sensitive to relocalization, necessitating a novel data analysis approach. This novel analysis of SPI screening data shows that regions of the cell are locally more sensitive to forced relocalization than previously thought. Furthermore, we found a set of associations that result in elevated SPB number and, in some cases, multi-polar spindles. Since hyper-proliferation of centrosomes is a hallmark of cancer cells, these associations point the way for the use of yeast models in the study of spindle formation and chromosome segregation in cancer.

This technical note describes some Bayesian procedures for the analysis of group studies that use nonlinear models at the first (within-subject) level – e.g., dynamic causal models – and linear models at subsequent (between-subject) levels. Its focus is on using Bayesian model reduction to finesse the inversion of multiple models of a single dataset or a single (hierarchical or empirical Bayes) model of multiple datasets. These applications of Bayesian model reduction allow one to consider parametric random effects and make inferences about group effects very efficiently (in a few seconds). We provide the relatively straightforward theoretical background to these procedures and illustrate their application using a worked example. This example uses a simulated mismatch negativity study of schizophrenia. We illustrate the robustness of Bayesian model reduction to violations of the (commonly used) Laplace assumption in dynamic causal modelling and show how its recursive application can facilitate both classical and Bayesian inference about group differences. Finally, we consider the application of these empirical Bayesian procedures to classification and prediction. •We describe a novel scheme for inverting non-linear models (e.g. DCMs) within subjects and linear models at the group level•We demonstrate this scheme is more robust to violations of the (commonly used) Laplace assumption than the standard approach•We validate the approach using a simulated mismatch negativity study of schizophrenia•We demonstrate the application of this scheme to classification and prediction of group membership

This article concerns the Bayes and frequentist aspects of empirical Bayes inference. Some of the ideas explored go back to Robbins in the 1950s, while others are current. Several examples are discussed, real and artificial, illustrating the two faces of empirical Bayes methodology: \"oracle Bayes\" shows empirical Bayes in its most frequentist mode, while \"finite Bayes inference\" is a fundamentally Bayesian application. In either case, modern theory and computation allow us to present a sharp finite-sample picture of what is at stake in an empirical Bayes analysis.

One of the most common analysis tasks in genomic research is to identify genes that are differentially expressed (DE) between experimental conditions. Empirical Bayes (EB) statistical tests using moderated genewise variances have been very effective for this purpose, especially when the number of biological replicate samples is small. The EB procedures can, however, be heavily influenced by a small number of genes with very large or very small variances. This article improves the differential expression tests by robustifying the hyperparameter estimation procedure. The robust procedure has the effect of decreasing the informativeness of the prior distribution for outlier genes while increasing its informativeness for other genes. This effect has the double benefit of reducing the chance that hypervariable genes will be spuriously identified as DE while increasing statistical power for the main body of genes. The robust EB algorithm is fast and numerically stable. The procedure allows exact small-sample null distributions for the test statistics and reduces exactly to the original EB procedure when no outlier genes are present. Simulations show that the robustified tests have similar performance to the original tests in the absence of outlier genes but have greater power and robustness when outliers are present. The article includes case studies for which the robust method correctly identifies and downweights genes associated with hidden covariates and detects more genes likely to be scientifically relevant to the experimental conditions. The new procedure is implemented in the limma software package freely available from the Bioconductor repository.

We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood and variational class that characterize the convergence rates. Under similar “prior mass and testing” conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors.