Computing the gradient of a function provides fundamental information about its behavior. This information is essential for several applications and algorithms across various fields. One common application that requires gradients are optimization techniques such as stochastic gradient descent, Newton's method and trust region methods. However, these methods usually require a numerical computation of the gradient at every iteration of the method which is prone to numerical errors. We propose a simple limited-memory technique for improving the accuracy of a numerically computed gradient in this gradient-based optimization framework by exploiting (1) a coordinate transformation of the gradient and (2) the history of previously taken descent directions. The method is verified empirically by extensive experimentation on both test functions and on real data applications. The proposed method is implemented in the $\texttt{R} $ package $ \texttt{smartGrad}$ and in C$ \texttt{++} $.

Background: The parameter uncertainty in the six-dimensional health state short form (SF-6D) value sets is commonly ignored. There are two sources of parameter uncertainty: uncertainty around the estimated regression coefficients and uncertainty around the model’s specification. This study explores these two sources of parameter uncertainty in the value sets using probabilistic sensitivity analysis (PSA) and a Bayesian approach. Methods: We used data from the original UK/SF-6D valuation study to evaluate the extent of parameter uncertainty in the value set. First, we re-estimated the Brazier model to replicate the published estimated coefficients. Second, we estimated standard errors around the predicted utility of each SF-6D state to assess the impact of parameter uncertainty on these estimated utilities. Third, we used Monte Carlo simulation technique to account for the uncertainty on these estimates. Finally, we used a Bayesian approach to quantifying parameter uncertainty in the value sets. The extent of parameter uncertainty in SF-6D value sets was assessed using data from the Hong Kong valuation study. Results: Including parameter uncertainty results in wider confidence/credible intervals and improved coverage probability using both approaches. Using PSA, the mean 95% confidence intervals widths for the mean utilities were 0.1394 (range: 0.0565–0.2239) and 0.0989 (0.0048–0.1252) with and without parameter uncertainty whilst, using the Bayesian approach, this was 0.1478 (0.053–0.1665). Upon evaluating the impact of parameter uncertainty on estimates of a population’s mean utility, the true standard error was underestimated by 79.1% (PSA) and 86.15% (Bayesian) when parameter uncertainty was ignored. Conclusions: Parameter uncertainty around the SF-6D value set has a large impact on the predicted utilities and estimated confidence intervals. This uncertainty should be accounted for when using SF-6D utilities in economic evaluations. Ignoring this additional information could impact misleadingly on policy decisions.

Neighborhood variations in depression, an important aspect of the overall mental health burden, have been linked both to environmental context (e.g., area crime, neighborhood cohesion), and to area socio-demographic composition. Previous models seeking to explain such spatial variations in mental health, such as those based on Bayesian disease mapping, follow a standard approach defined by: spatially stationary effects of area predictors; predictor effects neglecting potential spatial spillover; and a spatially structured residual to account for unmodelled spatial dependencies. In a study of depression incidence in England neighborhoods, we consider the gains from an alternative strategy, allowing nonstationary environmental impacts; spillover effects of environmental factors, and a non-stationary spatial intensity. We focus particularly on impacts of socio-behavioral environments, namely neighborhood cohesion and crime. We find these to be major influences on neighborhood depression incidence, and also find major gains in model performance by explicitly considering non-stationarity and spillovers. Allowing context heterogeneity, varying spatial intensity and spillover are shown to enhance the impacts of socio-behavioral environments on depression incidence, and such findings have broader relevance to disease mapping regression. Public health policy framing may therefore need to be tailored to locally specific environmental impacts, and to inter-agency collaboration across arbitrary boundaries.

Neighborhood variations in depression, an important aspect of the overall mental health burden, have been linked both to environmental context (e.g. area crime, neighborhood cohesion), and to area socio-demographic composition. Previous models seeking to explain such spatial variations in mental health, such as those based on Bayesian disease mapping, follow a standard approach defined by: spatially stationary effects of area predictors; predictor effects neglecting potential spatial spillover; and a spatially structured residual to account for unmodelled spatial dependencies. In a study of depression incidence in England neighborhoods, we consider the gains from an alternative strategy, allowing nonstationary environmental impacts; spillover effects of environmental factors, and a non-stationary spatial intensity. We focus particularly on impacts of socio-behavioral environments, namely neighborhood cohesion and crime. We find these to be major influences on neighborhood depression incidence, and also find major gains in model performance by explicitly considering non-stationarity and spillovers. Allowing context heterogeneity, varying spatial intensity and spillover are shown to enhance the impacts of socio-behavioral environments on depression incidence, and such findings have broader relevance to disease mapping regression.

The integrated nested Laplace approximations (INLA) method has become a widely utilized tool for researchers and practitioners seeking to perform approximate Bayesian inference across various fields of application. To address the growing demand for incorporating more complex models and enhancing the method's capabilities, this paper introduces a novel framework that leverages dense matrices for performing approximate Bayesian inference based on INLA across multiple computing nodes using HPC. When dealing with non-sparse precision or covariance matrices, this new approach scales better compared to the current INLA method, capitalizing on the computational power offered by multiprocessors in shared and distributed memory architectures available in contemporary computing resources and specialized dense matrix algebra. To validate the efficacy of this approach, we conduct a simulation study then apply it to analyze cancer mortality data in Spain, employing a three-way spatio-temporal interaction model.