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In the realm of large-scale statistical inference, empirical Bayes methods have proven to be fundamental tools for estimating prior distributions from observational data. Traditionally, these methods have been developed under the assumption of independence between observations, which simplifies the calculation of likelihood and allows for well-known asymptotic properties. However, in many real-world applications, data exhibit significant correlations, as occurs in time series, medical images, or spatial measurements. Ignoring this dependence can lead to biased estimates and a loss of efficiency. A recent approach proposes using the composite marginal likelihood (CML) estimator, which deliberately omits correlations in the likelihood calculation but manages to maintain an optimal convergence rate in weighted Hellinger distance, with a rate of order $n_*^{-1/2}$, where $n_*=n/\kappa_0$ is the effective sample size, depending only on the number of observations $n$ and the spectral radius $\kappa_0$ of the correlation matrix. This property is remarkable because it demonstrates that, even when the dependence structure is disregarded, CML remains nearly optimal under a wide range of correlations.
From a practical perspective, this result opens the door to applications in domains where dependence is unavoidable. For example, in Bayesian linear regression, where the prior on the coefficients is estimated from the least squares estimator, CML allows for robust inferences even when errors are correlated. Similarly, in nonlinear single-index models, prior estimation can be performed using a corrected de-biased gradient step, avoiding the complexity of the full likelihood. These techniques are especially relevant in business environments where large volumes of correlated data are handled, such as in demand forecasting, sensor analysis, or industrial process optimization.
At Q2BSTUDIO, we understand that the ability to extract knowledge from complex data is a differentiating factor for organizations. Therefore, we offer artificial intelligence for businesses that integrates advanced statistical methods, such as those based on empirical Bayes, to improve the accuracy of predictive models and decision-making. Our teams develop custom applications that incorporate these techniques, adapting to the particularities of each business. Additionally, we combine these capabilities with AWS and Azure cloud services to scale inference processes, and with business intelligence services such as Power BI to visualize results in an accessible manner. The implementation of AI agents and cybersecurity solutions complements our ecosystem, ensuring that models are not only accurate but also secure and efficient.
Research on empirical Bayes with dependent observations not only has profound theoretical implications but also provides concrete tools to address real-world problems. By combining this knowledge with a custom software approach, companies can build inference systems tailored to their specific needs, overcoming the limitations of standard methods. At Q2BSTUDIO, we are committed to transferring these advances into practical solutions that generate tangible value, whether through process optimization, pattern detection, or decision automation.
