This paper essentially considers Gaussian process (GP) methods for regression analysis in longitudinal studies. The model is indeed a reasonable choice for description and prediction of phenomena involving repeated measurements in which there is evidence of heterogeneity among batches of measurements. First, we provide all the theoretical and practical details behind our modeling strategy. Then, we study the main properties of the model using simulated data. And finally, we apply analyze an AIDS clinical study developed by the AIDS Clinical Trials Group.

Considerando la flexibilidad y aplicabilidad del modelado Bayesiano, se revisan y resumen las principales características de un modelo jerárquico bajo el supuesto usual de intercambiabilidad: presentamos la estructura probabilística del modelo, todos los niveles involucrados en ´el y la distribución condicional completa de cada parámetro del modelo. En este modelo, permitimos que la media de la segunda etapa del modelo tenga una dependencia lineal con un conjunto de covariables mediante un enfoque de regresión. Además, se describe y deriva completamente el algoritmo de muestreo de Gibbs utilizado para obtener muestras de este modelo jerárquico. El estudio de caso es uno en el que caracterizamos en profundidad la superficie media de la temperatura de la mar registrada por 86 dispositivos en el mar Mediterráneo, clasificados por el tipo de dispositivo y la ubicación descrita por la latitud y la longitud. El modelo jerárquico se ajustó considerablemente bien a este conjunto de datos. Los hallazgos derivados de esta aplicación incluyen la descripción de las medias y la variabilidad dentro y entre las temperaturas registradas, evidencia de precisión similar entre dispositivos, diferencias entre tipos de dispositivos y buenas cualidades de predicción del modelo. Finalmente, se prueba la capacidad de predicción del modelo para cada tipo de dispositivo utilizando datos de la Oficina Nacional de Administración Oceánica y Atmosférica.

In this paper, we illustrate in-depth Bayesian hierarchical statistical modeling approaches. Bayesian hierarchical modeling provides a robust framework for analyzing spatial data, accommodating complex dependencies, making possible incorporating external knowledge into the analysis. To do so, we consider a dataset from 80 stations in the Venezuelan state of Guárico consisting of accumulated monthly rainfall in a time span of 16 years. The spatial correlation is modeled by using a Mat´ern correlation function with a fixed smoothness parameter. Following Banerjee et al. (2014), we examine two fully Bayesian parametric approaches: One of them static, based on a hierarchical model with latent variables; and the other spatiotemporal, based on the dynamic framework given in West and Harrison (2006). Both alternatives are sensible ones, but due to the nature of the data, the dynamic model is more appealing since it gives a complete spatiotemporal characterization of the response variable.

In this paper, a Bayesian spatial voting model is applied for the first time to characterize the legislative behavior of the Senate of the Republic of Colombia 2006-2010. The analysis is carried out based on the plenary nominal votes of the Senate. The estimation of the model is done using Markov chain Monte Carlo (MCMC) algorithms. The estimated ideal points provide empirical evidence supporting a latent non-ideological feature (opposition--non-opposition) underlying the senators vote. Additionally, the relationship between the parapolitics scandal and the legislative electoral behavior of senators is analyzed through a logistic model, both Bayesian and frequentist. The results indicate a significant relationship between being or having been involved with the parapolitics scandal and the legislative behavior of the senators from the period 2006-2010.

En este documento, ilustramos en profundidad varios modelos Bayesianos para datos de series temporales. Para ello, consideramos un conjunto de datos sobre la serie de cambios semanales de la tasa de interés de la madurez constante de los bonos del Tesoro de EE. UU. a 3 años, desde el 18 de marzo de 1988 hasta el 10 de septiembre de 1999. Consideramos dos enfoques completamente Bayesianos: uno de ellos estático, basado en un modelo autorregresivo de orden 3 como en Prado and West (2010), y otros dos modelos de variación temporal, basados en el marco dinámico de West and Harrison (1999). Uno de estos modelos dinámicosestá especialmente diseñado para detectar valores atípicos. Ambas alternativas son sensatas, pero debido a la naturaleza de este conjunto de datos, la alternativa de modelamiento dinámico es más atractiva ya que proporciona una caracterización completa de la respuesta.

Longitudinal data analysis is essential when the response variable is measured repeatedly on the same observational unit over time. Traditionally, parametric methods have been employed to estimate coefficients that define the relationship between the linear predictor and the response variable. However, these methods fail when the assumptions regarding the response variable and the model's random components are violated, or when the expected value of the response variable (or its transformation via a link function) cannot be expressed as a known function of the fixed and random effects. In such cases, parametric models may yield conclusions that deviate significantly from the dataset's average trends. Non-parametric regression techniques, which replace fixed parameters with time-dependent smoothed local functions, offer a powerful alternative for longitudinal data analysis. These methods, known as dynamic coefficients or parameters, allow us to establish a more flexible functional relationship between the response variable and the covariates. In this study, we propose estimation and inference techniques for generalized non-parametric dynamic coefficient models, particularly for count response variables. We illustrate our approach through its application in analyzing the effect of viral load on CD4 cell count in HIV/AIDS patients undergoing antiretroviral therapy, as well as in predicting COVID-19 cases.

Statistical modeling of networks enables us to fully characterize the entire system as well as make predictions regarding link formation. Latent models address these tasks by incorporating non-conditional dependencies through random effects. A notable example is the Bayesian spatial process-based model, which is particularly useful for avoiding overfitting issues that may arise in latent distance space models. In this paper, we provide the computational implementation of the model and evaluate its goodness of fit and predictive performance using synthetic networks. The model demonstrates strong capabilities in replicating network statistics and estimating the corresponding latent surface. We also propose an alternative fitting approach using a case-control algorithm. Based on the estimated log-likelihood, the model exhibits good performance in terms of prediction as well as goodness of fit.