Brooks and Gelman propose a generalized method for monitoring convergence in iterative simulations by comparing variances across multiple chains. They emphasize the importance of assessing convergence through statistical analysis, particularly by comparing inferences from different sequences. The method involves calculating a potential scale reduction factor (PSRF), which compares the variance of pooled data with the variance within each chain. A PSRF close to 1 indicates convergence, while a value significantly greater than 1 suggests that further simulations are needed. The authors also discuss multivariate extensions of the PSRF to assess convergence of multiple parameters simultaneously. The original method by Gelman and Rubin (1992a) involves running multiple chains and comparing their variances to estimate convergence. They suggest discarding the first half of the simulation data to avoid burn-in and using the second half to assess convergence. The PSRF is calculated as the ratio of the pooled variance to the within-chain variance. However, the authors note that this method can be misleading if the starting points are not sufficiently overdispersed. They also correct for sampling variability in the variance estimates, using a factor of (d+3)/(d+1) instead of the previously used d/(d-2). The authors propose an iterative graphical approach to monitor convergence by plotting the PSRF over time. This allows for a visual assessment of convergence and helps identify when the chains have stabilized. They also suggest alternative measures, such as interval-based diagnostics and moment-based diagnostics, which can be used to assess convergence without assuming normality. These methods are particularly useful for highly non-normal distributions. The authors also discuss multivariate extensions of the PSRF, which involve comparing the covariance matrices of the pooled and within-chain data. The multivariate PSRF (MPSRF) is defined as the maximum root statistic of the covariance matrices, which provides a measure of convergence for multiple parameters simultaneously. The MPSRF is shown to be an upper bound for the maximum of the univariate PSRFs, making it a useful tool for assessing convergence in high-dimensional models. The authors illustrate these methods using examples, including a Weibull regression in censored survival analysis and a bivariate normal model with a non-identified parameter. These examples demonstrate the importance of monitoring convergence for all parameters in a model, even when only certain parameters are of interest. The authors conclude that a combination of graphical and numerical diagnostics is necessary to accurately assess convergence in iterative simulations.Brooks and Gelman propose a generalized method for monitoring convergence in iterative simulations by comparing variances across multiple chains. They emphasize the importance of assessing convergence through statistical analysis, particularly by comparing inferences from different sequences. The method involves calculating a potential scale reduction factor (PSRF), which compares the variance of pooled data with the variance within each chain. A PSRF close to 1 indicates convergence, while a value significantly greater than 1 suggests that further simulations are needed. The authors also discuss multivariate extensions of the PSRF to assess convergence of multiple parameters simultaneously. The original method by Gelman and Rubin (1992a) involves running multiple chains and comparing their variances to estimate convergence. They suggest discarding the first half of the simulation data to avoid burn-in and using the second half to assess convergence. The PSRF is calculated as the ratio of the pooled variance to the within-chain variance. However, the authors note that this method can be misleading if the starting points are not sufficiently overdispersed. They also correct for sampling variability in the variance estimates, using a factor of (d+3)/(d+1) instead of the previously used d/(d-2). The authors propose an iterative graphical approach to monitor convergence by plotting the PSRF over time. This allows for a visual assessment of convergence and helps identify when the chains have stabilized. They also suggest alternative measures, such as interval-based diagnostics and moment-based diagnostics, which can be used to assess convergence without assuming normality. These methods are particularly useful for highly non-normal distributions. The authors also discuss multivariate extensions of the PSRF, which involve comparing the covariance matrices of the pooled and within-chain data. The multivariate PSRF (MPSRF) is defined as the maximum root statistic of the covariance matrices, which provides a measure of convergence for multiple parameters simultaneously. The MPSRF is shown to be an upper bound for the maximum of the univariate PSRFs, making it a useful tool for assessing convergence in high-dimensional models. The authors illustrate these methods using examples, including a Weibull regression in censored survival analysis and a bivariate normal model with a non-identified parameter. These examples demonstrate the importance of monitoring convergence for all parameters in a model, even when only certain parameters are of interest. The authors conclude that a combination of graphical and numerical diagnostics is necessary to accurately assess convergence in iterative simulations.