New visualizations for Monte Carlo simulations

In Monte Carlo simulations, samples are obtained from a target distribution in order to estimate various features. We present a flexible class of visualizations for assessing the quality of estimation, which are principled, practical, and easy to implement. To this end, we establish joint asymptotic normality for any collection of means and quantiles. Using the limit distribution, we construct $1- \alpha$ level simultaneous confidence intervals, which we integrate within visualization plots. We demonstrate the utility of our visualizations in various Monte Carlo simulation settings including Monte Carlo estimation of expectations and quantiles, Monte Carlo simulation studies, and Bayesian analyses using Markov chain Monte Carlo sampling. The marginal-friendly interpretation enables practitioners to visualize simultaneous uncertainty, a substantial improvement from current visualizations.