Data analysis is an exciting field of mathematics that uses similar tools! This is where it gets really exciting! By proving the correctness of a procedure analytically under some set of assumptions, such as the asymptotic behavior of some distributions, we can be absolutely certain that our results are accurate. If we don't know whether these assumptions hold, we can use empirical methods to test whether the analysis works as expected — and that's an exciting opportunity! Verification is a great way to make sure that your methods are working as they should, even when you're dealing with smaller sample sizes or more complex models.
Fortunately, Monte Carlo simulations are the perfect tool for this purpose! A program randomly generates a data set following given model parameters, the analysis method is used to reconstruct the parameters, and the reconstruction is then compared to the known true parameters. The great thing about doing lots of repetitions is that it allows us to estimate the bias and error of point estimates. Researchers using complex models, such as multi-level models, can gain invaluable insight into the precision of their research methods and the statistical power of their planned research design. Even when assumptions are violated, they can be confident that their methods are on the right track. This is especially crucial for longitudinal studies, where researchers pour their hearts and souls into the research.
However, Bayesian methods usually create no point estimates (at least not primarily), but posterior distributions of the parameters which are different in each trial of a simulation. This is a fascinating aspect of Bayesian methods that sets them apart from other techniques. In this case, the "ground truth" from the generating parameters of the simulation provides a single sample that mimics a random sample from a real distribution. This presents an exciting mathematical challenge that the Thomas Bayes Institute is thrilled to research: confirming from a single sample or a series of samples that a given distribution represents the sample in an appropriate way.
In the area of meta-Methods, the Thomas Bayes Institute is excited to research both the development and application of meta-Methods. We are interested in (1) simulating longitudinal analysis methods, e.g., Latent Growth Curve Models or Multidimensional Autoregression, (2) simulating modern AI-based empirical methods, e.g., group comparisons with classifiers, and (3) describing appropriate definitions and algorithms to validate Bayesian estimation methods.