To design many Bayesian studies, the sample size is chosen to attain sufficient power to reject a null hypothesis or a desired probability that an interval estimate is sufficiently narrow. Determining the minimum suitable sample size for studies with complex models typically requires intensive Monte Carlo simulation. One simulation-based strategy to consider the sample-size space involves estimating sampling distributions of posterior summaries, such as posterior probabilities or interval estimate lengths, at various sample sizes. Another strategy instead explores these sampling distributions via stochastic root finding over the sample-size space, where the entire sampling distribution of posterior summaries is not estimated at any given sample size. The performance of these competing approaches can be assessed using their Monte Carlo sampling distributions of sample size recommendations across repeated implementations. In this paper, we theoretically derive these Monte Carlo sampling distributions for various Bayesian sample size determination approaches. To promote best practices for Bayesian sample size determination, we conduct extensive numerical studies to estimate these Monte Carlo sampling distributions for the sampling distribution estimation and exploration approaches to assess their bias, precision, and sensitivity to tuning parameters.

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