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I am currently doing Bayesian Inference by using the BayesAB R library. I've observed that that sample size has a big impact on the posterior distribution.
When doing Frequentist Hypothesis Testing it seems that everyone agrees on how to determine the minimum sample size required to observe a certain effect based on the power of the experiment.
However, is this a valid approach when using Bayesian Inference? If not, is there something equivalent for Bayesian Hypothesis testing?
There is something corresponding to power analysis for Bayesians, it is called preposterior analysis. The preposterior expectation is the prior expectation of the posterior. One example can be found here. Another post where this concept is used is Minimizing variance of an estimator under sampling cost penalty.
Some other links: Is power analysis necessary in Bayesian Statistics? and Power analysis from Bayesian point of view.
Here is a stored google scholar search
One of the more obvious approach is to simulate with model parameters drawn from your prior beliefs, then draw your Bayesian inference and assess what proportion of simulations looks they way you'd like them to look.
Alternatively, you could make frequentist style assumptions (at least with respect to the size of the quantity of interest - perhaps reflecting the uncertainty about nuisance parameters is rather useful) and then simulate your planned Bayesian analysis.
