I’m just wondering if Bayesian statistics would be applied consequently from the first study to the last if this makes a meta-analysis obsolete.
For example, let’s assume 20 studies which have been done at different timepoints. The estimate or distribution of the first study was done with a uninformative prior. The second study uses the posterior distribution as the prior. The new posterior distribution is now used as prior for the third study and so on.
At the end we have an estimate which contains all the estimates or data which have been done before. Does it makes sense to do a meta-analysis?
Interestingly, I suppose that changing the order of this analysis would also change the last posterior distribution, respectivly, estimate.
What you are describing is called Bayesian updating. If you can assume that subsequent trials are exchangeable, then it won’t matter if you updated your prior sequentially, all at once, or in different order (see e.g. here or here). Notice that if previous experiments influence your future experiments, then also in the case of classical meta-analysis there would be a dependence that is not taken into consideration (if assuming exchangeability).
It makes perfect sense to update your knowledge using Bayesian updating, since it’s simply another way of doing it, then using classical meta-analysis. The question if it makes the traditional meta-analysis obsolete, or not, is opinion based and depends if you are willing to adopt Bayesian viewpoint. The most important difference between both approaches is that in Bayesian case you explicitly state your prior assumptions.