When using posterior information from a pilot study to inform priors in my main experiment, I have been advised to “broaden” the SDs of the point estimates, e.g., by using twice the SD for the prior compared with the posterior. Does anyone know a source I could cite where they use this approach? And should I also adjust the shape parameter? I would be very thankful for any help. Thanks a lot!
I’m not familiar with the advice that you double the standard deviation of your pilot posteriors for the new priors. I can’t find anything giving that advice with a cursory search. However, it seems unnecessary to me if the pilot is using the same model, priors, experimental design, and sample population as the experiment since, ostensibly, your likelihood should already contain the uncertainty due to small sample size and pass it along to the posterior.
I update my priors by having MATLAB fit the posterior distributions with distributionFitter, which returns distribution parameters and an NLL for the fit. This can be exploited for both centered and noncentered prior distributions. That will tell you if you need to change the distribution type (e.g. normal vs lognormal), shape, central tendency or spread for your new prior.
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Hi @stanbeginner interesting question, I have not heard this recommendation but perhaps the person who advised you to do so would know a reference or a reason for doing so.
Inflating the SD suggests to me that the advisor doesn’t think the two experiments are identical. If they are then as mentioned by @Corey.Plate it would be unnecessary.
By the way if you do end up using the posterior as prior (exactly) it might be easier to combine the data and analyse in one model, since those approaches would give the same posterior anyway.
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Multiplying SD of normal corresponds to power prior approach, see this paper [2404.02453] Exploring the Connection Between the Normalized Power Prior and Bayesian Hierarchical Models
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