

Priors for residual covariances in BSEM 

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Dear Bengt, In your paper "Bayesian Structural Equation Modeling with Crossloadings and Residual Covariances: Comments on Stromeyer et al", it is indicated that the prior for the residual covariance matrix can be set to IW(dD,d) with d=100 (for sample size near 500), d=1000 (for sample size near 5000), etc. This magnitude of degrees of freedom seems to be very informative and I am not getting the motivation quite well. Why is the choice of degree of freedom (d) based on sample size? In other statistical literature (e.g. Lesaffre and Lawson, (2012) Bayesian Biostatistics), it is indicated that to obtain a minimally informative inverse Wishart distribution, d should be taken approximately equal to p where D is a pbyp matrix such that we have the prior as IW(D, p) or IW(D, p+1). The latter choices are based on the number of indicators rather than sample size. 


Intuitively speaking ... having a prior IW(dD,d) is like adding d observations that have a variance covariance D. If you are aiming for a certain/constant level of "informativeness" the bigger the sample size is the more of these "prior added observations" you want to have so that the observed sample has the same level of dominance over the prior. 


Many thanks for your response, Tihomir. 

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