Suppose that we know the values of population parameter and so can consider informative prior distributions. Then, we set Bayesian priors for all these parameters like, I S | y1@0y2@1y3@2y4@3; [I] (a); [S] (b); I (c); S (d); I with S (e);
a ~ N(100, 10); b ~ N(10, 3); c ~ N(500, 200); d ~ N(50, 25); e ~ N(-18, 15);
One more question! As recognized, the population parameter values cannot be easily known in real circumstance. So, if one wants to set (weakly??) informative priors based on the past literatures and do following steps: 1) From the meta analysis, one found that the mean and variance of intercept factor were respectively about 100 and 10. 2) then, set the prior like, a ~ N(100, 10).
I got reviewer's comment related to choice of priors. As knew, normal prior is often used for mean (fixed effect) parameter estimates, while inverse-Gamma and inverse-Wishart priors are commonly used for the variance/covariance. Although I set the normal priors for variance/covariance, the reviewer pointed out that the prior distribution of a variance should not be normal.
First, I cannot set normal priors for variances at all? Secondly, if yes, I need to set IW or IG for these parameters. How can I do it when considering above example?
I (c); S (d); I with S (e); model priors: c ~ N(500, 200); d ~ N(50, 25); e ~ N(-18, 15);
OR, model priors: c ~ IW(S, degree of freedom) S here is positive definite of matrix, right? What value of S can be this example?
Prof. Muthen, I am slightly confused with the IW prior as to how the mean and variance are calculated for this prior because the first parameter seems like a matrix?
1.) For the IW(S,df) prior, as the first parameter S is a matrix, I understand setting it to 1 (as in the above example for parameters d & e) implies identity matrix; but what does 0 (in above example for parameter e) stand for? So when I set the first parameter 'S' to 1, does it imply the mean is 1 and if I set 'S' to 0 does it imply the mean is 0?
2.)And if I set S to 0, from the formulas in both the papers for variance calculations of IW, as the numerator has 'S', does it mean the variance is 0 as well?
My apologies if I am missing something very obvious here. Please advice.
Prof. Muthen, Thank you very much for a clear simple explanation;it is becoming more clearer now.
In a similar manner is there a simple way to know/calculate the variance as well?
For my current analysis what I did was to create another separate Mplus script where I inputted different values for IW(S,df) S & df parameters and got in the output the values of Mean and variance for the IW prior. And then I used the proper S and df for my main analysis. Is this work around ok? I am asking this because Mplus output for same S & df gives slightly different Variance values? For eg below for IW(.9,52):