May I know your idea about difference between these two situation in Bayesian parameter estimation?
1-we repeat the test N times and calculate variance (and perhaps co- variance of the data) and then use it as the fix value in likelihood calculations.
2-we leave the elements of the variance (co variance) flexible and let the bayesian determine them after sampling. In the other word, we estimate them along with the other parameters.
which one is more reasonable?
I do not think there is a simple answer to your question. the more parameters you need to estimate the more data you will, in general, need. Have you checked the section on estimating hyperparameters in Gaussian Processes? e.g. in the link below.
http://www.gaussianprocess.org/gpml/
Hi all,
If you fix the variance-covariance matrix, you assume there is no uncertainty in it, which is very likely to change the estimation of the rest of your model: I am pretty sure that you are going to underestimate the uncertainty in the estimation of other parameters (giving confidence intervals that are too narrow) and depending on what kind of model you are fitting, it might also bias the posterior modes / means.
To estimate everything sounds more reasonable to me. If your model doesn't converge or gives inaccurate estimates when you don't fix the var-cov matrix, it probably reflects the uncertainty present in your data.
(then, of course in can sometimes make sense to fix (part of) a var-cov matrix, for instance to test it against some structure, to estimate models with undefined residual variance...)
Thank you all for your responses!
Timothée as you said, the Bayesian finds the variance parameters but the covariance parameters does not converge.
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