I talked to Kevin and he suggested a simpler straight forward Bayesian Analysis for the result. Following is the gist:

• Since Shear Loss Angle's contribution is so little to the coatings' brownian noise, there is no point in trying to estimate it from our experiment. It will be unconstrained in the search always and would simply result in the whatever prior distribution we will take.
• So, I accepted defeat there and simply used Shear Loss Angle value estimated by Penn et al. which is 5.2 x 10^-7.
• So now the Bayesian Analysis is just one dimensional for Bulk Loss Angle.
• Kevin helped me inrealizing that error bars in the estimated noise are useless in bayesian analysis. The model is always supposed to be accurate.
• So the log likelihood function would be -0.5*((data - model)/data_std)**2) for each frequency bin considered and we can add them all up.
• Going to log space helped a lot as earlier probablitis were becoming zero on multiplication but addition of log likelihood is better between different frequencies.
• I'm still using the hard condition that measured noise should never be lower than estimated noise at any frequency bin.
• Finally, the estimated value is quoted as the most likely value with limits defined by the region covering 90% of the posterior probability distribution.

This gives us:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Now this isn't a very good result as we would want, but this is the best we can report properly without garbage assumptions or tricks. I'm trying to see if we can get a lower noise readout in next few weeks, but otherwise, this is it, CTN lab will rest afterward.

Analysis Code