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 Wed May 13 18:07:32 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis Fri May 15 12:09:17 2020, aaron, DailyProgress, NoiseBudget, Bayesian Analysis Fri May 15 16:50:24 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis Fri May 22 17:22:37 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis Mon May 25 08:54:26 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis with Hard Ceiling Condition Tue May 26 15:45:18 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis Thu May 28 14:13:53 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis Sun May 31 11:44:20 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized Mon Jun 1 11:09:09 2020, rana, DailyProgress, NoiseBudget, Bayesian Analysis Finalized Thu Jun 4 09:18:04 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized Thu Jun 11 14:02:26 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized Mon Jun 15 16:43:58 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 14th Tue Jun 23 17:28:36 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 22nd (as I turned 26!) Wed Jun 24 21:14:58 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 24th Fri Jun 26 12:38:34 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized, Adding Slope of Bulk Loss Angle as variable
Message ID: 2574     Entry time: Fri May 22 17:22:37 2020     In reply to: 2573     Reply to this: 2575
 Author: anchal Type: DailyProgress Category: NoiseBudget Subject: Bayesian Analysis

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:
$\huge \Phi_B = 8.9^{8.9}_{3.7} \times 10^{-4}$

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

 Attachment 1: CTN_Bayesian_Inference_Analysis_Of_Best_Result.pdf  147 kB  Uploaded Sat May 23 11:23:52 2020
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