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Entry  Wed May 13 18:07:32 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis CTN_Bayesian_Inference_Analysis_Of_Best_Result.pdf
    Reply  Fri May 15 12:09:17 2020, aaron, DailyProgress, NoiseBudget, Bayesian Analysis 
       Reply  Fri May 15 16:50:24 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis 
          Reply  Fri May 22 17:22:37 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis CTN_Bayesian_Inference_Analysis_Of_Best_Result.pdf
             Reply  Mon May 25 08:54:26 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis with Hard Ceiling Condition CTN_Bayesian_Inference_Analysis_Of_Best_Result_Hard_Ceiling.pdf
                Reply  Tue May 26 15:45:18 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis CTN_Bayesian_Inference_Analysis_Of_Best_Result.pdfCTN_Bayesian_Inference_Analysis_Of_Best_Result_Hard_Ceiling.pdf
                   Reply  Thu May 28 14:13:53 2020, anchal, DailyProgress, NoiseBudget, Bayesian Analysis CTN_Bayesian_Inference_Analysis_Of_Best_Result_New.pdf
                      Reply  Sun May 31 11:44:20 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized CTN_Bayesian_Inference_Final_Analysis.pdf
                         Reply  Mon Jun 1 11:09:09 2020, rana, DailyProgress, NoiseBudget, Bayesian Analysis Finalized 
                         Reply  Thu Jun 4 09:18:04 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized CTN_Bayesian_Inference_Final_Analysis.pdf
                            Reply  Thu Jun 11 14:02:26 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized CTN_Bayesian_Inference_Final_Analysis.pdf
                               Reply  Mon Jun 15 16:43:58 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 14th CTN_Bayesian_Inference_Final_Analysis.pdf
                                  Reply  Tue Jun 23 17:28:36 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 22nd (as I turned 26!) CTN_Best_Measurement_Result.pdf
                                     Reply  Wed Jun 24 21:14:58 2020, Anchal, DailyProgress, NoiseBudget, Better measurement on June 24th 
                               Reply  Fri Jun 26 12:38:34 2020, Anchal, DailyProgress, NoiseBudget, Bayesian Analysis Finalized, Adding Slope of Bulk Loss Angle as variable CTN_Bayesian_Inference_Final_Analysis_with_Slope.pdf
Message ID: 2571     Entry time: Wed May 13 18:07:32 2020     Reply to this: 2572
Author: anchal 
Type: DailyProgress 
Category: NoiseBudget 
Subject: Bayesian Analysis 

I did this analysis last with bare-bones method in CTN:2439. Now I've improved this much more. Following are some salient features:

  • Assuming Uniform prior distribution of Bulk Loss Angle since the overlap with Penn et al. is so low that our measurement is inconsistent with theirs ((5.33 +- 0.03) x 10-4 )if we take into account their extremely low standard deviation associated to bulk loss angle.
  • Assuming Normal Distributed prior distribution for Shear Loss Angle matching Penn et al. reported value of (2.6 +- 2.6) x 10-7. This is done because we can faithfully infere only one of the two loss angles.
  • The likelihood function is estimated in the following manner:
    • Data cleaning:
      • Frequency points are identified between 50 Hz to 700 Hz where the derivative of Beat Note Frequency noise PSD with respect to frequency is less than 2.5 x 10-5 Hz2/Hz2..
      • This was just found empirically. This retains all low points in the data away from the noise peaks.
    • Measured noise Gaussian:
      • At each "clean" frequency point, a gaussian distribution of measured beat note frequency noise ASD is assumed.
      • This gaussian is assumed to have a mean of the corresponding measured 'median' value.
      • The standard deviation is equal to half of the difference between 15.865 percentile and 84.135 percentile points. These points correspond to mean +- standard deviation for a normal distribution
    • Estimated Gaussian and overlap:
      • For an iterable value of Bulk and Shear Loss Angle, total noise is estimated with estimated uncertainty. This gives a gaussian for the estimated noise.
      • The overlap of two Gaussians is calculated as the overlap area. This area which is 0 for no overlap and 1 for complete overlap is taken as the likelihood function.
      • However, any estimate of noise that goes above the measured nosie is given a likelihood of zero. Hence the likelihood function in the end looks like half gaussian.
      • The likelihood for different clean data points is multiplied together to get the final likelihood value.
  • The Product of prior distribution and likelihood function is taken as the Bayesian Inferred Probability (unnormalized).
  • The maximum of this distribution is taken as the most likely inferred values of the loss angles.
  • The standard deviation for the loss angles is calculated from the half-maximum points of this distribution.

Final results are calculated for data taken at 3 am on March 11th, 2020 as it was found to be the least noise measurement so far:

Bulk Loss Angle: (8.8 +- 0.5) x 10-4.

Shear Loss Angle: (2.6 +- 2.85) x 10 -7.

Figures of the analysis are attached. I would like to know if I am doing something wrong in this analysis or if people have any suggestions to improve it.

The measurement instance used was taken with HEP filter on but at low. I expect to measure even lower noise with the filters completely off and optimizing the ISS as soon as I can go back to lab.


Other methods tried:

Mentioning these for the sake of completeness.

  • Tried using a prior distribution for Bulk Loss Angle as a gaussian from Penn et al. measured value. The likelihood function just became zero everywhere. So our measurements are not consistent at all. This is also because the error bars in their reported Bulk Loss Angle are extremely
  • Technically, the correct method for likelihood estimation would be following:
    • Using the mean (\mu) and standard deviation (\sigma) of estimated total noise, the mean of the measured noise would be a gaussian distribution with mean \mu and variance \sigma^2/N where N is the number of averaging in PSD calculation (600 in our case).
    • If standard deviation of the measured noise is \sigma_m, then (N-1)\sigma_m^2/\sigma^2would be a \chi^2_{(N-1)} distribution with N-1 degrees of freedom.
    • These functions can be used to get the probability of observed mean and standard deviation in the measured noise with a prior distribution of the total estimated noise distribution.
    • I tried using this method for likelihood estimation and while it works for a single frequency point, it gives zero likelihood for multiple frequency points.
    • This indicated that the shape of the measured noise doesn't match well enough with the estimated noise to use this method. Hence, I went to the overlap method instead.

Analysis Code

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